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By Jeff Parker

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The Complete Mathematics of the Cyclic Addition Cylinder© Copyright 2015 Jeff Parker

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The Complete Mathematics of the Cyclic Addition CylinderContents

Introduction …4

Structure and Design …9

Common Multiple …13

Patterns …17

Sequence …24

Cyclic Addition Step by Step …27

Cyclic Addition ToolKit …35

Cyclic Addition Cylinder Laws …42

A Journey from the Hindu-Arabic Number System

to Cyclic Addition Whole Number …50

Cyclic Addition Cylinders …54

Tier 1 Common Multiple 1

Tier 1 Common Multiple 68

Tier 1 Common Multiple 19Tier 1 Common Multiple 50Tier 1 Common Multiple 43Tier 1 Common Multiple 26Tier 2 Common Multiple 7 4 CylindersTier 3 Common Multiple 49 4 Cylinders

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Introduction

First and foremost this is concentrated Cyclic Addition Mathematics on the “Cylinder”. There

are two other recent books in the last couple of years. The first “A New Invention: Cyclic

Addition Mathematics that repairs and perfects An old Invention: Number”. This is a

reference book. Written before the Cylinder was discovered. The second book “A Prophetic

Design for Number from Cyclic Addition Mathematics”. This contrasts the ‘current-day’

Base 10 Place Value Number with Cyclic Addition Mathematics.

On the CD-Rom are these 2 Books and an assortment of Cyclic Addition Mathematics:-

practical Workbooks, Guidebooks, Slideshows, Make your own paper and pdf Cylinders,

Older Cyclic Addition Laws Books, Cyclic Addition Wheels, A brief 7 page snapshot of

What is Cyclic Addition, and this book “The Complete Mathematics of the Cyclic Addition

Cylinder”.

The Cyclic Addition Cylinder or collection of Cylinders has an aim and a purpose. The aim is

to perfect Whole Number. The purpose is to reach the schooling system in Adelaide to begin

transformation of Whole Number. It is strange that this knowledge of Cyclic Addition

Mathematics has only been found, recorded and made public approximately 1400 years after

the original invention of Hindu-Arabic Numerals forming modern ‘current-day’ Number.

This book draws on all the available and discovered knowledge of Cyclic Addition and the

“Cylinder”. Briefly what comes before the Cylinder. Lower Primary the circular 21 star

Object Count, Middle Primary Mini-Wheels onto NumberGrids, Upper Primary Counting

and Basic Place Value with Wheels, then the Workbook with Advanced Place Value and

Remainder. Significant use and Mathematical mastery of the Cyclic Addition Step by Step

Templates leads to a foundation and preparation for the Cylinder. See pdf book “A New

Invention”.

Cyclic Addition Mathematical Laws began as a lengthy work passed onto Secondary Schools

and Colleges about 4 years ago. Since then, addressing also a Primary School audience,

Cyclic Addition has been simplified.

So that one might concentrate on Mathematical Number rather than English, the Cyclic

Addition 5 Steps and the ToolKit was invented. The 5 Step by Step Mathematical actions are

Counting, Place Value, Move Tens, Remainder and 7×Multiple. These are acted upon

number to master Whole Number. The intricacies of which are found at length in the 2 books

above. The ToolKit, comprises 7 elements. These are the Wheel, Pattern, Operation + × ̶ ÷,

Sequence, Circle, Common Multiple and this book Cylinder. The complexities are made

simple and universal by applying some or all of the ToolKit with each of the Cyclic Addition

5 Steps.

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Briefly, the mathematical cause and motivation for the Cylinder and Cyclic Addition as a

whole is to:- strengthen Place Value positions of a Number, strengthen sequence of numerals

forming a Number, unify Operations + × ̶ ÷ with Number, bringing together an ordered

Cylinder of Numbers for the sake of Mathematics, unify knowledge for one Common

Multiple, discover infinite patterns and order to Whole Number, Rational Number and

Fibonacci Number, bring Number together with the perfection of Circle and Spiral, make

highly practical for all to use, improve the original invention whilst maintaining existing

order of Whole Number, provide a schooling subject of Whole Number for all, laying the

foundation to broaden the strand of Number in school/college to receive the perfect

Mathematics of Cyclic Addition.

This book roughly has 7 parts. These being Structure & Design, Common Multiple, Pattern,

Sequence, Cyclic Addition Step by Step, general use of the ToolKit and finally guideline

mathematical Laws. The subtopics are often prefaced by a coin of phrase to aid memory and

practical application.

These 7 Parts are briefed.

Structure & Design

A look at how to build a Cylinder, why this shape, history of Cyclic Addition enabling this

form to be created, practicalities of use, a quick contrast with paper Cylinder and longer

Count pdf form for PC and tablet, how to read a Cylinder and generally apply Patterns and

Maths to it. Parts of a Cylinder include Count, meshed Spirals in 2 directions, Ring for a

Circle of Number on the Cylinder, and Vertically aligned Number on the Cylinder.

Common Multiple

The advantages of persisting with a whole Cylinder of Counts and their higher Tier collection

of Cylinders and 2 higher Tiers and so on. Having an equal eye without prejudice or

preference for all members of a Cylinder. How just a simple ToolKit Wheel can form a whole

Cylinder. Proving the Common Multiple perfect with the discipline of Cyclic Addition

Maths. A little theory of the basis of using Common Multiples 1 to 69 by 1’s. And their

higher Tiers. The Common Multiple and it hierarchy of Tiers and Wheels are essentially

infinite, however the first 7 Tiers are addressed practically. And once again that ‘strangeness’

of reading Number of a higher Tier that transcends the lower Tier.

Pattern

Patterns of a certain Common Multiple and their higher order Tiers. Patterns of a Wheel.

Patterns of components of Step by Step Cyclic Addition. These include Counting, Place

Value, Move Tens, Remainder and 7×Multiple. Patterns with Structure of all Counts for 1

Cylinder and Patterns for moving across 4 Cylinders of a higher Tier. Patterns in a Ring.

Patterns in a Spiral of Counts, and Spirals next to each other. Patterns with Vertically aligned

Number. Patterns with that ‘strangeness’ serving a higher order. Patterns that Mirror each

other. Patterns of a roughly perfect nature like Scale, Magnitude and Proportion across

Common Multiples. Patterns of like Common Multiple for example 3 and 3×n (n integer).

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And the discovery of Patterns that emerge by following Cyclic Addition Step by Step

Mathematics.

Sequence

Part of the ToolKit and essential to the workings of Cyclic Addition Step by Step Maths. The

generally accepted Counting direction is with addition around the Wheel Sequence. Reading

numerals in sequence left to right and right to left. Sequence of Spirals on the Cylinder.

Remainder Sequences on the Cylinder throughout. Wheel Sequence following a Circle in

both directions. Positioning any Number on a Cylinder with a V and Wheel Sequence.

General Sequence in applying pieces of the ToolKit to serve Step by Step Cyclic Addition.

Cyclic Addition Step by Step

The method and discipline of the 5 Steps of Cyclic Addition are about 7 years old. Gradually

refining each Step culminating into the Cylinder. Remember there might be Cyclic Addition

work required before venturing into the Cylinder. Each of the Steps use a unique

mathematical emphasis of the Wheel and Circle. Other elements of the ToolKit contribute to

perfecting the 5 Steps. The detailed Mathematics of these 5 Steps won’t be shown in this

book. Rather refer to the 2 major text identified above.

Let’s look at how the 5 Steps Counting, Place Value, Move Tens, Remainder and 7×Multiple

act on the Cylinder. Step 1: Counting Sequences Spiral around and down the Cylinder, every

Cylinder, in both directions. These Counts are spaced with Addition by the Wheel, again all

in rotation Sequence around the Wheel. Step 2: Place Value brings the unification of the 6

Wheel members to build each Count. Creating a Place Value Set for each place value position

requires Wheel Mathematics. Place Values interweave and Add to equal the Count. Step 3:

Move Tens to Units allows any Place Value outside the Units to be moved to the Units

position. This is to determine a Remainder the next step. Step 4: Remainder from the Wheel

bridges the Count to its underlying 7×Multiple. Step 5: The 7×Multiple is simply the next

higher order of the same Common Multiple. All the while unifying the hierarchy of Tier 1

with Tier 2. The 7×Multiple and Remainder prove perfect the creation of the Count Sequence

and furthering the knowledge of the Common Multiple.

The Cylinder automatically presents the Counting Step. However the paper Cylinder can be

veiled to Count along and improve the receive of this Step. The PC and Tablet pdf for 7

Cycles or 42 Counts can be also positioned to again act with Addition, revealing the next

Count in turn.

Acting mathematically with the Cylinder can be for all 5 Steps, or just a Count with Wheel,

or build Basic Place Values, or present a Remainder Sequence for each Cycle of 6 Counts.

Simply proving the Common Multiple with each Count in either Ring or Spiral Sequence is a

good start. So there is established Mathematics acting on the Cylinder to promote order and

Mathematics with Whole Number.

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ToolKit

Common Multiple, Pattern and Sequence are briefed above. Leaving Wheel,

Operation + × ̶ ÷ and Circle. The Wheel is in a fixed state of

‘Common Multiple’× ‘ 1 3 2 6 4 5 ’× 7(n-1) . The Common Multiple is 1 to 69 by 1’s. The

Tier is n. The multiple of 1 is always at the top of the Wheel. The rest in clockwise Sequence

in hexagonal points around the Wheel. Having this certainty of Wheel Structure presents

mathematical flexibility with the 5 Steps. The Wheels for the first 7 Tiers are detailed on the

CD-Rom with unification of Rational Number, Exponential Number and Whole Number. 7

Tiers of Wheels for each Common Multiple is only a practical teaching limit and is

essentially infinite. The Wheel is the first Ring of a Tier 1 Cylinder. The Wheel gives a

Pattern of a Remainder Sequence each Ring other than end of Cycle. The journey of Cyclic

Addition starts with Wheel and Circle and made perfect with Cylinder.

All Operations + × ̶ ÷ are in unity. Count with the Wheel is absolutely with + and × together.

The Remainder Step is always ̶ following the simple formula of ‘Count ̶ Remainder =

7×Multiple’.The Wheel Structure and the Place Value Step form whole number ÷. In general

Addition + is with Counting, Place Value and Remainder Mathematics. Across Cylinders of

like Common Multiple and Scale within the same Cylinder contribute to Multiplication ×.

Patterns of spacing a Wheel Number between 2 Spiral Count Sequences also gives

Subtraction ̶ . The way the Cylinder is constructed with Cyclic Addition Maths prepares one

to act with the next higher Tier and Order of the same Common Multiple. This is especially

from Pattern and Operations + × ̶ ÷. Arithmetic with Number is hundreds of years old.

However the presentation of Cylinder Number is only about a year old.

The Circle as a verb is an action of rotation in either direction. The Wheel and Cylinder are

made for Rotation. The advantage of a paper 3D Cylinder is its Rotation with Cyclic

Addition. Again all 5 Steps use and apply the Circle. Counting in either direction around a

Wheel. Place Value selection to build the Count is with a partial Sequence of the 6 number

Wheel with Rotation. Move Tens to Units applies clockwise Sequence of the Wheel from one

Number to the next moving a place value position to the right. Remainder has circular Laws,

when searching for the single member from the Wheel. Due to the stability of the Wheel

these Laws allows efficient Mathematics around the Wheel for any and every Wheel.

Mathematical Laws

A detailed book “Laws within a Number Universe” was written about 6 years ago. The pdf is

on the CD-Rom. This book brings authority of a mathematical nature to Cyclic Addition.

Many, many Laws bring the subject matter together with guidelines and practical

methodology. Since then, in the two recent books above, the TookKit was formed within the

Cyclic Addition Step by Step environment.

So the Part in this book “Mathematical Laws” is to bring a foundation and creation of the

Cylinder into a mathematical light and authority. Without this historical authority of Law the

Cylinder would be just Pattern and Number. However the intentional direction of Whole

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Number, with the Cyclic Addition Cylinder, requires Law to preserve and protect Number for

all who use it.

The Laws are grouped into general laws and each of the 5 Steps of Cyclic Addition.

Established arithmetical mathematics applied to Cyclic Addition Number is assumed

knowledge. Where that arithmetic is applied in a Cyclic Addition framework a Cyclic

Addition Law is used.

Laws are excluded from Pattern. So that there is freedom within the Cylinder to explore the

Common Multiple and Cyclic Addition Number. When to use a Pattern, in what Sequence to

apply a Pattern, how to guard the accuracy of a Number with Pattern is left to the creativity of

the Mathematician. Also when to leave the Tier 1 Hierarchy and venture into higher Tier 2

and beyond is left to the curiosity of the individual. This gives way to Cyclic Addition being

taught and learnt as something of awe and wonder rather than the rigidity of tireless

mathematical Laws.

History

Cyclic Addition as new has a brief history from original creation. It began about the turn of

the century so essentially only 15 years old. Basic arithmetic is a few hundred years old. To

measure the relevance and importance of Cyclic Addition to ‘current-day’ Number is left to

the Teacher, Student and Mathematician.

Cyclic Addition is still in a growth phase. There is an awesome amount of work in progress

till today. An intention of this book is to generate interest in, make practical the knowledge

and available to all. The most recent works are available on the CD-Rom and Google Play

Books under ‘Cyclic Addition’.

Follow along with a practical ‘Cylinder’ found in the Appendix at the back of the book.

Common Multiple 1 to 6, 7 and 49 are shown. Print out a Cylinder on A4 size paper, roll and

tape to begin Circle and Number. Or load the pdf Cylinders onto a Tablet, Laptop or PC.

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Structure & Design

The Cylinder and mastery of its Mathematics is the pinnacle of Cyclic Addition. Once the 5

Steps of Cyclic Addition are practised enough use of the Cylinder can commence. As the

Cylinder is the complete Common Multiple, practise with singular Counts to start with is

recommended.

The Cyclic Addition Wheel is only 1 Common Multiple. All Counts from that Wheel

generate knowledge and Pattern for that Common Multiple.

A little history on the Cylinder. A spreadsheet was used for 6 clockwise Wheel Counts and 6

anti-clockwise Counts. Once both sets were staggered all Counts clockwise were found in all

Counts anti-clockwise. Thus with some skilful spreadsheeting the paper and pdf Cylinder was

born to unite both directions now forming Spirals around the Cylinder.

Making a Tier 1 Cylinder from scratch requires Counting around the Wheel from all 6 starts.

There are 2 directions to the Spiral Counts making 6×2=12 Counts in all. Follow along with a

paper Cylinder Common Multiple 1.

A Spiral Count Sequence starts from the Wheel and increments by the next Wheel number

Clockwise to the Spiral right and Anti-Clockwise to Spiral left. In fact all Tier 1 Cylinders

follow Wheel Sequence with Step 1: Counting. A Spiral no matter the Tier Hierarchy of a

Common Multiple always presents Step 1: Counting.

Tier 2, Tier 3 and beyond have 4 Cylinders to completely present all possible Counts. Tier 2

Cylinders have a total of 28 Counts. The first Cylinder has 12 Counts=6×2. This Cylinder is

perfectly 7× the Tier 1 Cylinder. The second and third Cylinders starts from the first and

second possible 7×Multiple forming 3×2×2=12 Counts. These two 7×multiples, 49 and 98,

originate from a Tier 1 Count joining the Tier 2 Count without Remainder. See “A New

Invention: Cyclic Addition” Counts at the back of book for a visual presentation of Tier 2 and

Tier 3 for a cycle of Common Multiple 1. Every Count made with Tier 2 has its origins with a

single Common Multiple. See Wheels pdf on the CD-Rom. The fourth Cylinder is smaller in

diameter, being a 6 pointed Circle rather than the common 12 pointed. It begins with Tier 1

Counting and connects to Tier 2 without Remainder. Forming 2×2=4 Counts. A Total of

12+6+6+4=28 Counts for all Tier 2 Cylinders.

Generating any Cyclic Addition Count can be with either a lower Tier connecting to the next

higher Tier without Remainder or simply the higher Tier alone. This is a Cyclic Addition

Law. So one requires a foundation to support the structure of a Cylinder. The simplicity of

the Cylinder shows only a single Wheel. This allows concentration on one Tier of a certain

Common Multiple at one time.

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Tier 3 Cylinders have a total of 42 Counts. One Cylinder is 7× the first Tier 2 Cylinder. The

second and third Cylinders have exactly the same increments of Tier 3 Wheel, however they

start at the first two Tier 4 Number. For example with Tier 3 for Common Multiple 1 these

two Cylinders start at 343 and 686 respectively. Again these starts are generated by Tier 1,

Tier 2 and Tier 3 Cyclic Addition Mathematics. Thus forming 6×2×2=24 Counts. The fourth

Cylinder of Tier 3, is in the same form as the smaller Tier 2 Cylinder, however there are 3

Counts and 2 small Spiral directions forming 3×2=6 Counts. A Total of 12+12+12+6=42

Counts. For curiosity and Cyclic Addition Law note how all 21 clockwise Counts are made

from Common Multiple 1 Hierarchy in back of “A New Invention” pdf book.

Tier 4 Cylinders are perfectly 7× Tier 3 Cylinders. Likewise Tier 5 Cylinders a perfectly 7×

Tier 4 Cylinders. And so on up the Wheel Hierarchy. See Wheels pdf on CD-Rom.

The 69 Common Multiples and Hierarchy shown visually on the Wheels pdf is the basis and

foundation for Complete Number. Potentially there is an infinite number of Wheels in the

Hierarchy, however for practical sake only the first 7 Tiers are shown. Briefly this foundation

is from joining Whole Number Wheels, with Rational Number Pure Circular Fractions and

corresponding Exponentials. See relevant text on CD-Rom for detail.

Any Cylinder is formed by just 1 Wheel. Raise from Tier 1 to Tier 2 and the Wheel becomes

7×Wheel numbers of the lower Tier to follow the higher Tier. The Wheel has perfect qualities

of preservation. Acting on the Wheel with Operations + × ̶ ÷ and the Wheel Sequence is

preserved. See pdf book “A New Invention”. Like the mathematical purity of the Wheel is

unaffected by Maths acting upon it. For a complete list of Wheels see pdf book on CD-Rom.

The Wheel supports all the Mathematics of the Cyclic Addition 5 Steps. Each Step acts

uniquely upon the Wheel to strengthen the Wheel and Cylinder with Maths. Such perfection

from a simple circular 6 number Wheel.

There are 4 visual structures to a Cylinder. The first is clockwise Spirals. The second is anti-

clockwise Spirals. The third is Ring. The fourth is Vertically aligned Number.

The Spirals on all Cylinders mesh both clockwise Counts with anti-clockwise Counts. This is

a brilliant structural feature of the Cylinder. Each Spiral revolves from one Ring to the next

Ring Count by Count. Most Count Number belong to two, opposing direction, Spirals. The

exception is Tier 2. The next Count in a Spiral is always the next Ring downward. So to

follow Cyclic Addition Step by Step one follows a single Spiral at a time.

From an accentuation of the analogue clock, Cyclic Addition has maintained an emphasis on

clockwise Counting. This stature is no longer required as all 12 Counts of Tier 1, 6 Counts ×2

directions, should be seen with an equal eye without prejudice or preference. Rising through

the Hierarchy of Tiers one should also see all possible Counts for any particular Tier this

way.

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A Spiral Count has a length. This is measured in Cycles or every 6 Counts representing a

whole revolution around the Wheel. Typically all Spirals, except for the smaller Ring

Cylinder, are marked with a Ring of six identical 7×Multiples. This is often termed an ‘end-

of-cycle’. Typically a paper Cylinder and paper templates are 32 Counts or just over 5 cycles.

This was for easy reference and citing a handheld Cylinder Number. The pdf for Tablets,

Laptops and PCs is 7 Cycles. Most have the ability to view a pdf with visual perfection. It

was deemed long ago and could quite possibly be challenged that 7 cycles is enough

Mathematics for any 1 Tier. After completion of that Tier the next Tier is made available.

A Tier 2 Spiral has 3 of the Cylinders with what look like patches of missing Number. Look

closely at the start and follow the Spirals to understand how they mesh on half the number of

Spirals. Following on the pdf rather than paper has its advantages. And scrolling to and fro

from the other Tier 2 Cylinders is simple.

Meshing and Patterns of a Spiral is found in a further applicable Chapter.

A Ring is a Circle around the Cylinder of 6 Numbers. One must pass through every Ring to

Complete the Count. Thus in essence the next Ring is the next Count. The ring can be seen as

a Circular slice through any Cylinder. Thus the Circle contributes again to Cyclic Addition.

There are always 6 Rings in vertical Sequence forming a Cycle of Counts no matter the Tier.

A Tier 2 Common Multiple has 4 Cylinders. Following the order of above the first Cylinder

always has 6 numbers each Ring. The second and third Cylinders have a Structure of 4, 4, 5,

4, 4, 6 numbers each Ring for any Cycle of Rings. The smaller Cylinder has a Structure of 2,

3, 3, 2, 3, 3 numbers each Ring for a Cycle of Rings.

A Tier 3 Common Multiple has 4 Cylinders. The first, second and third Cylinders have 6

numbers each and every Ring. The smaller fourth Cylinder has 3 numbers every Ring. The

Rings on the smaller Cylinder can be grouped into pairs of Rings for Pattern making

purposes.

A Tier 4 and above Common Multiple follows the Structure of Tier 3 Cylinders.

The Ring, for a Tier 1 Cylinder, has the Wheel Sequence for the first top Ring. A Pattern is

shown from one Ring to the next Ring. Three Counts in rotation Sequence appear in both the

first and second Rings, the second and third Rings, the third and fourth Rings and the fourth

and fifth Rings. The end of Cycle Ring has 6 identical numbers forming a 7×Multiple equal

to the Count.

The Ring has a Cyclic Addition Step 4: Remainder Sequence of the Wheel. As the Wheel is

the Remainder Sequence for the 7×Multiple. The Wheel guides the Sequence of Number

around a Ring. So Ring works closely with Remainder and Wheel. This applies to all

Cylinders, even the pairs of Rings in the Tier 2 and 3 smaller Cylinder.

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The Circular Ring guards the mathematical accuracy of a Spiral Count Sequence. Any

Remainder out of kilter is shown by the Pattern of Remainders in any Ring.

The Ring brings near Common Multiple Counts together in a Circular Pattern. This is

important for Scale and Count Sequence position on the whole Cylinder. Letting you know

where you are within the entire 7 Cycle Count. Simple and always true.

Further Patterns and Cyclic Addition Step by Step of the Ring is discussed in a later Chapter.

The fourth visual Structure of a Cylinder is the Vertically aligned Number. There are 12

Vertically aligned Number on most Cylinders, the exception is the smaller Ring Tier 2 and

Tier 3 Cylinder. This has just 6 vertically aligned Number.

Counting in a Spiral down and around the Cylinder one returns to the same vertical point

every 2 Cycles. The smaller Cylinder every 1 Cycle.

The Vertically aligned Number assists in the Rotation of the Cylinder. Patterns of the same

vertically aligned Remainder Sequence helps. The first and fifth Rings, of any Cycle, have

the same Remainder Pattern. The second and fourth Rings also have the same Remainder

Pattern. The third Ring simply sits in the middle of the others, staggering the Remainder

Pattern mirrored above and below it within the one Cycle.

The other Vertically aligned Patterns that aid rotation of the Cylinder are found in Patterns.

The pdf Cylinder is a flattened cylinder. Where a Spiral of Counts runs off the page joining

the other side of the page. Essentially the pdf Cylinder has diagonals for Spirals. Across the

page for Rings and straight down the page for vertically aligned Number. Each, both the pdf

and paper Cylinders, have their own merits. Obviously the pdf is simpler to copy and pass

onto others. The Math of the pdf and paper Cylinders are thus deemed identical. Visually the

Circle around the paper Cylinder has a higher wow factor.

Look back at the 14 pages of Cylinders. There are 8 Common Multiples; 1 to 6 Tier 1

Cylinders, Common Multiple 7=1×7 for Tier 2 and Common Multiple 49=1×7×7 for Tier 3

Cylinders. Thus 6+4+4=14 Cylinders for easy reference in this “Complete Cylinder” book.

The rest of the Cylinders on the CD-Rom show all Common Multiples for Tier 1, Tier 2 and

Tier 3. Anything beyond that, Tier 4 and above, use the ‘Make your own Cylinder xls file’.

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Common Multiple

At the heart of every Cylinder is the Common Multiple. A Tier 1, 7 Cycle Cylinder presents

all multiples of the Common Multiple from 1 to 147. This is accomplished with just 1

Cylinder presenting 12 Counts. A Tier 2, 7 Cycle Cylinder presents all multiples of the

Common Multiple from 7 to 1029 in 7’s. This is discovered with 4 Cylinders presenting 28

Counts. A Tier 3, 7 Cycle Cylinder presents all multiples of the Common Multiple from 49 to

7203 in 49’s. This Structure and Design, mastering Tier 1 then moving to Tier 2, mastering

Tier 2 then moving to Tier 3, and so on up the Hierarchy of a chosen Common Multiple, is

Cyclic Addition Law.

Counting beyond 7 Cycles is possible, but practically speaking rising to the next Tier is a

wiser use of mathematical time. What one is shown with Cyclic Addition in a given Common

Multiple and Tier within the 7 Cycle boundary represents sufficient Mathematics. Tier 2

Multiples of 7 supporting the Common Multiple perfectly part and present the lower Tier 1

mathematics of the same Common Multiple. Likewise Tier 3 multiples of 49 with Tier 2.

Let’s immerse ourselves in the study of the Common Multiple. There are Patterns that prove

the Common Multiple is present within the Count. There is knowledge on ‘reading’ a

Number to receive the Common Multiple. The 5 Steps of Cyclic Addition all preserve the

Common Multiple. Discussion of these 3 Topics shows significant completeness of the

Cyclic Addition Common Multiple.

Patterns proving Common Multiple.

The Cylinder is perfect for focusing on one Common Multiple at one time. As mentioned all

other multiples, where the Common Multipleבother multiple’=Count always, serve the

mathematical number of the Common Multiple.

For example Common Multiple 1 shows all Counting Numbers. Tier 2 or the more familiar

Common Multiple 7 shows a higher order of the same Common Multiple 1.

For example Common Multiple 2 use a Pattern of even units. Tier 2 Common Multiple 14

and Tier 3 Common Multiple 98 use a higher order Sequence of numerals forming a Number.

Tier 3 asks for +2 units to receive 10 tens.

For example Common Multiple 3 use sum of digits =3,6 or 9. There is a sum of digits Pattern

every Cycle and every Count Sequence. Tier 2 Common Multiple 21, shift the Sum of Digits

to 1 unit and 2 tens. Tier 3 Common Multiple 147 shows further perfection of the three’s sum

of digits. A familiar Pattern with Cyclic Addition order and hierarchy.

A further example Common Multiple 4 use units and tens structure to show 4’s. A 0,4 or 8 in

units with an even tens or a 2 or 6 in units with an odd ten. Tier 2 Common Multiple 28

places the structure of 2’s in the tens and 8’s in the units. Like the 28=4×7 for Tier 2 the role

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of the ×7 perfectly shows the inherent assumptions with Common Multiple 4 and allows

exploration into the realm of Whole Number 4 with an order that creates and completes.

A further example Common Multiple 5 with 5 or 0 units. Tier 2 Common Multiple 35, simply

places a 3 in the tens with the stable 5 units. The Tier 3 Common Multiple 245=5×7×7

simply places the many factored 24 tens against the units 5.

A further example Common Multiple 6 with evens and sum of digits together. Tier 2

Common Multiple 42 shifts 4 units and places them in the tens producing 42. This has

enough place value changes in the Common Multiple 6 to warrant a higher Tier 2. The Tier 3

Common Multiple 294 looks at the completion of 29 tens and 4 units. Asking for the

presentation of Tier 1 Common Multiple 6 to receive 30 tens.

A further Common Multiple 7 has a Tier 1 Cylinder. Thus can stand on its own without any

support from Common Multiple 1. Following the Wheel of Common Multiple 7, which is

‘7 21 14 42 28 35’ with Place Value and Cyclic Addition Mathematics. The Wheel of

Common Multiple 1, which is ‘1 3 2 6 4 5’, is used for the Cylinder of 1’s. Declaring a

Remainder and 7×Multiple=7’s as one progresses. An important distinction is the Wheel

always follows the Cylinder. Tier 2 Cylinder commands a Tier 2 Wheel for all of

Mathematics with it.

Other Common Multiples have either 1 or 2 factors that make proving the Common Multiple

present with the Count simple. However rising to Tier 2 and Tier 3 Mathematics requires a

swift and accurate basis of Common Multiple 7 and 49’s. For example Common Multiple

36=9×4. Proving both factors is simple. Rising to Tier 2 Common Multiple 252=36×7

requires proving 252=9×4×7 all three factors. Or apply Cyclic Addition Step by Step

Mathematics guarantees accuracy. Patterns of the factors 9 and 4 together that show higher

order with a higher Tier. This makes for interesting exploration into the Common Multiple, in

this case the 36. So long as one proves the Common Multiple perfect and accurate, creativity

is encouraged with such a new study of Cyclic Addition.

Reading a Number with Common Multiple

The Common Multiple are usually given to 1 to 69 by 1’s and a Tier number. For example

Common Multiple 2 Tier 3 is the same as Common Multiple 98, but for terms of reference

the Counts belonging to the Wheel ’98 294 196 588 392 490’ belong to Common Multiple 2

higher order. This becomes important with higher Tiers as one is shown richer Patterns and

far more intricate Sequences of numerals forming Number all belonging to one Common

Multiple. Thus navigation around a Wheel, along a Cylinder, in the midst of Cyclic Addition

Place Value Step the Common Multiple remains consistent no matter the Tier. Also one

learns to grapple with that mysterious knowledge gleaned from a higher Tier that cannot be

comprehended by the lower Tier. Like ascending skyward in a balloon one sees further, as

one rises like the Tier, the comprehension and mastery extends further throughout Whole

Number.

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Reading a Count Number, on any Cylinder, one should group numerals into paired Sequence.

For example 21952= 2352units+1960tens=392×56 Common Multiple 8 Tier 3 or Tier 4

Cylinder. Look at the story of the Number with pairs of numerals 21=3×7, 19=19×1,

95=19×5, 52=13×4. Once the Tier 1 of Common Multiple 21, 19 and 13 are mastered this

Number has a depth of truth and mathematical Pattern than ‘current-day’ Base 10 place value

Number.

There is in progress an experimental and exploratory method of reading a Number and

recognising the Common Multiple and Tier. This requires Whole Number of a Cylinder and

Tier to be mastered to a point where instant recognition is possible.

The completeness of just 69 Common Multiples within a Cyclic Addition and ToolKit

framework shows all that Whole Number purports to be. The 69 Common Multiples can be

said to present numerals 1 to 6 (in the tens) to all numerals 1 to 9 and 0 (in the units). So for

example the Common Multiple 37 presents the 3 to the 7. Showing complete whole number

of the 3 to the order and receive of number 7. So Count Number and Count Sequences

generated by this Common Multiple all share this like knowledge of the example 3 presented

to the 7. Note how Common Multiple 1 to 9 stand to represent Whole Number on their own.

So the nature and secrets of a Number form from viewing literally the Common Multiple.

Their Patterns, their Sequences, their Order, their multi-dimensional inter-relationships

amongst all Whole Number can be illuminated by this light upon Common Multiple.

How long is the journey with Cyclic Addition. A Tier 1 Common Multiple has 12 Counts,

Tier 2 28 Counts, and Tier 3 42 Counts. Each Cylinder of 7 Cycles is 7×6=42 Rings in

length. There are 69 Common Multiples that with Cyclic Addition maturity should be seen

with equal eye. Thus a Count Total of 82×42×69=237636 or about a ¼ of a million. The

number of words in the bible amount to approximately 750,000 or about ¾ of a million. So if

reading Number in a Cylinder form was as familiar as English language Cyclic Addition

would be well within the reach of many. To protect and preserve this ancient invention:

Whole Number. A new Cyclic Addition Number System could easily supplement the existing

‘current-day’ Base 10 place value Number.

The journey of each Common Multiple from 1 to 69 and its hierarchy is essentially unlimited.

Each Common Multiple has a mathematical art and science awaiting discovery. Improving

human Patterns with light and sound makes the world go round.

The 5 Steps of Cyclic Addition preserve the Common Multiple

Step 1: Counting on a Tier 1 Cylinder begins with a member from the Wheel. Continuing in

rotation Sequence with just members from the 6 number Wheel. All 6 members of a Wheel

share the same Common Multiple thus so to all Counts from the same Wheel. Tier 2 and Tier

3 Counting may begin with a 7×Multiple from the next higher Tier. However all Counting

from then on uses just one Wheel ensuring all Counts are from the same Common Multiple.

16

Step 2: Place Value uses Wheel Patterns and Circle to create each place value in the units,

tens and hundreds. All members of all the Place Value Sets forming a Count Number are

from the one Wheel. Thus conclusively proving the Count is from the Common Multiple of

the Wheel. Place Value builds Sets in each place value position, which may overlap into

other place value positions. All Place Value Sets add to equal the Count. Another aspect of

Addition and identical Common Multiple throughout.

Step 3: Move tens to units moves all Place Value Sets outside the units to the units. This is

performed by the mathematics of the clockwise Remainder Sequence around the Wheel. A

Place Value member from the Wheel in tens position becomes the next clockwise member

from the same Wheel and placed into the units position. Since all members of the Wheel

share the same Common Multiple, regardless of place value position, Move Tens preserves

the Common Multiple.

Step 4: Remainder adds all the Place Values in the units together, applying Remainder

Patterns and Laws, leaving a single member from the Wheel. The solitary member is the

difference between the Count and the nearest 7×Multiple. As the calculation of a Remainder

is only from the Wheel members the mathematics of Remainder preserves the Common

Multiple. Yet another form of Addition and Subtraction with Wheel members thus acting on

the Common Multiple Count. All the while preserving the Wheel Sequence and Common

Multiple.

Step 5: 7×Multiple is created from ‘Count ̶ Remainder = 7×Multiple’. Simply put as the

Count and Remainder preserve the 7×Multiple, thus from Subtraction formula the 7×Multiple

also preserves the Common Multiple. In fact from Cyclic Addition Step by Step the next

higher Tier is initially discovered by declaring the 7×Multiple each Count.

The Wheels pdf on the CD-Rom has 1 Common Multiple per page, complements of 69 on

facing pages, unification of Rational Number, Exponential Number with Whole Number. All

the while focusing on the Common Multiple. Hints on how that mathematical Art and

Science of Number work is shown by a perusal of these Wheels and Common Multiples.

In summary, many Common Multiples from 1 to 69 by 1’s rely solely on these Cyclic

Addition 5 Steps to maintain Order, search for the 7×Multiple, and all the while preserving

the Common Multiple throughout the Cylinder.

Having a stable and permanent Common Multiple in any Tier makes for perfect unification of

Count Sequences on the Cylinder. All inter-relationships and, in a later Chapter, Patterns

maximise the opportunity to discover and explore to one Common Multiple. Leading to

higher Tiers and higher perfection with that Common Multiple.

17

Patterns

Patterns of each Common Multiple

Continuing from the previous Chapter each of the 69 Common Multiples have Patterns. The

Patterns begin at a basic and fundamental level with Tier 1. As one climbs to a higher Tier 2

and Tier 3 Order, the same Common Multiple becomes like a fractal. No matter how deep

one delves into a Common Multiple the Pattern making remains infinite.

For example the simple Common Multiple 1 shows all Counting Numbers 1 to 147 to length

of the Cylinder. That is the Pattern. Tier 2 or Common Multiple 7 shows all multiples of 7

again to the length of the Cylinder. Tier 2 reaches 1029, thus all the 3 digit 7’s. Where a

longer Cylinder of Tier 1 would continue this Pattern, Tier 2 shows the whole lot with the

higher Order 7’s. The Tier 2, 3 digit Numbers, show at least 2 pairs of numerals that describe

the 7’s. Tier 3 reaches 7203, thus most Counts are 4 digit, to the length of the Cylinder. The

Tier 3 Counts show an even higher Order to ‘include all Counting Numbers’ with Pattern.

See Appendix Common Multiple 49, the last 4 Cylinders in this book.

Patterns with the Wheel

The Wheel has a stable and consistent structure of ‘Common Multiple’× ‘1 3 2 6 4 5’×7(n-1) .

Where the Common Multiple is from 1 to 69 by 1’s, and ‘n’ is the Tier. Also easily missed is

the Wheel is circular, right through the 2 text books of ‘A New Invention’ and ‘A Prophetic

Design’. Mathematics of Operations + × ̶ ÷ acting on the Wheel preserve the Circle and

Sequence of the Wheel. The Circle gives rise to the perfect 5 Steps of Cyclic Addition.

Producing mathematics and all the while preserving the Circle and Sequence. Circling around

the Wheel produces a resisting and repeating Pattern of Addition each Cycle.

The Wheels Order shows how the first 6 multiples of the Common Multiple, are in Cyclic

Addition Wheel Sequence. These 6 Wheel members show a hint of Pattern with the Common

Multiple. For example Common Multiple 2 Tier 2 Wheel ‘14 42 28 84 56 70’ shows the 4, 2

and 8 in both units and tens, and a Pattern of 4 5 6 7, 0 1across 4 numbers. Also all evens, in

a new Sequence, contrasting to Tier 1, around the Wheel.

Patterns within Cyclic Addition 5 Steps

Step 1: Counting joins all Count Sequences together with plain and simple Addition around

the Wheel. Continuous Addition forming a Count Sequence increments Number by a Wheel

member. Addition is a major feature with Cyclic Addition. The interlocking of numerals of a

Count with Addition reveals the Count’s Sequence of numerals forming Number and Pattern.

Step 2: Place Value uses all the parts of the Wheel to create a Place Value Set Pattern. There

are 270 clockwise Pattern choices to form a Place Value Set. The Set is from Wheel

Sequence Pattern making to match to the Count’s, units, tens and hundreds. Each Set

preserves again the Wheels Circle and Sequence. See the 2 Text book pdf on the CD-Rom.

18

The Workbook pdf has many advanced Place Value fill-it-ins to explore this Step 2: Place

Value. This Step certainly strengthens the Wheel with plenty of Circular Patterns.

Step 3: Move Tens uses the fact that the Wheel is the Remainder Sequence of the next higher

Tier. To show the simplicity of this Sequence with ‘current-day’ zero consider the number

1,000,000. Move the 1 to a place value position right and use the next Wheel member

‘1 3 2 6 4 5’ clockwise around the Sequence. Notice what’s given away with each place value

movement. This illustration is also made clearer by a long division table of 1÷7 and follow

just the Remainder. Moving all Place Value Sets to the units and forming a final Remainder

is all that Cyclic Addition uses. The 7×Multiple is only applied with the final ‘units’

Remainder. The Pattern of the Wheel Sequence is regularly used and mathematically applied

with this Step.

Step 4: Remainder converts all Place Values in the units to a single Remainder. Notice all

Patterns of any two Wheel members forming 1 Remainder are always obedient to a Circular

Pattern. Likewise 3 and 4 Wheel Members also have Circular Patterns. These are discovered

with use. See Workbook pdf Remainder which continues on from Advanced Place Value. In

fact the Step 3: Move Tens and Step 4: Remainder often combine into Mathematics and

Pattern around the Wheel.

Step 5: 7×Multiple joins the next Tier to the Count. This applies to every Count including

the 7×Multiple with the same Cylinder and the same Common Multiple. The 7×Multiple can

be checked against the next higher Tier Wheel. The higher Tier number amongst the lower

Tier Counts introduces the higher Order alongside every Cylinder Count. The end of Cycle is

usually a Count that has no Remainder thus is equal to a 7×Multiple. As a whole revolution

around the Wheel equals 21×Common Multiple, regardless of direction, every 6th Count for

all larger Cylinders is marked by a complete Ring of identical 7×Multiple. As a Cyclic

Addition Cylinder is perused the position of a Count is often referenced by the distance from

the last end of Cycle 7×Multiple.

The formula again of ‘Count ̶ Remainder = 7×Multiple’ shows the Count’s position, place

and Order amongst partner Counts. Within one Cycle, or 6 Rings, there are 3 consecutive

7×Multiples. These Remainders Pattern, as mentioned further on, around the 7×Multiple as

the Spiral Count progresses. So the role of the 7×Multiple introduces the next higher Tier to

the lower Tier Count. Many a time showing a glimpse of strength in Pattern and Common

Multiple not shown in the lower Tier Counts with Remainder.

Patterns within the Structure have 5 Parts. The bi-directional Spirals running down and

around the Cylinder. The Ring circling around to form each new Count. The Vertically

aligned Counts. And 4 Multiple Cylinders forming Tier 2 and Tier 3 Common Multiples.

Patterns with Spiral

Clockwise Spirals are the Cyclic Addition Count Sequences. The first and third Counts in a

new Cycle Add to a 7×Multiple. The fourth and fifth Counts also Add to a 7×Multiple. Spiral

19

members 6 Counts apart differ by exactly 21×Common Multiple. The Spiral Count Sequence

has a Remainder Sequence of 5 Remainders and a 7×Multiple. For example Common

Multiple 7, starting a Count at 42 has a Remainder Sequence of 42 21 7 14 35 ̶ . Note this

Pattern visually around the Wheel, any Wheel template will do, as it is frequently used and

relied upon for review of Cylinder Counting. Note also 2 Clockwise Spirals running next to

each other differ by a Wheel Member. Follow this difference with pairs of diagonals across

the clockwise Spiral. A simple feature to strengthen a Wheel member against the Count

Sequence. So one can apply Cyclic Addition 5 Steps to any Spiral and any Tier and any

Cylinder.

Anti-Clockwise Spirals are Cyclic Addition Count Sequences in the opposite direction.

Running down and around to the left of a Cylinder. The first and second Counts in a new

Cycle Add to a 7×Multiple. The third and fifth Counts also Add to a 7×Multiple. There are 12

Remainder Patterns on a Cylinder, one for each Count Sequence. The Anti-Clockwise 6 are

the exact reverse of the Clockwise 6. In fact a Clockwise Count and anti-Clockwise Count

sharing the third Ring have identically reversed Remainder Sequences. Remainder Patterns

with Spiral are a fundamental feature of the Cylinder, protecting the Count Sequence from

error and skipping over or missing or doubling up Counts. Like the Clockwise Counts the

pairs of Anti-Clockwise Counts Spiralling down and around the Cylinder differ by a Wheel

member.

Note the sheer strength of the Cylinder Shape. A roping and meshing of all Clockwise Spirals

with all Anti-Clockwise Spirals. Each Count Number has a Clockwise Count Sequence and

an Anti-Clockwise Count Sequence. Any 6 consecutive Clockwise Counts, or a Cycle,

include all Counts running in the Anti-Clockwise direction. This feature alone shows the

highest order of Pattern and creation of a certain Cylinder. This unites the 36 Counts forming

1 Cycle of the Tier 1 Cylinder. Seeing all of these Counts with equal eye without prejudice or

preference is a strong mathematical talent. Where the whole Cycle of 6 Rings is greater

Maths than the sum of the parts (Counts).

Patterns with Ring

There are 6 consecutive Rings each Cycle meshed into diagonals to serve the Spiral Counts.

Each Rings has a Remainder Pattern of exactly the Wheel. All 5 Rings that have Remainders

display this quality. The 6th Ring or Cycle end has no Remainder or is simply equal to a

7×Multiple. Thus a circular Ring serves the circular Wheel. The Wheel Circle and Sequence

are completely preserved by Mathematics acting upon the Cylinder.

Each Ring with Remainders has 6 unique Count Number. 3 Counts in rotation Sequence in

the first Ring are found in the second Ring. Simple Pattern and also applied to the second and

third Rings, the third and fourth Rings and the fourth and fifth Rings. The other 3 members of

the Ring relative to the next Ring differ by 7בCommon Multiple’. This Pattern also applies

to consecutive pairs of Rings from the first Ring to the fifth Ring. Often it’s helpful to guide a

Spiral half way through with an entire Ring or two. Presenting Remainders and shared

20

Counts in Sequence with the next Ring. The perfects the Rotation of the Cylinder in the midst

of Cyclic Addition.

The Counts that make up a Ring can be spread across 1, 2 or 3 7×Multiples in any one Ring.

To aid finding where within a Cycle the Mathematics of a Ring belongs, rotation sequence of

a Remainder Pattern is necessary. The first and fifth Rings within a Cycle share not only the

Vertically aligned Remainder Sequence but also only have 1 7×Multiple each. This stands to

reason as the first Ring and fifth Ring are only 1 Count away from an end of Cycle,

7×Multiple. The second and fourth Rings also share Vertically Aligned Remainder

Sequences and have 2 7×Multiple each Ring.

To study the perfection of the Wheel Sequence start at the top of the Cylinder and literally

zigzag in any Spiral diagonal for 1 Cycle. Note the increment and pattern of 6 members from

the original Wheel. All Wheel members are use once, no matter the direction or travel of the

zigzag. The Pattern shows the Wheel member moving in either direction from a start.

A simple Ring Pattern to finish is opposite sides of any 1 Ring or three Count Number apart

Add to a 7×Multiple. In fact this Addition shows only 1 7×Multiple for all 3 Counts on

opposite sides of the Cylinder. Look at the Wheel Addition on all 6 Ring members and as one

moves to the next Ring the opposites, on any 1 Ring, add to the next 7×Multiple. So the fifth

7×Multiple can be found with opposites on the fifth Ring.

Patterns with Vertically aligned Counts

Following on from Ring Patterns and rotation Remainder Sequence Patterns are Vertically

aligned Number on the Cylinder. The first and fifth Rings present a Vertical difference of

2ב7×Multiple’ or 14בCommon Multiple’. This Pattern runs through all Cycles of a

Cylinder. The second and fourth Rings present a Vertical difference of 1ב7×Multiple’ or just

simply 7בCommon Multiple’. Showing how the Counts in a Cycle submit to the higher

Order 7×Multiple.

The next Pattern presents Addition of Vertically aligned Number rather than Subtraction.

Look at a Cylinder and move 3 numbers down or 6 Rings. Add both and these Add to a

7×Multiple. Move diagonally down one Ring and Add the number 3 down or again 6 Rings.

This Addition yields the next 7×Multiple in Sequence. One can follow along a Cylinder with

this Vertically aligned Number with any 2 Count Sequences to yield sequential 7×Multiple.

Pick any end of Cycle 7×Multiple and reflect two numbers, above and below, this Ring. The

Vertically aligned Number must perfectly mirror the end of Cycle Ring. Add both to equal

2בend of Cycle 7×Multiple’. This is an amazing feat of Cylinder Mathematics as once again

all 36 Counts can serve the higher Tier 7×Multiple. Literally play with this ‘mirror’ along the

end of Cycle. Add both by a zigzag or by two diagonal Spirals one backward (up) and one

forward (down).

21

As the Cylinder is Circular the final Pattern is with Circle. Each Count Spiral revolves around

the Cylinder to return to the same Vertically aligned point every 12 Counts or 2 Cycles, thus

every Count separated by 6 Vertically aligned Number or 12 Rings differs by exactly 6×7×

‘Common Multiple’ apart.

These Patterns reveal a direct relationship that a Common Multiple has with its next higher

Order or Tier. For example Common Multiple 2 Tier 1 serves Common Multiple 2 Tier 2 or

Common Multiple 14. This relationship is built on observable Patterns with any Cylinder and

any Common Multiple from 1 to 69.

Thus the Cylinder contributes to the connectivity and unity of all Tiers with the one Common

Multiple. Supporting the theory and practise of Cyclic Addition 5 Steps to mathematically

search for the perfect Remainder and 7×Multiple from the Hierarchy of Tiers.

As the Cyclic Addition Cylinder invention is relatively new, these Patterns prove the

Common Multiple Counts perfect by submitting them to a higher Tier of the same Common

Multiple. Contributing greatly to the concentration of effort on 1 Common Multiple at a time.

Thus the theory of Cyclic Addition discussed and illustrated to great lengths in the 2 texts ‘A

New Invention’ and ‘A Prophetic Design’ are brought to a pinnacle with the Cylinder.

Pattern within the Count Number

In any given Count on a Cylinder one reads the Count with the Common Multiple and

Remainder in mind. The Remainder has a position around the Wheel of ‘Common Multiple’

× ‘1 3 2 6 4 5’. Note the position in the circular ‘1 3 2 6 4 5’ and apply Mathematics to the

Count of that multiple. This often, if not always, presents knowledge on how to read and

interpret that Count Number in unison with the Common Multiple and other Cylinder Counts.

This Patterning of a Count’s numerals, forming a Whole Number, is a simple matter as the

Remainder is part Cyclic Addition 5 Steps. This reading of a Count with a Remainder is like

a child beginning to Count and make sense of all the numerals. One requires practise to

master this Art and Science of forming Common Multiple Pattern.

When a Whole Number is without Cylinder completeness give as many Remainders as

possible to unify the Number with Cyclic Addition Mathematics. For example the Year 2013

has Remainders of 4, 11, 18, 165 and 305. The Year 2014 has Remainders of 5, 12, 19, 152

and 159. The Year 2015=31×13×5 has Remainders of 6, 13, 20, 62 and 195. Year 2016 lots...

Pattern with any Count above all should be in unison with the Sequence of numerals from left

to right forming a Number. As Pattern with Cyclic Addition 5 Steps give all

Operations + × ̶ ÷, and the Cylinder gives Operatiosn + × ̶ ÷, one might conclude that

reading a Number, any Count Number, one applies all Operations + × ̶ ÷. First give Addition

of consecutive left to right numerals, then Subtraction or spacing of numerals again from left

to right, then look for Multiplication and Division. Some higher Order Cyclic Addition

contribute to the Sequences of Rational Number. Read the 2 pdf text books iterated above.

22

A Parallel of Word and Number. A Word a Count, a Sound a Pattern, a Sentence a Cycle, a

Paragraph a Cylinder, a Chapter a Common Multiple and a Story Number.

Pattern with higher Tier 2 and Tier 3 Hierarchy

The Structure of Tier 2 and Tier 3 has been briefed. The Patterns follow. The Tier 2 Cylinder

has the first Cylinder exactly 7× the Tier 1 Cylinder. This Cylinder beholds all Patterns

discussed above. The second and third Cylinders have that ‘missing’ Numbers where each

Cylinder generates only 6 Counts rather than the Tier 1 12 Counts. The Patterns of the Tier 1

Cylinder can still be applied however not with the blank Number spaces. Note with these 2

partial Cylinders that multiples of 3 have a Count in both clockwise and anti-clockwise

directions. Other Counts only have one direction.

The smaller Cylinder has only 6 Vertically aligned Number where as the Tier 1 has 12. These

all follow a simple and very convenient Pattern. The Remainder Sequences of a Count

Sequence follows the Wheel. Each Vertically aligned Number has the same Remainder. As

one Rotates the small Cylinder the Remainder Pattern Sequence is the Wheel Sequence. This

Cylinder, from Counting with Cyclic Addition Laws, has only 4 Counts however relative to

the like Tier 3 Cylinder only 1 Count in 9, or 2 each Cycle of 6 Rings, is missing. Thus this

makes for good navigation around the smaller Cylinder. Note with this smaller Cylinder all

Vertically aligned Number are 7×’Common Multiple’ apart. Making its connection with the

next higher Tier simple to follow.

Tier 3 Cylinder has 3 complete Cylinders with 12 Counts each. And the smaller Cylinder of 6

Counts. Let’s jump to the cross Cylinder Mathematics. Start by seeing where the Cylinder

commences from. Is it from the Tier 3 Wheel, or the first or second Tier 4 7×Multiple, or the

smaller Cylinder. Align all 4 Cylinders either on the template spreadsheet or 4 paper

Cylinders to show matching Ring by Ring mathematics. The smaller Cylinder should be

viewed in pairs of Rings (3 Counts × 2 Rings = 6 Counts) to match against the larger

Cylinders. Match a Ring across all 4 Cylinders where one large Cylinder only matches 4 of

the 6 numbers in the Ring. This is termed Ring, Ring, Ring, partial Ring. Note simply the end

of Cycle 7×Multiple has no match as the Ring itself has 6 identical Counts equalling the

7×Multiple. Between the ‘Ring, Ring, Ring, partial Ring’ is two large Cylinders with

identical Rings and the third large Cylinder with just an end of Cycle. Call this ‘Ring, Ring’.

The smaller Cylinder simply moves down to the next ‘pair’ of Rings. Moving down Ring by

Ring there is a Pattern of alternate ‘Ring, Ring, Ring, Partial Ring’ and ‘Ring, Ring’. The

Partial Ring is the third Ring in any Cylinder Cycle.

As one climbs the Cyclic Addition Hierarchy from Tier 2 to Tier 3 and above this cross

Cylinder mathematics binds and makes certain the Spiral Count Sequence. Each Cylinder

protects the accuracy of Count Sequence with all other Cylinders. This Mathematical feat of

Number and Geometry continues with higher Tier 4 and above. So this ‘Cyclic Addition

Number System’ has its foundation with the ‘5 Steps’ and the ‘Cylinder’ Mathematics.

23

When one uses the Cyclic Addition Cylinder to navigate and explore Whole Number do so

with these 5 aspects. The bi-directional Spirals, the Ring, the Vertically aligned Number and

the higher Tier cross Cylinder mathematics. This in essence, in truth and in Mathematics does

justice to the Completeness of Whole Number.

Mathematics that can act upon Number to form Number is made complete with Cyclic

Addition. This pathway has that ‘strangeness’ with awe and wonder that suggests that “This

requires professional presentation to protect and preserve it for as long as we all use

Number”.

24

Sequence

Mathematically and numerically Sequence often takes a back seat to other more pronounced

Arithmetic. The way unique Students finds a Difference between two Numbers may vary

from their mathematical experience. Sequence is introduced this Chapter to describe the

Maths necessary to master the Cyclic Addition Cylinder.

Cyclic Addition grew with simple single Count Sequences made from scratch. A Circular

Wheel of 6 Number at the top of the page and the Clockwise Count for 7 Cycles underneath.

Today, we’re spoilt by the perfection, Pattern and simplicity of the Cylinder. Technology has

made the pdf Cylinder simple to make and simple to distribute to Schools.

Thus let’s look into Cyclic Addition 5 Steps and their Sequence of Mathematics.

Step 1: Counting begins by selecting a Wheel members and cumulative Addition with each

Wheel member in Sequence either in a clockwise or anti-clockwise direction. This constancy

of Sequence with the Wheel direction creates Circle with Number. The Count Sequence

increments by the Wheel’s Circular Sequence. When one closely inspects a Common

Multiple Tier 3 Set of 4 Cylinders, one will find 6 Counts with all Wheel members that

follow any single Count Number. i.e. All Counts, after about the first Cycle or so, are

Counted 6 times and the following Count is unique. This was the historical path to finding all

possible Counts with a Wheel. Now this has been replaced by the perfection and integrated

Cylinder. Also note with the higher Tier 3 Counting that any 3 consecutive Counts with any

Count Sequence is unique. Look for this Pattern in the Appendix Common Multiple 49, the

last 4 Cylinders.

Step 2: Place Value is made with mini-wheels. A Mini-Wheel is 1 to 5 numbers in Sequence

around the Wheel. With any single mini-wheel a rotation Sequence of between 1 to 5

numbers can be formed. The Addition of the Place Value Set makes up the Place Value Step.

As mentioned there are only 270 clockwise Place Value Sets to choose from for any given

Wheel. A Place Value Set is matched to the units of a Count, and another Place Value Set

then matched to the tens, and if the Count is high enough to the hundreds position as well.

This discipline and search for a perfect Place Value selection from the Wheel knots and binds

the Wheel Sequence even further than the Count Sequence alone. A Place Value Set can

range from 1× ‘Common Multiple’ to about 20× ‘Common Multiple’. Most Place Value Sets

overlap significantly into other place value positions to the left.

Step 3: Move Tens shifts all Place Values outside the units, typically the tens, to the units.

Accomplished by merely rotation Sequence around the Wheel. Again the Maths is shown in

the 2 major pdf text. This Maths works with all Wheels, in all cases. So the Wheel Sequence

is again relied upon in another mathematical aspect. This stability of the Wheel promotes

genuine creativity whilst obedient to natural Cyclic Addition Laws, in a later Chapter. This

Step can be combined with Step 4: Remainder for more efficient Wheel and Circle Maths.

25

Step 4: Remainder uses the inherent Wheel Patterns to form a single number Remainder.

Again a Remainder is always from the Wheel or nothing as the end of cycle show. The

Sequence of the Wheel allows this perfect Circular Patterns of Remainder to form. For

example pairs of Place Values matched anywhere on the Wheel convert always to a single

Remainder. Any three Place Values also follow Patterns from the Sequence of the Wheel.

Thus the Wheel Sequence serves Circle and Remainder. Bear in mind the aim of the Step by

Step Cyclic Addition is to connect and unify each Counts with its higher Order.

The Remainder Pattern can also be viewed as a Sequence of Remainders each Cycle. Note

the familiar Pattern around the Wheel used with the larger Cylinders for all Tiers. And anti-

clockwise Counts have a Sequence of Remainders in reverse of the clockwise Counts.

Step 5: 7×Multiple is the final Step in the Sequence of 5 Steps. Sometimes the Remainder is

obvious. Following a simple 5 number Remainder Pattern and then the end of Cycle, right

through a Count Sequence one merely concentrates on receiving how each Count connects to

the next Tier 7×Multiple. The 7×Multiple can be checked against the Wheel for the next

higher Tier. Again the higher Tier Wheel Sequence follows the form of the lower Tier.

Sequence with Reading a Number

Count Number can be cited from either left to right or the opposite right to left. Basically this

is done subconsciously with Addition and Subtraction performed in the right to left direction.

For Pattern making and recognising the Common Multiple one ‘should’ stay with the

traditional print direction of left to right. This left to right Sequence of numerals forming a

Count should be able to be cited in groups of numerals to enhance Pattern and memory of the

Count Number.

Each Common Multiple has a constant Pattern to aid recognition. We’ve briefed Common

Multiple 1 to 10. Note with Common Multiple 11 (and 33 22 66 44 55) the hopping over

numerals to prove a multiple of 11. Common Multiple 19 uses a division of 2 every place

value position moved to the right. Common Multiple 37 uses numerals 3 apart to make a

recognisable Pattern. So Sequence with a Count Number is far beyond simply ‘symbolising’

Number with one numeral at a time from left to right. The higher truths of Complete Number

await exploration and discovery.

Sequence with a Cylinder Count

The larger Cylinder has 12 Counts that all have identical end of Cycle 7×Multiples. Thus all

Counts and their Number join at the end of Cycle. This Count Sequence Cycle by Cycle

shows every Count Number’s place, position and Order amongst partner Counts. So the 30

Counts within a Cycle all join together with the same Number at the end of Cycle.

So mapping a Count is a simple (1) the Wheel that created the Count, which also gives

Common Multiple and Tier, (2) previous end of Cycle 7×Multiple and (3) the Wheel Count

from the last end of Cycle. This process maps any Count onto any Cylinder. Simple to keep

26

an overall guide as to which Wheel, which Cycle and which Count Sequence within the

Cycle.

Let’s show a Pattern called ‘V’ that has its home with ‘Sequence’. Take a Count, any Count

on a larger Cylinder and map or find the last Count clockwise and also the last Count anti-

clockwise. Continue mapping this Count back to the previous end of Cycle 7×Multiple. The

clockwise Count starts the Cycle with the last anti-clockwise Count. Move around the Wheel

clockwise until reaching the Count. The anti-clockwise Count is simply the reverse of the

clockwise Count. Try this with any Count on the larger 12 pointed Cylinder. This shows a

vertical symmetry of Counts in both directions and again works for all Counts.

A Count Sequence increases at a rate unique to the Wheel and Common Multiple. The

resistance of Counting by the rotation Wheel Sequence makes the Cylinder Scale from start

to finish. This scale allows the Student to completely receive how Number scales to

incrementing or increasing Number by Number.

The sheer variety of Pattern in the previous Chapter shows a high degree of creativity and

spontaneity can be used to navigate around a Cylinder. However a word of caution, always

make use of the Cyclic Addition 5 Steps ‘in Sequence’. These strengthen the Count Number

and Count Sequence. This is also mentioned in Laws Chapter.

27

Cyclic Addition Step by Step

The Cyclic Addition 5 Steps are well versed in both pdf text. Rather than repeating what’s

found in these 2 pdf books, let’s brief both books to enable an informed decision to study

them. Bearing in mind the distribution of these 2 pdf books was for Primary School Teachers

and Children. This Book ‘The Complete Mathematics of the Cyclic Addition Cylinder’ is

targeted at Secondary Schooling both Teacher and Student. To invite both Teacher and

Student make good the New Invention and Prophecy of Cyclic Addition. As it is new and

original one hopes that the works on the CD-Rom and this book receive a fair trial to do

justice to the ‘current day’ mathematical Number.

The pdf book ‘A New Invention: Cyclic Addition’ discusses the ToolKit immediately.

Delving then into three types of Number ̶ Whole, Rational and Fibonacci Number.

Progressing to a perfect starting point for Cyclic Addition by asking the Wheel to perform

basic Arithmetic. Then Circular Addition to begin (mini-) Wheel with Number. This, as

mentioned, is a Primary School starting point. The middle of the book is the 5 Steps of Cyclic

Addition (pre- Cylinder). There is 30 odd pages on the 5 Steps alone. So this book will

simply join the 5 Steps to the Cylinder. Also note the Cyclic Addition connection to Rational

Number in depth. As well as creating a Count, with multiple Tiers, to show valuable Cyclic

Addition Laws introducing all possible Counts for higher Tiers. This mapping of Tier 2 and

Tier 3 Counts can be overlooked with use of the Cylinder alone. Touching on the subject of

Number serving other strands of Mathematics. And finally the ‘What’s on the CD-Rom ?’.

Basically everything other than the more recent ‘A Prophetic Design’ and ‘pdf Cylinders’.

So the pdf book ‘A New Invention’ is like a historical reference document of Cyclic

Addition. The next pdf book ‘A Prophetic Design’ is a contrast of ‘current day’ Number with

Cyclic Addition including the Cylinder.

The content of ‘A Prophetic Design’ includes An Object Count with Circle and Sequence.

The mini-Wheels used with special NumberGrids with all Whole Number to about 100. To

practically introduce, at a Primary Level, Number with Circle, Sequence and Common

Multiple. A quick look at how the current day teaches Whole Number which is used as a

benchmark for all of the major Topics #1 to #12 that follow. These Topics contrast aspects of

the current day with Cyclic Addition Mathematics. Topic #11 and a later Chapter look

carefully at the Cylinder. And Topic #12 ‘A Pedagogy of Number’ presents a complete work

of Cyclic Addition in 15 pages with illustrations for a guide. Thus there is an awesome

amount of Teaching and study on ‘Cyclic Addition Mathematics’ in these 2 pdf books.

This page was written to guide the Secondary / College Teacher and Student to bear in mind

there is a very strong foundation for Cyclic Addition Mathematics outside the highly

numerical Cylinder. This perhaps contrasts well against the lengthy and wordy ‘Laws’ book

delivered to Secondary Schools a few years ago, approximately 2011-2012.

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Let’s begin the Cyclic Addition 5 Steps with the Cylinder.

Cyclic Addition Step 1: Counting

Step 1: Counting is Circular Addition around the Wheel in either direction. The Wheel should

be present in pen to paper, or a printed ‘Reference Page’ of a Common Multiple from the

Wheels pdf. Used to guide all Cyclic Addition 5 Steps. To join all Counts from a Spiral, in

either direction requires thus the Wheel and Addition from the Wheel to the next Count.

Giving just a Remainder Pattern each Cycle with a Spiral of Counts is only half the Common

Multiple Story!

The Spiral Spacing of a Wheel member with Addition shows exactly how closely united

Operations + and × have with Whole Number. Never to be underestimated. Addition all with

a Common Multiple is the first primary step with both Number schooling and Cyclic

Addition teaching from scratch. Addition to Number is likened to grammar with Words.

Circular Addition with the Wheel shows a continuous mathematical Story. The fact that there

are many Spirals all mathematically knotted and roped together adds to the perfection of this

Cyclic Addition Step 1: Counting.

The ToolKit with this Step 1: Counting has a purpose. To show Mathematics that supports

Whole Number and the creation of Cylinder Number. The 6 pointed Wheel becomes stronger

with use. And each Wheel contributes to the mathematical dexterity and creativity of other

Wheels. The Circular Wheel Sequence resists a list of current day times-tables used with

Multiplication. Note the Patterns of a Common Multiple from a Cylinder serving and

encouraging higher Tier Mathematics. Cyclic Addition uses Operations + and × together to

show the united strength of each Operation. This is unusual and rarely found in a Maths text

book. The Operations simply command two Number be smashed against each other to form

the resultant equality. Cyclic Addition asks that the ToolKit serves just Number. The Circle

acts on the Wheel for this Step 1: Counting, describing the subtle movement of Addition

around the Wheel rather than the Wheel rotating by itself. This provides stability and

constancy with Wheel Mathematics and is thus Cyclic Addition Law. The Sequence of Step

1: Counting is enhanced by the Sequence of Remainders in any 1 Spiral. Thus showing the

Counting Step, regardless of the Cylinder already producing the Addition for the Cylinder,

has its home with Addition.

Cyclic Addition Step 2: Place value

Step 2: Place Value also uses Wheel members with Addition. Again Addition and Number is

made perfect with the Wheel. A Place Value Set, being 1 to 5 numbers from the Wheel, is

Added together to form a Place Value, in either units, tens or hundreds. The Place Value Set

is made from 1 to 5 numbers around a mini-wheel. A Mini-Wheel is a Circular Sequence of

1, 2, 3, 4 or 5 numbers from the Counting Wheel. The Place Value Set can start anywhere

around the mini-wheel and finish at any point. The proviso being the Place Value Set is only

between 1 and 5 numbers in length. This action of finding a Place Value Set to fit each place

value position acts upon the Wheel Sequence and Circle. Examples and illustrations are

shown in both pdf books ‘A New Invention’ and ‘A Prophetic Design’.

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The Step 2: Place Value has 2 grades. The first is basic which uses simple 1, 2 or 3 numbers

in Sequence from the Wheel to form a ‘basic’ Place Value Set. The Advanced uses the 1, 2,

3, 4, or 5 numbers from the Wheel as iterated above. These two grades allow the Place Value

Step to be applied to either simple Cyclic Addition using basic or more complete Cyclic

Addition using advanced.

Thus the Place Value Step applies mini-wheels and their corresponding interwoven Place

Value Set to the 6 number Wheel. Applying the Sequence of the Wheel continuously to Add

and match to the Counts units, tens and hundreds positions. This effective Wheel

Mathematics uses a chosen group of a Place Value Set, from the possible combinations of

forming the Set. There are between 1× ‘Common Multiple’ and 20× ‘Common Multiple’

from choosing a Place Value Set to equal the Count’s units, tens or hundreds position. See

table of complete Place Value Sets each derived from a mini-wheel in the Guidebook pdf on

the

CD-Rom under the Place Value Chapter. This shows a complete table of Advanced Place

Value Sets for the Wheel ‘1 3 2 6 4 5’.

The Addition of the Place Value Set together with the Circular Wheel Mathematics forms the

unique Cyclic Addition action to preface Step 3: Move Tens and Step 4: Remainder. This

Mathematics asks the Student to accurately form the connection of a Place Value Set to the

Count. This connection perfectly surrounds the Wheel’s 6 members with possible choices

from the Circular Wheel. Testing all the while the correct Addition and structural foundation

support of the mini-wheel to guard the Sequence and Circle of the Wheel. This Mathematics

is tested in the Workbook pdf again on the CD-Rom.

There are about 3 times as many Place Value Set choices from 10×’Common Multiple’ to

20× ‘Common Multiple’ as compared to the simpler and traditional place value 1× to 9×

‘Common Multiple’. Thus Place Value Sets with a length of between 3 to 5 numbers have

many more choices than the simple ‘basic’ Place Value Sets.

So grappling and manipulating the Wheel to find an appropriate Place Value Set requires, all

the while, Law to preserve the Circle and Sequence of the Wheel. The Cylinder pdf can be

enlarged on a Tablet to show say a Cycle of Counts. This makes for more visually simple

Place Value Set creation. Or use the A4 paper Cylinder. Practise of this Step improves the

Structure of numerals forming a Count. Again, not to be underestimated, this Step shows how

Addition serves Number and Wheel to master the prevailing and working Cylinder. The

overlay of Place Value Sets into more left place value positions requires a little practise with

the reward of perfecting Number and Cyclic Addition as a whole.

The chosen Place Value Sets can be visually and tactilely matched to the Wheel and

immediately acted on with Step 3: Move Tens and Step 4: Remainder. Once the Addition of

all Place Value Sets equals the Count the next 2 Steps can be performed together.

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Cyclic Addition Step 3: Move Tens to Units

Step 3: Move Tens literally moves all Place Values in the tens and hundreds to the units. This

is accomplished by a movement around the Wheel one clockwise number for every place

value position outside the units. By matching Place Values to the Wheel, usually in Set

groups, and then rotating them around the Wheel one notch. The Step 4: Remainder might be

used before Step 3: Move Tens. Converting many Place Values, in say the tens, to a single

number Remainder for that place value position. Either method is mathematically acceptable

however the simplest conversion of Place Value Patterns to a Remainder is recommended.

The Mathematics of Step 3: Move Tens is simple. See the three tables of long division in pdf

book ‘A Prophetic Design’ in Topic #12 A Pedagogy of Number. These show all that’s used

to move a Place Value from the Wheel to the units position is the Wheel’s Circular Sequence.

Since all Place Values are derived from the Wheel, and the Wheel in Circular, this Step

always works. The 3 tables show long division for 5/7, 15/21 and 20/28. All that’s used from

this Mathematics is the Circular Remainder Sequence. The decimal answer, the multiple of

the 7, 21 and 28, and the drop down Zero are completely ignored. Only the Remainder is used

in Cyclic Addition. Note how simple this current day contrast is with these tables.

Thus for example the Wheel of Common Multiple 2 being ‘2 6 4 12 8 10’ is actually the

Remainder Sequence for 14. When moving Place Values through from hundreds to tens to

units all other Maths for the next Tier Common Multiple 14 is given away. Simply left. The

Remainder and the 7×Multiple are thus preserved as one moves to the right of a Count

Number. Then all those Place Values in the Units are Patterned to form a Step 4: Remainder.

So the Wheel Circle and Sequence are again preserved for Step 3: Move Tens. The visual

templates in the Workbook pdf show a preference of calculating a Remainder for Tens and

Units before combining both to a final single Number Remainder from the Wheel.

As this Step is only Clockwise and the Cylinder Counting to the Right Spiral is for Clockwise

Counting, this proves the clockwise Sequence of the Wheel. Otherwise mathematically there

is no preference in the direction of the Wheel Sequence. The Wheel supports a Clockwise

rotation for this Step 3: Move Tens and only this direction. Interesting as all other Cyclic

Addition Steps can be in both directions.

Note also generally the Common Multiple is also preserved at each movement of this Step 3:

Move Tens to Units. All Wheel members work in unison again to perform another facet of

Cyclic Addition Mathematics. Once the basics have been mastered, the members of the new

Common Multiple Wheel can be readily accessed in Circular Sequence Order.

Cyclic Addition Step 4: Remainder

Step 4: Remainder uses a Wheel member to join every Count from the Cylinder to a

corresponding 7×Multiple. All the while, preserving the Wheel Sequence, Circle and

Common Multiple. This gives the Count a place, position and Order within the Cylinder.

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The Remainder obeys the formula ‘Count ̶ Remainder = 7×Multiple’ again all sharing the

same Common Multiple. The 7×Multiple is from the next higher Tier. The Count in essence,

for the purposes of the Cylinder Mathematics, belongs to a single 7×Multiple. Remainder

Mathematics, following the previous Cyclic Addition Steps, makes this connection possible.

Let’s go on a journey to calculate a Remainder. Bear in mind the comprehensive and

illustrative works on ‘Remainder’ in the 2 pdf books ‘A New Invention’ and ‘A Prophetic

Design’.

The aim is to find a single number called ‘Remainder’ from all of the Place Values. There can

be anything up to 5 Place Values in the tens and 5 Place Values in the units. Work on one

place value position at a time, before applying Step 3: Move Tens. Start with the Tens and

match all Place Values to the Wheel. Eliminate the 7’s or Place Values adding to

7בCommon Multiple’. Leaving 1 or 2 Place Values to apply simple Remainder Laws to, to

receive the Remainder for that tens position. Repeat by matching all Place Values in the units

to the Wheel. Again eliminate 7’s leaving one or two Place Values. Then apply Remainder

Laws to receive the Remainder in the units position. This leaves two Remainders, Units and

Tens, apply Step 3: Move Tens to the Remainder in the Tens to move to units. Finally match

these two to the Wheel to receive the final Remainder. Immediately Subtract the Remainder

from the Count to equal the 7×Multiple.

There are many Remainder Patterns, specifically shown in the pdf book ‘A New Invention’,

showing all possible types of Place Value Sets. All of these Sets have examples on how to

eliminate 7’s to leave the Remainder. This knack of matching Remainders to the ‘Circular’

Wheel and eliminating 7’s, is made simple by the presence of a Wheel in conjunction with

the Cylinder. Even the basic Place Values in units, tens and hundreds may require a visual

Wheel to form this Cyclic Addition Step.

The Remainder has the same Pattern each Cycle of a Spiral Count. Usually there a 5 uniquely

placed Remainders Patterned around the Wheel and then a end of Cycle 7×Multiple. This

Pattern of Remainders each Cycle is simple to Remember. The anti-clockwise Spiral Counts

are in the reverse of clockwise Spiral Counts. In fact this Pattern is the same for all Tier 1

Counts, 24 of 28 Tier 2 Counts and 36 of 42 Tier 3 Counts. So the Mathematics of this

Remainder Pattern is frequently used. See pdf book ‘A New Invention’ with Wheel and

Remainder Pattern.

The Remainder is mathematically used to improve the receive of its corresponding Count.

The Remainder equals a position around the Wheel. The positions are ‘Common Multiple’×

‘1 3 2 6 4 5’. View all numerals of a Count in the light of the knowledge of the Remainder

being one of ‘1 3 2 6 4 5’ positions. This may seem strange to begin with, relative to old

current day methods of factoring a number, however persist and the Count reveals knowledge

on the Number and Common Multiple.

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There is a Cylinder Count with Common Multiple 1 in the pdf book ‘A Prophetic Design’.

This shows 1 Cycle of the Cylinder and its Remainder underneath. Note the Remainder

Pattern is ‘mirrored’ on the third Ring. Showing Counts in opposite directions sharing the

third Ring have Remainder Patterns in the reverse of each other. For example with the Wheel

‘1 3 2 6 4 5’ a clockwise Pattern of ‘1 4 6 5 2 – ’ and the reverse anti-clockwise Pattern

‘2 5 6 4 1 –’. This Remainder Patterning across the whole Cylinder contributes greatly to the

formation of many Structural Patterns discussed in Chapter ‘Patterns’ .

A template found in the Workbook pdf on the CD-Rom might be in order to structure the

Step by Step Cyclic Addition Mathematics with the Wheel present at the top. This guides the

Teacher and Student to follow the 5 Steps in Sequence. Once practised, the leap to the

Cylinder with accompanying Wheel by the side, one can master Cylinder Mathematics.

Also from a humble start of a single Count at a time the Wheel was ‘always’ present.

Perfecting the Circular Mathematics and Laws of Remainder was a constant and sculpting

process. Having the Cylinder answers readymade one ‘should’ avoid the temptation to ignore

these Cyclic Addition Steps 1 to 4. Always be prepared to instantly apply the Wheel to form

Place Values, Basic and Advanced, and sculpt the Wheel to find that Remainder. A stronger

Wheel makes for a stronger Cylinder, each Step perfecting a unique aspect of its

Mathematics. Remember the Wheel Sequence is also the Remainder Sequence for each Ring

on a Cylinder other than end of Cycle. Again A stronger Wheel makes a stronger Cylinder.

A final point is the universality of the Wheel. A Tier 1 Wheel shows scale across other Tier 1

Common Multiples. Place Values sculpted with a simple Tier 1 Wheel assists in other Tier 1

Wheels and higher Tier 2 and Tier 3 Wheels. Likewise Step 3: Move Tens and Step 4:

Remainder. The inherent Remainder Mathematics of all Cyclic Addition Wheels is in unison

regardless of how many Place Values, or a longer length of numerals from a Common

Multiple Wheel and Cylinder of a higher Tier.

This unification makes for a wondrous Cyclic Addition Number System to sit side by side the

existing ‘current day’ Whole Number. As its nature is perfect and complete.

Go over the 5 Step Wheel Mathematics contributing to each Count, from the 2 text pdf and

the Workbook pdf. And decide for yourself the sheer supremacy of Whole Number with the

Wheel. Then move the Student to the Cylinder… Thus Cyclic Addition remains as perfecting

Whole Number with Mathematics rather than a Pattern making exercise with the Cylinder

and Common Multiple. A Lesson that took the author a while to learn. Thankfully the 5 Steps

were invented before the Cylinder. This Maths is backed up by the Laws Chapter later.

Cyclic Addition Step 5: 7×Multiple

The Step 5: 7×Multiple is the closest next higher Tier Number from the Count. This follows

the familiar formula of ‘Count ̶ Remainder = 7×Multiple’. Thus every Count has a

7×Multiple. The end of Cycle Count, without Remainder, equals the 7×Multiple. Introducing

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the next higher Tier seamlessly with the current Tier. This Maths applies universally to all

Counts on all Cylinders.

The 7×Multiple is the final Step of Cyclic Addition. Culminating in a link to a higher Order

of the same Common Multiple. This link begins the next Tier Wheel Mathematics.

Introducing consecutive 7×Multiples, 3 every Cycle. As a Cycle equals 3×7בCommon

Multiple’. Note also the sheer perfection of Pattern making with the 7×Multiple shown in

Chapter Patterns. Many a Pattern serve the 7×Multiple. In one Ring, in consecutive Rings, in

Spirals, and in Vertically aligned Number, both with spacing and mirroring the end of Cycle

7×Multiple. Thus the Structure of the Tier 1 Cylinder prepares the beginning of the Tier 2

Cylinder. Again a seamless connection between the two Tiers of the same Common Multiple.

Every Cycle of a Cylinder has 6 Rings. The Counts on the first and fifth Rings have just 1

7×Multiple. As they are only a single Count away from the end of Cycle. The Counts on the

second and fourth Rings have 2 unique 7×Multiples around each Ring. The middle Count or

third Ring has 3 unique 7×Multiple. The end of Cycle or every 6th Ring, all 6 Counts equal a

7×Multiple. This is helpful as one Counts around a certain Spiral, the 11 other Counts on the

same Cylinder share this 7×Multiple Pattern.

As one Counts around the Spiral on Cylinder there is a distinct Pattern of 7×Multiple. From

the previous end of Cycle 7×Multiple, the first Count, in the new Cycle, has this as the

7×Multiple. Concentrate on the Remainder Sequence, if the Sequence increases the

7×Multiple is the same as the previous Count. If the Sequence decreases the 7×Multiple

increments to the next 7×Multiple. This applies universally to all Cylinders. Thus the

Remainder Sequence serves the Pattern of 7×Multiple each Cycle.

When the Common Multiple for Tier 1 and Tier 2 are the same. The 7×Multiple acts as a

gauge and signal to the Count. Bringing the Count into a realm of the next higher Tier Order.

Simple and subtle Mathematics connecting Tiers. This is performed as a Count at the end of

Cycle where all Counts on the Cylinder join to become a Count from the next higher Tier.

Interestingly the Tier 2 and Tier 3 larger Cylinders show all possible 7×Multiples as an end

of Cycle complete Ring.

There are 12 Remainder Sequence Patterns for one Cycle, there are 6 Patterns that are

mirrored on the third Ring. Thus 6 Clockwise Counts produce 6 7×Multiple Patterns. There

are identical Patterns found by Counting Anti-Clockwise.

Again any Count from a Cylinder has its Place, Position and Order with the ‘V’ Pattern from

the previous end of Cycle 7×Multiple. So any Count can be mapped onto a Spiral belonging

to a Cylinder.

A 7 Cycle Count on a Cylinder presents 1 Cycle of 7×Multiple from the next Tier Cylinder.

So the incrementing 7×Multiple on the lower Tier Cylinder, is put to Cylinder Spiral Count

Sequence in the higher Tier. Connecting a Cylinder to its next higher Tier by exact Number.

34

The Wheels pdf on the CD-Rom shows 7 Tiers of each Common Multiple from 1 to 69. This

can be used as a Reference Page for any Cylinder. Though a Circular Wheel is favoured and

mathematically shows Circular Maths far more clearly than a straight line Wheel. This Set of

Complete Wheels is an ideal reference to prove the incrementing 7×Multiple against the

higher Tier Wheel. The Wheel Common Multiple shows a beginning and authority to the

Cylinder Spiral Counts from where they are all created. Remember that the Wheels are

potentially infinite and the same Laws apply to higher Tiers as with the lower Tiers.

Thus let’s continue on from last Chapter with creating a new Cyclic Addition Number

System. The 7×Multiple is the focal point and way to perfect the Count on all Cylinders.

These 5 Steps of Cyclic Addition culminate with the 7×Multiple. Once proved perfect and

mapped onto the higher Tier Wheel the next Count commences.

The design and Structure of the Cylinder shows how all Cylinder Whole Number Pattern with

Mathematics the lower Tier to serve and unite the higher Tier 7×Multiple.

Bear in mind that a Count can be from Multiple Tiers. The way to mathematically express

Number on all Cylinders often requires Tier 1 and Tier 2 Counting to form a start on a Tier 3

Cylinder. This is excluded from the Cylinder pdf and paper Mathematics. However remember

that any Count can raise to a higher Tier without Remainder. Begin with a lower Tier, stop at

the end of Cycle where 7×Multiple=Count and continue with the next higher Tier. Likewise

again with the Wheel from the next higher Tier and the next and so on. Thus the Cylinder is

very efficient at presenting one Wheel’s Mathematics, however Cyclic Addition is far more

flexible than isolating one Wheel and one Cylinder to produce Number. The main reason the

Cylinder is as is, is to master a Common Multiple with a common Wheel. So there sits at the

back of Cyclic Addition this creativity and flexibility to move across Tiers with a single

seamless Count.

35

Cyclic Addition ToolKit

The ToolKit, constantly applied to Cyclic Addition, is comprised of 7 elements. These are

Wheel, Pattern, Operation +× ̶ ÷, Sequence, Circle, Common Multiple and Cylinder. This

book teaches the Cylinder. And the beginning Chapters discuss at length Common Multiple,

Pattern and Sequence. Thus leaving for this Chapter Wheel, Operation +× ̶ ÷ and Circle.

ToolKit Wheel

The Wheel completely belongs with the Cylinder. The Wheel creates the movement around

the 12 Spirals in both directions around the Cylinder. The Wheel creates the Addition

between Number that brings the numerical increment between all Number alive. This quality

of Number, taken for granted, the basics learnt at an early age, is given to many other Strands

of Mathematics and other Scientific subjects. So Addition with the Wheel is a natural course

for Cyclic Addition.

The Wheel has the fixed numerical form of ‘Common Multiple’× ‘1 3 2 6 4 5’× 7(n-1) . As

mentioned, its stability and constancy allows highly mathematical operations of the 5 Steps to

act upon it.

The Wheel is used for Step 1: Counting with circular Addition to create a Spiral of Counts.

The Counts interlace and mesh the Spirals together to form all Patterns of the Wheel

Common Multiple.

The Wheel is used for Step 2: Place Value with creating a Place Value Set directly from the

Wheel. All possible mini-wheels are used to create potential Place Value Sets. These mini-

wheels are smaller Circles of in Sequence number from the Wheel. Adding the Place Value

Set together receives the place value position of the Count. Progressing from units to tens and

if the Count is high enough hundreds. Higher Tier Place Value Sets overlap and intertwine

the place value positions of larger 3-4-5-6 numeral number. This roping of Place Values is

from just Wheel Numbers.

The Wheel is used for Step 3: Move Tens by the mathematics inherent within the Circular

Sequence of the Wheel. In fact, as mentioned the Wheel is the circular Remainder Sequence

for the next higher Tier Common Multiple. For example, Wheel ‘2 6 4 12 8 10’ is the

Remainder Sequence for Common Multiple 14. This mathematics allows Cyclic Addition to

move Wheel members through place value positions with just the Wheel. Again the Wheel

contributes to the complete Order and sheer numerical supremacy of the Cylinder Count.

Numerals forming a Number, go beyond just Pattern, the numerals are bound by Cyclic

Addition Laws. There is perfect Mathematics within each Counts numerals.

The Wheel is used for Step 4: Remainder by searching, with Law and Pattern, for a single

Number from the Wheel. Again Circular Patterns allow mathematical flexibility to find that

Remainder from up to, roughly 10 Place Values from the Wheel. The Laws are simple and

36

guide one to find a Remainder. The Laws are all Circular and universal amongst all Wheels.

Once a Remainder is chosen, this Wheel member further acts upon the Count’s Sequence of

numerals to enhance and illume the Number. Bringing the Common Multiple and Tier within

the reach of all those skilled with the Cyclic Addition Wheel.

The Wheel is used for Step 5: 7×Multiple with checking the formula and the finish of the

Step 4 ‘Count ̶ Remainder = 7×Multiple’. The 7×Multiple is always equal to a small Count

from the next higher Tier Wheel. The Reference Page in the Wheels pdf with 7 Tiers for each

Common Multiple is a handy guide. Although the Circular Wheel, like the Cylinder, shows

mathematics of the Circle with Number. The 7×Multiple is used as a flag for the end of

Cycle, where all 6 Counts on the Cylinder equal a 7×Multiple. The use of 7×Multiple within

the Cylinder, both end of Cycle and Remainder for every Count, shows that the next higher

Tier Order sits transparently underneath the black and white Cylinder. From the Patterns

chapter, again we find that the Cylinder has a multitude of Patterns that form 7×Multiples

from 2 Counts with Remainder, both using Addition and Subtraction. This Maths of the

7×Multiple is absolutely connecting Tiers of Cyclic Addition Mathematics.

From another standpoint, the Wheel needs to be enacted with the Cylinder and Cyclic

Addition Mathematics to be become one with Number. These 5 Steps act upon the Wheel

uniquely to serve all the Mathematics from the Wheel to the Cylinder. Thus the presence of a

Wheel is genuinely recommended with Cylinder use.

ToolKit Operation +× ̶ ÷

If one imagines the foundation of all numerical Number with Operation +× ̶ ÷ in a Cyclic

Addition framework. What’s missing? Staying with just Number, no other mathematics

acting on Number. Let it stand on its own merits with Operations +× ̶ ÷. This is the tall ask

that Cyclic Addition can fulfil.

Let’s brief the use of Operations +× ̶ ÷ with the 5 Steps, then look at how these complete

Number.

Operations +× ̶ ÷ with Step 1: Counting. Obviously Operation +, Addition in a cumulative

Spiral of Counts shows the heart of like Number connected to like. Not so obvious, the

Operation ×, Multiplication is shown by all Wheel members having exactly the same

Common Multiple. This, as shown above, applies to all Wheels. So both Operation + and ×

unite Cylinder Spirals of Number. Very rare to find mathematics that combines both.

Operations +× ̶ ÷ with Step 2: Place Value. A small, 1 to 5 number, Circle of Number from

the Wheel are Added, Operation +, together to form each Count. As one becomes familiar

with Wheel Mathematics, the Place Value Step shows Operation × with small Number and

Circle. By matching Place Value Sets to the Wheel for each position, units and tens and

hundreds, one can find the other multiple where ‘Common Multiple’× ‘Other Multiple’=

Count. Again universal Mathematics. This shows whole number, Operation ÷, Division. Also

37

showing the building of Place Value Sets, with staggered positions in units, tens and

hundreds, to create a Count with Operation +, Addition.

Operations +× ̶ ÷ with Step 3: Move Tens shows simply a Remainder Sequence for the next

higher Tier. The premise, using current day Maths, for this Step is Operation ÷, Division.

Where, like long division, the Remainder ÷ 7בCommon Multiple’ gives the next Number

around the Wheel for the next place value position to the right.

Operations +× ̶ ÷ with Step 4: Remainder. Literally the word remainder means what’s left

over after dividing a Place Value Set by the 7בCommon Multiple’. Each Place Value Set can

be mathematically added, Operation +, and the Remainder can determined. However with the

simplicity of the Circular Wheel this Maths is best achieved by matching a Place Value Set to

the Wheel and eliminating 7’s. Discussed in 5 Steps Chapter above. Once the Remainder is

found apply formula ‘Count ̶ Remainder = 7×Multiple’ introducing the nature of Operation ̶

, Subtraction within Cyclic Addition.

Operation +× ̶ ÷ with Step 5: 7×Multiple. The 7×Multiple is checked against the next higher

Tier Wheel. For a 7 Cycle Cylinder, this 7×Multiple reaches one complete Cycle around the

next higher Tier Wheel. Thus the 7×Multiple begins Operation +, Addition with the higher

Tier Wheel.

Let’s look now at Complete Number. The Cylinder has a myriad of Patterns upon it. Most,

mentioned in the Chapter Patterns, apply Operations + and ̶ , Addition and Subtraction. Note

carefully the contrast of Adding two numbers with opposite Wheel Remainders to reveal a

7×Multiple. And Subtraction with Number having the same Remainders to reveal again a

7×Multiple. This is Pattern making Cylinder Mathematics at its best, serving how Number is

all glued together with Operations +× ̶ ÷. Again Structurally from all parts of the Cylinder –

Ring, bi-directional Spirals and Vertically aligned Number.

Note also Wheels with a factor in Common show this factor by Wheel members and

Counting on the Cylinder. So factors that are shared across Cylinders are shown simply by

the strength of the Wheel and Cylinder.

Of Course the next higher Tier Wheel from the Counting Wheel is 7× greater. From the

Wheel formula again, ‘Common Multiple’ × ‘1 3 2 6 4 5’ ×7(n-1) , where n is the Tier of the

Cyclic Addition hierarchy. This shows by proof of Cylinder Mathematics and numerical

perfection that the 7× acting on a Common Multiple, perfectly returns that same Common

Multiple in a perfect higher Order. Laws and multiple Cylinders, forming Tier 2 and Tier 3,

attest to this fundamental Cyclic Addition truth.

There is Cyclic Addition Remainder Mathematics that suggests that a Cylinder can be acted

upon with multiples of a Wheel member. i.e. For Common Multiple 1, with Wheel

‘1 3 2 6 4 5’ one can act upon any Spiral and search for a Remainder from multiples of 1 to 6.

This introduces Wheel member Common Multiples to the original Common Multiple. The

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pinnacle of Cylinder follows the Cyclic Addition 5 Steps declaring a Remainder and

7×Multiple. The search for a Common Multiple is bound with Operation ×, multiplication.

So in summary, Operations +× ̶ ÷ all unite to form a cohesive Cylinder Number. This

purports that Number can only remain as Number with this unification of Cyclic Addition

ToolKit Operation +× ̶ ÷. Without it, as discussed in depth in the pdf book ‘A Prophetic

Design’, Number is smashed destructively against any Number, to form equalities, that

neither have anything in Common with the Number that was acted against it with an

Operation, nor Ordered Pattern and Law that can be depended upon to preserve and protect

the Numberness of Number, found on the Cylinder.

ToolKit Circle

Circling around the Wheel and Cylinder shows high Order Mathematics with Cyclic

Addition. The Wheel and Cylinder are Circular. So let’s examine what actions from the 5

Steps and the rest of the ToolKit use Circle. Keep the positions of the Wheel fixed in a

hexagon Pattern guided by a Circle in the middle of the 6 member Wheel.

Circle with Step 1: Counting. The Wheel remains still whilst Counting in either a clockwise

or anti-clockwise direction. Circling around the Wheel with each Count in turn. Each Spiral

on the Cylinder follows the Wheel direction. There are 12 Vertically aligned Number on most

Cylinders, thus a complete revolution around the Cylinder occurs every 2 Cycles or 12

Counts.

Circle with Step 2: Place Value. To form a Place Value Set for units, tens or hundreds a mini-

wheel is formed. Then a small 1 to 5 number Count is performed from this mini-wheel. A

Circular mini-wheel can be any 1 to 5 number Sequence from the Counting Wheel. Thus one

Circles around any of the possible 30 mini-wheels to find the Place Value Set.

For example let’s create all possible clockwise Place Value Sets for a total of 11. Use the

Circle and Wheel ‘1 3 2 6 4 5’. From this complete list find the mini-wheel that belongs to

each Set. 11= 1+3+2+5, 1+4+5+1, 1+6+4, 2+4+5, 2+5+1+3, 2+6+3, 3+1+3+1+3, 3+2+6,

3+2+5+1, 3+2+1+3+2, 4+5+2, 5+1+5, 5+1+3+2, 5+6, 6+3+2, 6+4+1, 6+5 all =11. Use the

Circle within the Circle of the Wheel for this Step. So this Place Value Set could be applied

to Count 21=1 ten+11units, 101=9tens+11units, 121=11tens+11units and so on. By applying

Sets creatively and mathematically obedient to Laws one receives the Circle of the Wheel.

Circle with Step 3: Move Tens. Every Place Value outside the units is matched to the Wheel.

Then moved around the Circle one number clockwise for each place value position from the

left to right. The Circle and Wheel Sequence shows how simple the Place Values are

preserved by this movement around the Wheel. A small Step acknowledging the position of

every Place Value in any Count. As the Wheel is the Circular Remainder Sequence for the

next higher Tier, preparing all Place Values to be moved to the units, follows this Maths of

Circle and Sequence.

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Circle with Step 4: Remainder. There are many Remainder Patterns that can be formed from

all the Place Values in the units or tens. In fact there are exactly 270 Patterns shown in the

Guidebook pdf. Essentially only 45 Patterns, of 1 to 5 numbers, rotated around the Wheel’s 6

number locations. The pdf book ‘A New Invention’ shows these on a table, and goes through

the Patterns of eliminating 7’s, to arrive at a final single Wheel number, being a Remainder.

Thus there is only 45 Patterns to fathom all Place Value Sets, for Common Multiple 1, from

there simply apply Circle with rotation to generate any other Set. Once familiar with the

Remainder Patterns, one can apply these to any Cyclic Addition Wheel. Even the learning

curve of Cyclic Addition is Circular.

Circle with Step 5: 7×Multiple. As one Counts around the Spiral the next incrementing

7×Multiple is revealed. This 7×Multiple is found by matching it to the next higher Tier

Wheel. This may involve up to a cycle of Counting with the higher Tier. Thus once familiar

with the lower Tier Wheel, the higher Tier being 7× the lower Tier begins to be used as a

Circular Wheel.

The 7×Multiple appears on the Cylinder at the end of Cycle. As all Counts with all members,

for one Cycle or complete Circle, equal 21בCommon Multiple’, thus a 7×Multiple. This

applies to all larger Cylinders and all Tiers.

Mesh a Cycle of Remainders with the 7×Multiple. An increasing Remainder yields a

7×Multiple equal to the previous Count. A decreasing Remainder yields the next 7×Multiple.

So the Counting Wheel shows this Remainder Pattern to aid finding the 7×Multiple. Note the

Circling Patterns around the Wheel. There are 6 like Remainder Patterns for any larger

Cylinder and 6 Remainder Patterns the reverse of clockwise Counts. These Remainder

Patterns form across Cylinders of a Tier 3 Common Multiple.

Circle with other elements of the ToolKit

The Circle shows the action of moving around the fixed structure of the Wheel. All 5 Steps

have a unique method of Circling around the Wheel to form Cyclic Addition Mathematics.

The Circle and Operation +× ̶ ÷ are intertwined with the Wheel. Counting in a Circle. Place

Value Set creation with mini-wheels shows a Circle within a Circle. Move Tens shows a

Remainder Sequence of like current day Division. Remainder Patterns formed by matching

Place Values to the Wheel.

The Circle and Pattern are shown in Place Value to create a Set and Remainder to reduce that

same Set down to a single Number Remainder. The Circle and Cylinder are shown by the

direction of a Spiral and by the Ring. The major role of the Circle around the Cylinder is

shown by the Ring Patterns. The Ring moves the Cylinder around in a Circle to check the

Remainder Sequence matches the Wheel. Vertically Aligning Rings with Rotation equality

shows Circular Patterns around the Rings.

The Circle and Sequence are perhaps best shown by the ‘Mysteries’ Chapter in pdf book ‘A

New Invention’. This shows an Operation +× ̶ ÷ acting on the Wheel and the Circle and

Sequence of the Wheel are always preserved. This is an important feature to allow creating

with Operations +× ̶ ÷ in the Law and Mathematics of producing the 5 Steps perfectly. As

40

the Wheel Circle and Sequence are made complete and mathematically stronger by

Operations +× ̶ ÷ acting upon them. As this is indeed “A New Invention”, this foundation of

Law requires a hearing and judgement, to enable Cyclic Addition to sit side by side current

day Number. The best way for Cyclic Addition to move forward is schooling this knowledge

and practise together with current day Number.

Circle and Rational Number

The pdf book ‘A New Invention’ has a couple of Chapter on Pure ‘Circular’ Fractions. These

fraction Sequences with a Remainder are created with one number and are always completely

Circular. So this interpretation of Rational Number uses Sequences of Numerals to form part

or whole of a Circular Sequence. Now, certain higher Order Whole Number with Cyclic

Addition, can match these partial Circular Sequences of Rational Number. This is an amazing

feat of Numerical Mathematics. The whole fraction can be formed by Whole Number. To

completely and exactly match Sequences of Whole Number, derived from the Cyclic

Addition Wheel, to Rational Number. Showing that there is that strangeness of numerical

sequences like 1÷7= .’142857’ and those higher Order Number that partially match these

Circular Sequences. What’s required is often a modification to a numeral to convert a Whole

Number into a Circular Sequence of Rational Number. As one reads longer Whole Number,

there is that presence of a start, a Pattern, a Common Multiple and a completion or

terminating numeral(s). So there is a school of thought that proposes that all Whole Number

is derived from Circle. This unifies the Cyclic Addition Wheel’s Whole Number with

Rational Number. An unexplored area of Number! See Wheels pdf and an old pdf ‘Pure

Circular Fractions 2008’.

If one begins the journey of Cyclic Addition, one starts with Number (or numeral), word to

express the Number, Circle and Sequence. This formed to beginning basically approaching

Number with Circle and Sequence in mind. Before Wheel Counting with 6 members, these is

a Primary graded study of ‘Circular Addition’. This uses partial Sequences of the Wheel

‘1 3 2 6 4 5’ to form 30 mini-wheels. These are again Circle with simple Sequence. Like

mini-wheels with the same Circle Total are grouped to form a Common Multiple. These can

be matched to an old fashioned NumberGrid in rows of the Common Multiple. See both pdf

books for details. Thus the Circle of a Wheel can completely show all component parts and

the Sequence and Circle of the original 6 member Wheel is still preserved. The author

reckons that Primary is Primary and Secondary is for the specialised Teachers and teenagers.

It is good to know that the basic fundamentals are unified Mathematics no matter whether

low Order or high Order. All Cyclic Addition Maths has a continuity taught at any and all

levels.

The ToolKit was invented to communicate a vast number of mathematical Laws from about 6

years ago. Note the terminology like ‘end of Cycle’ and ‘Vertically aligned Number’ require

almost a Glossary. However the many uses of these terms justify their inclusion. There are

ways to navigate through a Common Multiple and higher Tiers that will always be ‘work in

progress’, so this book is a start to preserve the accumulated knowledge and experience of

Cyclic Addition. Showing a rough path to explore Cyclic Addition Whole Number with the

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Cylinder. Putting Order before chaos many a time has a Mathematics origin. This book of

Cyclic Addition purports that the Cylinder and supporting Mathematics are the Tools

necessary ‘to preserve and protect’ Number.

The qualities of Whole Number shown with the Cyclic Addition Cylinder are parallel to the

most perfect Patterns in Geometry, and Scientific wonders from Nature. Thus one does

question whether Number has the qualities of a naturally forming organism. To be applied in

the real world to serve Mankind’s battle between man(person)made and that which is created

by Nature.

Based on the plethora and sheer dominance of our society by human made Number, those

intelligent enough to steer Man’s direction with Wisdom, by avoiding future pain and

suffering, require careful and planned decision making, to apply the best available course

based on the known rather than unknown.

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Cyclic Addition Cylinder Laws

The Laws book pdf on the CD-Rom was created about 6 years ago. To permanently record all

the accumulated Cyclic Addition actions in one consolidated work of Mathematical Laws for

Number. This provided an avenue to communicate with English the works of Cyclic

Addition. However to teach Cyclic Addition from a text required the last two pdf text ‘A

New Invention’ and ‘A Prophetic Design’. These two pdf text together with the practical

Cylinder and this text show a readiness to be incorporated into a Mathematics Curriculum.

These Laws thus are a guide and formation of Cyclic Addition Mathematics, rather than the

be all end all as the previous Laws book attempted to be. Those wanting to fathom the origins

of the Laws are welcome to the older pdfs on the CD-Rom.

The Structure of this Laws book includes (1) Cyclic Addition 5 Steps and (2) specialising in

the Cyclic Addition Cylinder. So that the foundation, and established wisdom of the 5 Steps

remains perfect and true to Law amongst all Cyclic Addition Mathematics.

Laws of Cyclic Addition 5 Steps

Laws Step 1: Counting

The word ‘Counting’ is a continuous incrementing Addition from Number around the Wheel.

The Wheel is a 6 number Circular Sequence.

The Wheel is always in the form of ‘Common Multiple’ב1 3 2 6 4 5’×7(n-1) . n=Tier number.

A Cycle of Counting is a 6 Number Count Sequence. Usually finishing at an end of Cycle.

A Number with Cyclic Addition has arithmetical qualities with Operations +× ̶ ÷.

A Number has place value positions describing each numeral from right to left.

A Wheel has a certain ‘Common Multiple’ from 1 to 69. All Wheel members contribute to

the study of a Common Multiple.

A complete Count with a single Common Multiple forms a Cyclic Addition ‘Cylinder’.

A Count Sequence of any Cycle of 6 numbers is unique to the Wheel and Cyclic Addition.

Counting with a Tier 1 Wheel presents all other multiples to the length of the Count where

the ‘Common Multiple’בOther Multiple’=Count. This is with all 5 Steps of Cyclic Addition.

At any moment during a Count a ‘Pattern’ can be formed. Patterns include the Count

Sequence itself, 6 Remainders each Cycle, a Common Multiple, or 7×Multiple formed from 2

Cylinder Counts.

Cyclic Addition is not here to question an established 1400 year old invention of Number,

rather to perfect Number with Cyclic Addition Mathematics.

All 12 Tier 1 Count Sequences for a Common Multiple are generated by the Wheel. There

are 6 Clockwise and 6 anti-clockwise Count Sequences. This is shown best by the Cylinder.

All Counts from 1 Wheel share the same Common Multiple. This perfects Patterns.

An end of Cycle of Counts is in most cases with a 7×Multiple=Count, without Remainder.

A Count Number has its place, position and Order amongst all Counts by its position on a

Cylinder or Tier of Cylinders.

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The Student of Cyclic Addition may review any Cylinder Count Sequence. Recommending

the review of an entire Spiral of Counts is performed at one duration of time. Aiding memory.

The Circle and Sequence of a Wheel are preserved by any of the 5 Steps of Cyclic Addition.

The 7 Cycle Count limit on the Cylinder is a guide to allow all Counts to be performed with

equal eye without prejudice or preference. All Counts require a go to receive the whole

Cylinder.

A Count can be with multiple Tiers. Begin at a lower Tier and connect the Count without

Remainder to the next higher Tier. Then apply Cyclic Addition with the higher Tier to all 5

Steps. Move to a higher Tier Wheel and all Cyclic Addition Mathematics acts upon this

higher Tier Wheel.

One can begin Cyclic Addition with Circular Addition using mini-wheels. These 30 mini-

wheels are created from the original Wheel ‘1 3 2 6 4 5’. Then grouped into like common

multiple. All of these mini-wheels train the beginner Counter with Circle, Cycle, Sequence,

Common Multiple all with Whole Number. See 2 recent pdf books on the CD-Rom.

Counting brings forth Mathematics of magnitude, scale, ratio, proportion, quantity and

estimation, exactness and preciseness.

Counting is the beginning of exploring other strands of Mathematics.

Laws Step 2: Place Value

Place Value builds a Set for each place value position in a Whole Number Count. This Set is

made from Circular Mathematics of the Counting Wheel.

A Place Value Set is 1 to 5 numbers counting clockwise with a mini-wheel. A mini-wheel is a

partial 1 to 5 number Sequence from a 6 number Wheel. A Place Value Set can begin at any

number around a mini-wheel and finish at any number. See Workbook pdf Chapter

‘Advanced Place Value’ on the CD-Rom.

A Place Value Set matches its total with the units of a Count. Then what remains is given to

another Place Value Set in the tens. Then, if the Count is high enough, a hundreds Place

Value Set. All Place Value Sets mesh with Addition to equal the Count.

There are only 30 mini-wheels to any Cyclic Addition 6 number Wheel. Any of these 30

mini-wheels can be used to create a Place Value Set.

There is only a potential 270 clockwise Place Value Sets possible for any 1 Wheel.

These laws provide a foundation to act creatively with any Wheel in this Step 2. All the while

this Step reinforces and preserves the Circle and Sequence of the Wheel.

Note illustrations and tables of Place Value Sets and how they’re derived in pdf book ‘Laws

within a Number Universe’ on the CD-Rom.

There are about 3× as many Place Value Sets between 10× to 20בCommon Multiple’

compared with 1× to 9× ‘Common Multiple’. So connecting any possible Place Value Set to

a Count is a Science and Art form. This choice leads to actively participating with the Wheel

members in a creative and Circular way. A brilliant way to build a Count.

Place Value Sets can and often do overlap into higher place value positions. For example

Count 525 with Wheel ‘7 21 14 42 28 35’ can use two Sets ’21 14 42 28’=105 in units and

‘7 21 14’=42 in tens. Note the overlap of 105 into tens and hundreds, and the overlap of 42

into hundreds.

Creating a Place Value Set can be viewed as Circular Addition with just 1 to 5 numbers.

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Traditional Names for place value positions like units, tens and hundreds are completely

preserved. Merely applying Cyclic Addition Mathematics to Number.

The Place Value Step selects Wheel members that Add to the Count. This action joins the

numerals forming a Number with Mathematics. A highly sought after goal accomplished by

simple Wheel Mathematics.

A Place Value Set with Common Multiple 1 Wheel ‘1 3 2 6 4 5’, like the Sets adding to 11

above, prepare one to act, with exactly the same Wheel member locations, with another

Common Multiple. For example any of the 17 possible Place Value Sets to make 11, can be

then used to make 22=11×2 with Common Multiple 2, and 33=11×3 with Common Multiple

3 and so on.

This awesome flexibility to choose a Set is then available to any Wheel and any Tier.

Laws Step 3: Move Tens

The Move Tens Step makes use of Traditional Names for place value positions units, tens and

hundreds. As these are familiar to all and ideal to involve them with this Step.

The tens Place Values are matched to the Wheel and then rotated one number clockwise and

literally placed in the units position.

The hundreds Place Values and matched to the Wheel and rotated two number clockwise and

placed in the units position. Higher Place Values rotate around the Wheel clockwise one

number for each place value position left of units.

This acknowledges each Place Value’s position within a Count with Wheel Mathematics.

As all Wheels are in the form of ‘Common Multiple’ב1 3 2 6 4 5’ this rotation Mathematics

of Step 3 Move Tens works universally.

The Cyclic Addition Step 3: Move Tens can be combined with Step 4: Remainder, performed

at the same time. The simplest and most accurate method is mathematically encouraged.

See Workbook pdf for ‘Remainder’ examples showing a template for this Step 3.

The pdf book ‘A Prophetic Design’ illustrates the Math of this Step.

The Mathematics of this Step 3: Move Tens shows yet further application of the Wheel’s

Circle and Sequence.

The Wheel Sequence is actually the Remainder Sequence of 7×’Common Multiple’. Maths

with the 6 number Remainder Sequence can be viewed as the Step 3:Move Tens function.

Laws Step 4: Remainder

The Cyclic Addition Step 4: Remainder searches for 1 member of the Wheel that shows the

difference between the Count and the nearest 7×Multiple.

This follows the universal formula of ‘Count ̶ Remainder = 7×Multiple’. At the end of Cycle

there is no Remainder, thus the Count = 7×Multiple. See Cylinder.

By matching Place Values to the Wheel and applying Patterns or Remainder Laws to the

Wheel one receives the single number Remainder, again from the Wheel.

A Place Value Set in the Tens having 1 Remainder can be combined with a Place Value Set

in the Units having another Remainder. Merely apply Step 3: Move Tens after the Remainder

for both has been found. Leaving two Remainders in the Units for simple final Remainder

calculation.

The Remainder Laws that apply to all Cyclic Addition Wheels are such.

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Two Number Place Values condensing down to a single Remainder.

Consider the space around a 6 number Wheel.

Two of the same number yields a single Remainder found two numbers clockwise.

Two numbers next to each other yields a single Remainder found 3 numbers clockwise.

Two numbers two apart yields a single Remainder found in the middle of both.

Two number three apart, or opposite sides of the Wheel, yields no Remainder as they add to a

7×Multiple.

Three Number Place Values. 3 examples only.

Three of the same number yields a single Remainder found the next number clockwise.

Three numbers in Sequence yields a single Remainder found the next number in rotation

clockwise.

Three numbers all spaced 2 apart yields no Remainder as they add to a 7×Multiple.

Practise making other Remainder Patterns with the simplest Wheel ‘1 3 2 6 4 5’.

With more Place Values in a position, search for Patterns that eliminate 7’s. The Laws pdf

has a complete Table of 270 possible Place Value Set and their corresponding Remainder.

This Table Patterns the Set in groups of 6 and in Sequence around the Wheel.

The searching for a Remainder can also be accomplished with Addition of the Place Value

Set and finding the Remainder from the nearest 7×Multiple. The Table above has this

Addition with Pattern and Remainder.

The Workbook pdf has a Chapter Remainder to practically apply many Place Value Sets with

a range of Common Multiples. Try Addition of the Set to form the Remainder as well.

The Step 4: Remainder mathematically acts upon the Wheel to preserve Circle and Sequence.

In a Cycle of 6 Counts, there are 6 unique Remainders. The end of Cycle, typically has a

7×Multiple=Count, with no Remainder. The other 5 Remainders have a Pattern. This Pattern

applies to most Cylinders and Cyclic Addition. It is important to master this Circular Pattern.

This Pattern of 5 Remainders and a 7×Multiple each Cycle of Counts remains the same for

the length of the one Spiral Count on a Cylinder.

Again this Pattern of Remainders, once mastered, prevents errors of double Counting a

Number on the Wheel, missing out a Count member of the Wheel, incorrect Place Value Sets

to Add to the Count, faulty application of the Move Tens Step. So the Pattern of Remainders

each Cycle protects the Count and consequent 7×Multiple. This is to perfect Cyclic Addition

Mathematics.

When applying Cyclic Addition to a Whole Number, any Whole Number, search effectively

for that Remainder and Whole Number takes on the role given by the Hierarchy and

Common Multiple.

Remainder Mathematics on the Cylinder is left to the next Topic in this Chapter.

A Remainder has a position around the Wheel ‘Common Multiple’×’1 3 2 6 4 5’. Use the

location around the Wheel as one of ‘1 3 2 6 4 5’. Then apply Mathematics of that number to

the Count. Often the Remainder’s position around the Wheel highlights and illumes

knowledge on how to read numerals in a Number with Cyclic Addition. This may take

practise from current day methods, however it yields that role that a number plays amongst

all. Also the Remainder itself may contribute like knowledge to the Count. This is a

mysterious part of Cyclic Addition. Where the Count is mapped to its nearest 7×Multiple.

See 2 pdf text ‘A New Invention’ and ‘A Prophetic Design’ for illustrations and guide.

46

Laws Step 5: 7×Multiple

This is the final Step culminating in the receive of the 7×Multiple. The next Count follows.

Once the Remainder is determined, and Subtracted from the Count the 7×Multiple appears.

This is from the universal formula ‘Count ̶ Remainder = 7×Multiple’.

Actually the 7×Multiple sits transparently underneath the Cylinder Count. Like the many

Patterns of the Cylinder serving and Addition or Subtraction of two Counts, in Pattern,

equalling a 7×Multiple. Consider the unity of these Patterns and Step 5: 7×Multiple.

The 7×Multiple is a Number that can be found by Counting around the next higher Tier

Wheel. For example Counting with Tier 1 Common Multiple 2 Wheel ‘2 6 4 12 8 10’

presents 7×Multiples from the Tier 2 Common Multiple 14 Wheel ’14 42 28 84 56 70’.

Typically the end of Cycle Count has no Remainder and equals a 7×Multiple. The Cylinder

clearly presents this as a Ring of 6 identical 7×Multiples.

The creation of the 7×Multiple unites the lower Tier with the next higher Tier for Cyclic

Addition. In fact the higher Tier Wheel should be present whilst performing Cyclic Addition

with the lower Tier Wheel. See Wheels pdf for the first 7 Tiers of all Common Multiples.

Cylinder Patterns of the 7×Multiple are detailed next Topic this Chapter.

The Count, Remainder and 7×Multiple all share the same Common Multiple.

The 7×Multiple acts as a unseen framework beneath the Count to enhance the Count’s

numerals forming Number, Pattern of the Common Multiple, and a Reference point for the

Count. So one is mathematically informed as to the Count’s place, position and Order

amongst other Counts on the Cylinder, and other Cylinders with the same Count.

The 7×Multiple thus aids scale, ratio, magnitude, size and proportion of the Count.

There is a maximum of 6 unique Counts with Remainder utilizing one 7×Multiple. The

seventh multiple is where the Count = 7×Multiple without Remainder.

All 7×Multiples are shown as an end of Cycle, Count = 7×Multiple, from Tier 2 and above.

The Order and Hierarchy of Cyclic Addition 5 Steps introduces and unites the next higher

Order with the lower order. So the skill in working with the lower Order is presenting the

higher Order in conjunction with the lower Order Cyclic Addition 5 Steps.

Before moving to a higher Order Wheel and Cyclic Addition, make sure of your confidence

in this presentation of the higher Order within the lower Order 5 Steps. This is the test of

Mathematical ability required to shift to a higher Order Wheel.

The range of 7×Multiple in a 7 Cycle Count is about a complete Cycle with the next higher

Tier.

One can introduce oneself with Patterns found in higher Tier Wheels. This shows the starting

point from the Wheel’s Reference Page from the Wheels pdf on the CD-Rom. These Wheels

are in straight line form and can be rewritten in a Circular Form for Cyclic Addition 5 Steps.

The whole search for the 7×Multiple, found on the next higher Tier Wheel, again preserves

the Circle and Sequence.

Note the number of occurrences of a 7×Multiple within a Spiral Count on a Cylinder. This is

akin to the Remainder Pattern each Cycle. This number of a distinct 7×Multiple repeats each

Cycle. There is either 1, 2, 3 or 4 identical 7×Multiple in any complete Cycle of 1 Spiral.

Finding an accurate 7×Multiple is a perfect sign as to all the Mathematics of the other Steps.

Again the Count choice, Place Value Set choice, Move Tens Circle, Remainder Patterning to

derive the 7×Multiple shows mastery of Cyclic Addition 5 Steps with this accuracy.

47

A simple parallel with complete Cyclic Addition Mathematics. Circular Addition with mini-

wheels Count and present a consecutive Common Multiple from a revolution around the

mini-wheel. Progressing to a Common Multiple 6 Number Wheel with Cyclic Addition

Mathematics. Now the 7×Multiple is introduced in a consecutive Order with the lower Tier

Cyclic Addition 5 Steps, and then made complete by Cyclic Addition with the higher Tier

Wheel. So Cyclic Addition all the way along, from beginner to advanced mathematician,

follows the same Order. This complete Order is fundamental to a complete Number System

that Cyclic Addition presents to the Teacher and student.

Thus the Cyclic Addition Mathematics is a journey that continually perfect the higher Order

by applying ToolKit Mathematics to the lower Order. The Mathematics of the lower Order is

adequate preparation to receive the next higher Order. From a glimpse of the Wheels pdf

showing 7 Tiers for each Common Multiple, one asks how to join consecutive Tiers. The

Cyclic Addition 5 Steps, the ToolKit and the Cylinder answers this connection completely.

The Mathematics of Cyclic Addition matures the Student’s Number, discarding guesswork,

gambling and magical tricks with Number, in preference to completeness of mastering the 5

Steps with the ToolKit.

After going through a few Cylinders with Cyclic Addition 5 Steps one realises the inherent

perfection of a Number System that is guided by such Laws. A Number System that provides

freedom and creativity to express Number, with beauty and perfection, by natural Laws.

These natural Laws guide and instruct the newcomer to Cyclic Addition, to effectively

challenge all prior beliefs, in how to use the 10 Hindu-Arabic Numerals forming Number.

Imagine finding these 10 numerals in an Order of Cyclic Addition 1000’s of years ago.

Etched into a stone tablet found at a historical monument. The transition to merge Cyclic

Addition Number with current day Number would be simple. Today with competing forces to

distract our attention away from the big picture, away from the whole view, the eternal view

of the subject matter, ones powers of truth, reason and discernment are limited by current day

propaganda.

Laws of Cyclic Addition Cylinder

Cylinder Creation and Form

Cyclic Addition and the Mathematical and Numerical ‘Cylinder’ are one.

Cyclic Addition Mathematics is only complete with the mathematical study of the Cylinder.

As mentioned in the Chapter Structure the Cylinder contains 4 visual forms. These are a

Spiral Clockwise, a Spiral anti-clockwise, a Ring and Vertically aligned Number.

The Cyclic Addition 5 Steps are applied to any or all of the Spirals forming a Cylinder.

Cyclic Addition recommends that the core activity of using a Cylinder is with any or all of

the 5 Steps of Cyclic Addition. And treat the Pattern making and exploration of a Common

Multiple as a supplementary activity. This allows consistent concentration on a Spiral with

48

the purpose of Cyclic Addition 5 Steps in mind. Sure Patterns form along the way however to

remain true to the long term use of the Cylinder one should always return to the 5 Steps.

The form of the Cylinder is a 7 Cycle Count with each Spiral down and around the Cylinder.

The paper Cylinder is about 32 Counts or 5 Cycles designed for an A4 piece of paper.

The form of the Cylinder has 6 members to a Ring, for Tier 1. The first, third and fifth Rings

in any Cycle are Vertically Aligned. So to the second, fourth and sixth Rings. These form 12

points Vertically aligned, equally spaced, all the way down the Cylinder.

The Tier 1 Common Multiple Cylinder has 12 Counts on 1 Cylinder created with 1 Wheel.

The Tier 2 Common Multiple Cylinder has 28 Counts on 4 Cylinders created with 1 Wheel.

The Tier 3 and above Common Multiple Cylinder has 42 Counts on 4 complete Cylinders

created with 1 Wheel.

The Tier 2 Wheel is exactly 7× Tier 1 Wheel. The Tier 3 Wheel is exactly 7× Tier 2 Wheel.

From the formula of any Wheel ‘Common Multiple’ב1 3 2 6 4 5’×7(n-1) . When n=Tier.

Each new Ring of 6 Numbers is the next Count. The smaller Cylinder, Tier 2 and up, has a

Ring of 3 Numbers, where the next Ring down is also the next Count.

The spacing of Counts on any Spiral is generated by the Wheel in either clockwise or anti-

clockwise direction.

Counting all Counts of a Tier with Cylinder(s) presents completeness with the Common

Multiple for that Tier.

Cylinder Patterns

The use of any Pattern upon a Cylinder(s) for a Tier n has no limitation of Law. These

Patterns guide the Teacher and Student to master Number by exploration into the Common

Multiple and Tier.

The Patterns listed here are a guide only to Cylinder Mathematics and the perfection of the

7×Multiple discovery by applying Patterns.

Each Ring other than the end of Cycle has a Remainder Sequence of the Wheel.

Each Spiral also has a Pattern of Remainder each Cycle. There are 6 Patterns for clockwise

Counts and 6 reverse Patterns for anti-clockwise Counts.

The Cylinder Clockwise Spirals present a 7×Multiple by adding the first and third Counts,

and the fourth and fifth Counts. The end of Cycle has no Remainder thus the

Count=7×Multiple.

The Cylinder Anti-Clockwise Spirals present a 7×Multiple by adding the first and second

Counts, and the third and fifth Counts. The same end of Cycle, as Clockwise Counting, has

no Remainder thus the Count=7×Multiple.

Any two Spirals running next to each other share a diagonal difference of a Wheel member

all the way down the Spirals.

The roping and meshing Pattern of clockwise Counts with anti-clockwise Counts each Cycle.

Where a single Spiral clockwise also forms part of 6 (or less) anti-clockwise Counts each

Cycle. This is a major feature of the strength of the Cylinder, all Tiers and all Cylinders.

A Count can be seen on most Cylinders as its distance from the last end of Cycle. This is

called the V Pattern. To Count in both directions to meet at one Count.

49

Patterns with Ring include finding the difference between the first and fifth Rings in any

Cycle. And finding the difference between the second and fourth Rings in any Cycle. These

Differences are 14בCommon Multiple’ and 7בCommon Multiple’ respectively.

Pattern of Mirroring Number both sides of a end of Cycle 7×Multiple. These Number, any

perfect pair of Counts Mirroring each other, Add to 2× ‘End of Cycle 7×Multiple’. This

Pattern pervades the whole Cylinder. Simply use the end of Cycle as a Mirror.

Pattern of 3 apart Vertically, or 6 Rings apart in the same Vertical alignment, Add and equal

a 7×Multiple. Adding one Ring down, again 3 vertically apart, increases the previous

7×Multiple by 7בCommon Multiple’. Again this Pattern works by following two Spirals, 3

vertically apart, down and around the Cylinder. In fact this Pattern applies the whole

Cylinder. One can ZigZag down a Ring at a time to receive the same Pattern.

Patterns on any 1 Ring show a 7×Multiple equal to opposites on the Cylinder, due to the

Remainder Pattern around the whole Ring being equal to the Wheel. In fact there are 3

identical 7×Multiple from this Pattern. Also the next Ring down, the Opposites Add to the

next 7×Multiple or increment by 7בCommon Multiple’. This Pattern applies to all larger

Cylinder across all Tiers.

Like the Whole Cylinder is universally checked and proofed, made perfectly accurate by

adhering to both Cyclic Addition Law and Patterns forming 7×Multiple.

Note carefully the Pattern which show Addition of two Counts with Remainders on opposite

sides of the Wheel Add to a 7×Multiple. Patterns which show Subtraction of two Counts with

the same Remainders from the Wheel, have a Difference of a 7×Multiple. Carefully study

which Patterns apply Addition and which apply Subtraction. And note how Cyclic Addition

Step 1: Counting is with Addition and Step 4: Remainder is with Subtraction.

Cyclic Addition 5 Steps shows the intricacies of any 1 Number. And the Cylinder shows all-

pervading wholeness of all Number belonging to 1 Tier of a certain Common Multiple. This

perfects any Whole Number and its unity with all Whole Number.

Patterns of the Common Multiple are left to the Mathematician to discover. To transcend the

current day limitations by exploring uncharted territory of Mathematical Whole Number.

As the Cylinder is Circular, Two vertically aligned Counts separated by 12 Rings is exactly

2×21בCommon Multiple’ apart or simply 2 whole Cycles.

If one loose a position around a Cylinder. Start at the first Count of the Spiral and literally

Spiral down and around to reach the last Count with Cyclic Addition.

See Chapter Patterns for details on methods of reading a Number and Common Multiple.

Note Patterns across Cylinders for Tier 3 and above. Patterns that share the same Ring across

4 Cylinders and Patterns that share the same Ring across 2 Cylinders. This brings to light the

multi-dimensionality of the Cylinder with higher Tiers. Remember one Tier one Wheel. So

the maximum number of occurrences of a Number in any 1 Tier is 6. Such that the next

Count following this Count is unique. Explore this in the two simple Tables of Tier 2 and

Tier 3 Clockwise Counts for 1 complete Cycle. Note how each Count is derived and unique.

This concludes the Laws of Cyclic Addition 5 Steps and the Cylinder. Those wanting to

peruse Cyclic Addition Laws see the Laws pdf on the CD-Rom.

50

A Journey from the Hindu-Arabic Number System to Cyclic Addition Whole

Number

The Hindu-Arabic Number System has certain qualities. For one it is simple and easy to use.

Using only 10 numerals to form any Whole Number. These Number are grouped by 10’s and

written together to form a Whole Number.

The Hindu-Arabic Number System has a simple Place Value System. See Table below. The

leftmost numeral is units×100, the next right numeral is tens×101, the next right is

hundreds×102, the next right is thousands×103, the next right is ten thousands×104 . Each

numeral within the Number has this Base 10 assignment. All numerals are written close

together, usually read from left to right.

Place Value

Units 1 2 3 4 5 6 7 8 9

Tens 10 20 30 40 50 60 70 80 90

Hundreds 100 200 300 400 500 600 700 800 900

Thousands 1000 2000 3000 4000 5000 6000 7000 8000 9000

By selecting, at the most, one place value from each row and Add them together any number

from 1 to 9999 can be formed. Simple. This Addition of, for example, 2000+300+40+5=2345

is called expanded notation and is taught in Primary Level. Thus with this method shown by

the Table above, the Zero has the role of a Place Value Holder. The 2 in the 2000 is held in

the thousands by the Zeros, likewise the 3 in the 300 is held in the hundreds by Zeros, and the

4 in the 40 is held in the Tens by a single Zero. This aids definition of the Structure of our

now ancient Hindu-Arabic Number System.

The Naming conventions for Whole Number follow this Zero Place Value Holder. The 2345

is likely to be named two thousand (2000), three hundred (300), and forty (40) five (5). So the

words are like the expanded notation and run together or Add together to form a Whole

Number.

Whole Number also has arithmetical qualities. Any Whole Number can be Added,

Subtracted, Multiplied or Divided by any other Whole number to form yet again another

Number. So Whole Number maintains its original qualities of Number when an

Operation +× ̶ ÷ is acted upon the Number it returns a Number. This preservation of these

qualities above is the major success in its prolific use around the globe.

Let’s begin our journey from Hindu-Arabic Number System to Cyclic Addition. There are in

use today other bases for Number, Binary with zeros and ones, Hexadecimal with 16 symbols

acting in each place value position. Let’s follow the Table above and use Base 7. The

established Names for each place value position are now no longer Base 10. So all that can be

identified is their position location in a Base 7 Number.

51

Place Value

position 1 1 2 3 4 5 6

position 2 10 20 30 40 50 60

position 3 100 200 300 400 500 600

position 4 1000 2000 3000 4000 5000 6000

A Number for example 2345 (Base 7), looking like the Base 10 Number above, has a decimal

Number equal to (5×70 )+(4×71 )+(3×72 )+(2×73 ) = 5 + 28 + 147 + 686 = 866. Rather difficult

to interpret or translate into a familiar Base 10 form. However the Arithmetical qualities of

Number are preserved for Base 7. So a Base 7 Number can be formed, but its simplicity and

practical use are limited.

Let’s keep the Table of Base 7 Number for as moment and change the Number in Position 2,

3, and 4 back to Base 10. The Table looks something like this.

Place Value

position 1 1 2 3 4 5 6

position 2 7 14 21 28 35 42

position 3 49 98 147 196 245 294

position 4 343 686 1029 1372 1715 2058

Now the first thing to notice is our positional System with each numeral from the left to right

has disappeared. However our Base 10 positions for each Number on the Table have

returned. Thus 1372=1000+300+70+2. So the established place value mechanism still works

with this converted Table. And of course our mathematical Arithmetical qualities revert back

to Base 10 standard. Also note that the Table can successfully form any Whole Number from

1 to 2400, using at most 1 number from each position. This preserves Base 10 Number as any

Number can be created. A very important feature of the existing Number System.

What to do about out Place Value mechanism that is so widespread and simple to use. Let’s

preserve the Tables mathematical qualities above and simply change to Sequence of each

position. And in fact change their linear increasing nature to a Circular Nature. Let’s create

the first Four Tiers of Common Multiple 1 Cyclic Addition. Essentially 4 Wheels with the

simplest Common Multiple 1.

Wheel

Tier 1 1 3 2 6 4 5

Tier 2 7 21 14 42 28 35

Tier 3 49 147 98 294 196 245

Tier 4 343 1029 686 2058 1372 1715

Where are we. Creating any Base 10 Number is possible from selecting only, again 1 number

from each of the Wheels. And then Add them together like expanded notation. So Base 10

Number is preserved along with its arithmetical qualities. How then do these Wheels form a

Place Value mechanism as strong or stronger than our existing Number System ?

52

Over the last 15 years the 5 Steps of Cyclic Addition were formed. Creating a way to connect

The Tier 1 Wheel with Tier 2 Wheel above. To join the Tiers with Mathematics required the

Cylinder(s) of all possible Counts for a Tier and presenting a ‘Remainder’ to join the current

Tier with the next Tier. Contrast the first Table of Base 10 Number with the Table just above.

The Counting in Step 1 creates a unification of Counting Numbers {1, 2, 3, 4, 5, 6, 7…}

placed on the Cylinder for stronger Mathematics. Using Operations + and × together. The

Counting with Cyclic Addition reinforces that multiples of 1 need not be Counted unto

infinity. Although with Common Multiple 1 Wheel all Counting Numbers can be created, the

Cyclic Addition System encourages one to form Patterns and Knowledge about a Common

Multiple. Successive Tiers of Wheels and their Cylinders raise this knowledge to a higher

level.

Place Value Step 2 makes use of the Base 10 established position assignments of units, tens

and hundreds. To make the most out of the existing Number System. The Wheel, and just the

6 Numbers from the Wheel, can form Patterns with Circle and Sequence. These so called

Place Value Sets match to a position and literally create the Number on the Cylinder from the

Wheel. There are usually 1, 2 or 3 Place Value Sets forming a Number. These Sets have the

capability to overlap into multiple sequential place value positions. This greatly contributes to

the flexibility, simplicity and perfection of a Number’s place value positions. Thus our major

feature of the existing Hindu-Arabic Number System, with its simple Base 10 place value

positions, using the Zero as a place marker, improves to show the ‘complete’ use of all

numerals, 1 to 9 and the 0, and applicable place value positions.

Move Tens Step 3 actually shows the Wheel to be the Remainder Sequence for the next

higher Tier Common Multiple. A Remainder Sequence is Circular and describes the

movement of Cyclic Addition Place Value(s) through hundreds and tens to units. There are

many unique Remainder Sequences, however Cyclic Addition uses just the 6 number Wheel.

For basically its efficiency and mathematical perfection. So this Step reinforces the work

performed in Step 2 and moves all Place Value Sets around the Wheel and into the units. This

Step prepares the Count to receive a Remainder, the next Step.

Remainder Step 4 shows a single Wheel member to join the next higher Tier to the current

Counting Tier. Following our simple formula of ‘Count ̶ Remainder = 7×Multiple’. The

Remainder yields knowledge about the Count and gives a Count its place, position and Order

amongst partner Counts. So a Cylinder of Counts is unified in Pattern and Mathematics, via

use of the Remainder, with its 7×Multiple. Thus our Table above only works with one Tier at

a time, to produce a Cylinder(s). Moving to Tier 2 Wheel creates Number, on a new set of 4

Cylinders, where all Counts share the new higher Tier Common Multiple that perfects the

lower Tier of the same Common Multiple.

7×Multiple Step 5 shows basically the beginning of the next higher Tier with the current Tier.

With any Spiral of Counts the 7×Multiple is revealed in an incrementing Order. When the

higher Tier begins, the 7×Multiple of the lower Tier is put to Sequence and Pattern. This

53

7×Multiple appears frequently with Pattern making on all of the Cylinders. Thus reinforcing

the Structure of the Cylinder to join these consecutive Tiers with Mathematics. This is

fundamental to Cyclic Addition and the joining mechanism of one Tier to another.

Thus our potential new Cyclic Addition Number System has 69 Common Multiples and an

infinite number of Tiers. All original Mathematical qualities are preserved. And also the new

Wheel Circle and Sequence, right through all 5 Steps and Cylinder Mathematics. See the

Wheels pdf on the CD-Rom for a detailed look at each Common Multiple.

The Cyclic Addition Number System has strong axioms for the creation these Wheels. Where

Whole Number is unified with Rational Number, Exponential Number and Fibonacci

Number. This unification brings new light and knowledge to our existing Base 10 Number.

The Cyclic Addition Cylinder(s) show extremely high unity with Whole Number. This unity

is with Wheel that created it and mathematics that supports it. Encompassed with many

Pattern and Circular actions that are distinctly mathematical. Bringing the home of Numerical

Mathematics out of a destructive realm into a ‘officially’ creative one.

Cyclic Addition, as with the premise on current day Number, is infinite. And brings one

quickly and effectively back to the structural formation of Number. Building Counts with

Wheel Mathematics strengthens qualities of Number awaiting those challenged by exploring

and discovering new frontiers with plain ol’ Number.

Cyclic Addition serves Number. The ToolKit above was invented specifically for that

purpose, to serve Number. A natural progression for Number so that Number retains its

highest Mathematical qualities, which is where the chapter began. And places no limitation as

to how Number, created from Cyclic Addition, can be used with any other form whatsoever.

The home for Number will always be Cyclic Addition Mathematics. From here Number is

preserved to enable serving the myriad upon myriad of practical applications.

The Cylinder is quick to make and with a simple Common Multiple easy to use. The author

strongly recommends Cylinder Mathematics with Number and Circle. This is an original

work and to date nothing else like it on the planet. So, like the author, one asks how best to

preserve it. The school system of South Australia was chosen. Under the assumption that the

Teacher is the best Student.

The author can be contacted via email, for any contributions, and recommendations, any

future considerations, any errors found, or any suggestions for the future of Cyclic Addition.

email jeff132645(at)hotmail.com

A final parallel between nature and number. Frozen water forms crystals of perfect hexagonal

structures. The more perfect (or blessed) the water the more perfect and patterned the crystal.

This is a scientific study by Emoto, M. Likewise hexagonal shaped Cyclic Addition Wheels

are made perfect by rising to a higher Tier. The journey of mankind’s Number is infinite.

1 3 2 6 4 5

4 5 8 10 9 6

9 6 11 12 15 10

11 12 15 17 16 13

15 17 16 20 18 19

21 21 21 21 21 21

27 25 26 22 24 23

31 30 27 25 26 29

33 36 31 30 27 32

38 37 34 32 33 36

41 39 40 36 38 37

42 42 42 42 42 42

43 45 44 48 46 47

46 47 50 52 51 48

51 48 53 54 57 52

53 54 57 59 58 55

57 59 58 62 60 61

63 63 63 63 63 63

69 67 68 64 66 65

73 72 69 67 68 71

75 78 73 72 69 74

80 79 76 74 75 78

83 81 82 78 80 79

84 84 84 84 84 84

85 87 86 90 88 89

88 89 92 94 93 90

93 90 95 96 99 94

95 96 99 101 100 97

99 101 100 104 102 103

105 105 105 105 105 105

111 109 110 106 108 107

115 114 111 109 110 113

68 204 136 408 272 340

680 612 408272 340 544

612 408 748 816 1020 680

1156 1088 884748 816 1020

1020 1156 1088 1360 1224 1292

1428 1428 14281428 1428 1428

1836 1700 1768 1496 1632 1564

1700 1768 19722108 2040 1836

2244 2448 2108 2040 1836 2176

2176 2244 24482584 2516 2312

2788 2652 2720 2448 2584 2516

2856 2856 28562856 2856 2856

3264 3128 31962924 3060 2992

3536 3468 32643128 3196 3400

3672 3876 35363468 3264 3604

4012 3944 37403604 3672 3876

4216 4080 41483876 4012 3944

4284 4284 42844284 4284 4284

4352 4488 44204692 4556 4624

4556 4624 48284964 4896 4692

4896 4692 50325100 5304 4964

5032 5100 53045440 5372 5168

5304 5440 53725644 5508 5576

5712 5712 57125712 5712 5712

6120 5984 60525780 5916 5848

6392 6324 61205984 6052 6256

6528 6732 63926324 6120 6460

6868 6800 65966460 6528 6732

7072 6936 70046732 6868 6800

7140 7140 71407140 7140 7140

7208 7344 72767548 7412 7480

7412 7480 76847820 7752 7548

1083 1121 1102 1178 1140 1159

1197 1197 1197 1197 1197 1197

969 912 1007 1026 1083 988

1007 1026 1083 1121 1102 1045

817 855 836 912 874 893

874 893 950 988 969 912

1577 1539 1558 1482 1520 1501

1596 1596 1596 1596 1596 1596

1425 1482 1387 1368 1311 1406

1520 1501 1444 1406 1425 1482

1311 1273 1292 1216 1254 1235

1387 1368 1311 1273 1292 1349

1881 1919 1900 1976 1938 1957

1995 1995 1995 1995 1995 1995

1767 1710 1805 1824 1881 1786

1805 1824 1881 1919 1900 1843

1615 1653 1634 1710 1672 1691

1672 1691 1748 1786 1767 1710

2109 2071 2090 2014 2052 2033

2185 2166 2109 2071 2090 2147

285 323 304 380 342 361

399 399 399 399 399 399

171 114 209 228 285 190

209 228 285 323 304 247

19 57 38 114 76 95

76 95 152 190 171 114

779 741 760 684 722 703

798 798 798 798 798 798

627 684 589 570 513 608

722 703 646 608 627 684

513 475 494 418 456 437

589 570 513 475 494 551

2850 2950 2900 3100 3000 3050

3150 3150 3150 3150 3150 3150

2550 2400 2650 2700 2850 2600

2650 2700 2850 2950 2900 2750

2150 2250 2200 2400 2300 2350

2300 2350 2500 2600 2550 2400

4150 4050 4100 3900 4000 3950

4200 4200 4200 4200 4200 4200

3750 3900 3650 3600 3450 3700

4000 3950 3800 3700 3750 3900

3450 3350 3400 3200 3300 3250

3650 3600 3450 3350 3400 3550

4950 5050 5000 5200 5100 5150

5250 5250 5250 5250 5250 5250

4650 4500 4750 4800 4950 4700

4750 4800 4950 5050 5000 4850

4250 4350 4300 4500 4400 4450

4400 4450 4600 4700 4650 4500

5550 5450 5500 5300 5400 5350

5750 5700 5550 5450 5500 5650

300 200 25050 150 100

750 850 800 1000 900 950

500 450 300200 250 400

600 750 500450 300 550

550 600 750 850 800 650

1050 1050 1050 1050 1050 1050

1350 1250 1300 1100 1200 1150

1650 1800 1550 1500 1350 1600

1550 1500 1350 1250 1300 1450

1900 1850 1700 1600 1650 1800

2050 1950 2000 1800 1900 1850

2100 2100 2100 2100 2100 2100

2451 2537 2494 2666 2580 2623

2709 2709 2709 2709 2709 2709

2193 2064 2279 2322 2451 2236

2279 2322 2451 2537 2494 2365

1849 1935 1892 2064 1978 2021

1978 2021 2150 2236 2193 2064

3569 3483 3526 3354 3440 3397

3612 3612 3612 3612 3612 3612

3225 3354 3139 3096 2967 3182

3440 3397 3268 3182 3225 3354

2967 2881 2924 2752 2838 2795

3139 3096 2967 2881 2924 3053

4257 4343 4300 4472 4386 4429

4515 4515 4515 4515 4515 4515

3999 3870 4085 4128 4257 4042

4085 4128 4257 4343 4300 4171

3655 3741 3698 3870 3784 3827

3784 3827 3956 4042 3999 3870

4773 4687 4730 4558 4644 4601

4945 4902 4773 4687 4730 4859

43 129 86 258 172 215

172 215 344 430 387 258

473 516 645 731 688 559

645 731 688 860 774 817

1161 1075 1118 946 1032 989

903 903 903 903 903 903

1333 1290 1161 1075 1118 1247

1634 1591 1462 1376 1419 1548

1419 1548 1333 1290 1161 1376

1763 1677 1720 1548 1634 1591

1806 1806 1806 1806 1806 1806

387 258 473 516 645 430

1482 1534 1508 1612 1560 1586

1638 1638 1638 1638 1638 1638

1326 1248 1378 1404 1482 1352

1378 1404 1482 1534 1508 1430

1118 1170 1144 1248 1196 1222

1196 1222 1300 1352 1326 1248

2158 2106 2132 2028 2080 2054

2184 2184 2184 2184 2184 2184

1950 2028 1898 1872 1794 1924

2080 2054 1976 1924 1950 2028

1794 1742 1768 1664 1716 1690

1898 1872 1794 1742 1768 1846

2574 2626 2600 2704 2652 2678

2730 2730 2730 2730 2730 2730

2418 2340 2470 2496 2574 2444

2470 2496 2574 2626 2600 2522

2210 2262 2236 2340 2288 2314

2288 2314 2392 2444 2418 2340

2886 2834 2860 2756 2808 2782

2990 2964 2886 2834 2860 2938

390 442 416 520 468 494

546 546 546 546 546 546

234 156 286 312 390 260

286 312 390 442 416 338

26 78 52 156 104 130

104 130 208 260 234 156

1066 1014 1040 936 988 962

1092 1092 1092 1092 1092 1092

858 936 806 780 702 832

988 962 884 832 858 936

702 650 676 572 624 598

806 780 702 650 676 754

805 798 777 763 770 791

777 763 770 742 756 749

735 735 735 735 735 735

693 707 700 728 714 721

665 672 693 707 700 679

651 630 665 672 693 658

616 623 644 658 651 630

595 609 602 630 616 623

588 588 588 588 588 588

581 567 574 546 560 553

560 553 532 518 525 546

525 546 511 504 483 518

511 504 483 469 476 497

483 469 476 448 462 455

441 441 441 441 441 441

399 413 406 434 420 427

371 378 399 413 406 385

357 336 371 378 399 364

322 329 350 364 357 336

301 315 308 336 322 329

294 294 294 294 294 294

287 273 280 252 266 259

266 259 238 224 231 252

231 252 217 210 189 224

217 210 189 175 182 203

189 175 182 154 168 161

147 147 147 147 147 147

105 119 112 140 126 133

77 84 105 119 112 91

63 42 77 84 105 70

28 35 56 70 63 42

7 21 14 42 28 35

805 798 826 819

784 784 784 784 784 784

777 763 742 756

742 756 714 721

721 742 700 679 714

693 700 679 672

679 672 658 651

637 637 637 637 637 637

595 609 630 616

567 574 595 609

553 532 567 574 595

532 525 546 553

511 504 532 525

490 490 490 490 490 490

483 469 448 462

448 462 420 427

427 448 406 385 420

399 406 385 378

385 378 364 357

301280273238259

315301280273

322336315301

196196196196196196

343343343343343343

231238210217

259252231238

12691112154133

168154175189

133126168154

8491112105

494949494949

63708491

840 854 875 861

833 833 833 833 833 833

812 819 791 798

791 798 777 770

770 791 756 749 728

756 749 728 714

728 714 693 707

686 686 686 686 686 686

644 651 665 672

630 623 644 651

602 581 623 644 609

581 567 609 602

546 560 581 567

539 539 539 539 539 539

518 525 497 504

497 504 483 476

476 497 462 455 434

462 455 434 420

434 420 399 413

315350329287308

357350329336

378371357350

392392392392392392

245245245245245245

273287266252

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182189210203

210203231224

989898989898

119105126140

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.

.

756 777

798 812 791

707 728 721

749 763 742

658 679 672

714 693

609 630

651 665 644

560 581 574

602 616 595

511 532 525

567 546

462 483

504 518 497

413 434 427

455 469 448

364 385 378

420 399

315 336

357 371 350

266 287 280

308 322 301

217 238 231

273 252

168 189

210 224 203

567763

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133140119

4221

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49 147 98 294 196 245

441 294 539 588 735 490

196 245 392 490 441 294

735 833 784 980 882 931

539 588 735 833 784 637

1323 1225 1274 1078 1176 1127

1029 1029 1029 1029 1029 1029

1617 1764 1519 1470 1323 1568

1519 1470 1323 1225 1274 1421

2009 1911 1960 1764 1862 1813

1862 1813 1666 1568 1617 1764

2107 2205 2156 2352 2254 2303

2058 2058 2058 2058 2058 2058

2499 2352 2597 2646 2793 2548

2254 2303 2450 2548 2499 2352

2793 2891 2842 3038 2940 2989

2597 2646 2793 2891 2842 2695

3381 3283 3332 3136 3234 3185

3087 3087 3087 3087 3087 3087

3675 3822 3577 3528 3381 3626

3577 3528 3381 3283 3332 3479

4067 3969 4018 3822 3920 3871

3920 3871 3724 3626 3675 3822

4165 4263 4214 4410 4312 4361

4116 4116 4116 4116 4116 4116

4557 4410 4655 4704 4851 4606

4312 4361 4508 4606 4557 4410

4851 4949 4900 5096 4998 5047

4655 4704 4851 4949 4900 4753

5439 5341 5390 5194 5292 5243

5145 5145 5145 5145 5145 5145

5635 5586 5439 5341 5390 5537

343343 343 343 343 343

5733

5194 5292 5243

1078 833

882 931 1078 1176 1127 980

1519 1470

1862 1813 1666 1568 1617 1764

1225 1274

539 588

539 588 735 833 784 637

784 637 882 931

392 490 441 637

1372 1372 1372 1372 1372 1372

1666 1568 1617 1421

1078 1176 1127 1323

2205 2156

2401 2401 2401 2401 2401 2401

1666 1911

2205 2156 2009 1911 1960 2107

2352 2254 2303 2107

1960 2107 1862 1813

3136 2891

2940 2989 3136 3234 3185 3038

2597 2646

2597 2646 2793 2891 2842 2695

2842 2695 2940 2989

2450 2548 2499 2695

3577 3528

3920 3871 3724 3626 3675 3822

3283 3332

3430 3430 3430 3430 3430 3430

3724 3626 3675 3479

3136 3234 3185 3381

4263 4214

4459 4459 4459 4459 4459 4459

3724 3969

4263 4214 4067 3969 4018 4165

4410 4312 4361 4165

4018 4165 3920 3871

5194 4949

4998 5047 5194 5292 5243 5096

4655 4704

4655 4704 4851 4949 4900 4753

4900 4753 4998 5047

4508 4606 4557 4753

5635 5586

5341 5390

5488 5488 5488 5488 5488 5488

5537

5439

5782 5684

686 686 686 686 686 686

882 931 1078 1176 1127 980

735 833 784 980 882 931

1225 1274 1421 1519 1470 1323

1127 980 1225 1274 1421 1176

1715 1715 1715 1715 1715 1715

1421 1519 1470 1666 1568 1617

2205 2156 2009 1911 1960 2107

2009 1911 1960 1764 1862 1813

2548 2499 2352 2254 2303 2450

2303 2450 2205 2156 2009 2254

2744 2744 2744 2744 2744 2744

2695 2597 2646 2450 2548 2499

2940 2989 3136 3234 3185 3038

2793 2891 2842 3038 2940 2989

3283 3332 3479 3577 3528 3381

3185 3038 3283 3332 3479 3234

3773 3773 3773 3773 3773 3773

3479 3577 3528 3724 3626 3675

4263 4214 4067 3969 4018 4165

4067 3969 4018 3822 3920 3871

4606 4557 4410 4312 4361 4508

4361 4508 4263 4214 4067 4312

4802 4802 4802 4802 4802 4802

4753 4655 4704 4508 4606 4557

4998 5047 5194 5292 5243 5096

4851 4949 4900 5096 4998 5047

5341 5390 5537 5635 5586 5439

5243 5096 5341 5390 5537 5292

5831 5831 5831 5831 5831 5831

5537 5635 5586 5782 5684 5733

6125 6027 6076 5880 5978 5929

.

.

931

5047

4018

2989

1960

4851 4900 4998

4949 5096

5194 5243 5341

5292 5439 5390

4263 4410 4361

4508 4557 4655

4606 4753 4704

3577 3724 3675

3822 3871 3969

3920 4067

4165 4214 4312

2891 3038

3136 3185 3283

3234 3381 3332

3479 3528 3626

2450 2499 2597

2548 2695 2646

2793 2842 2940

1764 1813 1911

1862 2009

2107 2156 2254

2205 2352 2303

1176 1323 1274

1421 1470 1568

1519 1666 1617

245294147

1969849

1078 1127 1225

980833

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