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© Copyright 2019 M J Cooper In Accordance With Title Page – Oregon USA Paper 5. Passage Reconstruction M. J. Cooper Version 2.0 (October 21, 2019) Copyright © 2019 M. J. Cooper, Oregon, USA All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, write to the publisher, addressed ʺAttention: Permissions Coordinator,ʺ at the address below. Publisher: [email protected]

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Page 1: Paper 5. Passage Reconstruction M. J. Cooper Version 2.0 ......2019/10/05  · Passage, whereas it is 125.4 to 127.6ʺ above at the south side; this is 1.1 to 3.3ʺ higher. The roof

© Copyright 2019 M J Cooper In Accordance With Title Page – Oregon USA

Paper 5. Passage Reconstruction

M. J. Cooper

Version 2.0 (October 21, 2019)

Copyright © 2019 M. J. Cooper, Oregon, USA

All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in

any form or by any means, including photocopying, recording, or other electronic or mechanical

methods, without the prior written permission of the publisher, except in the case of brief

quotations embodied in critical reviews and certain other noncommercial uses permitted by

copyright law. For permission requests, write to the publisher, addressed ʺAttention: Permissions

Coordinator,ʺ at the address below.

Publisher: [email protected]

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© Copyright 2019 M J Cooper In Accordance With Title Page – Oregon USA 1

5. Passage Reconstruction

Before reconstructing the design of the interior passages and chambers of the Pyramid, it is

necessary to understand some observations and tools that will help with reconstructing the

passages. These are:

The effect of subsidence on the Pyramid

How the passages and pathways are measured

More Inductive Metrology

Effects of Subsidence on the Great Pyramid

The purpose of this section is to show that the Pyramid has been subject to subsidence, and the

extent of the subsidence is modeled. In Chapter III of ʺGreat Pyramid, Its Divine Messageʺ

Davidson discusses the impacts of subsidence on both the interior and exterior of the Great

Pyramid. Limestone, which forms the Nile Valley, has many subterranean caverns deep within it.

These caverns collapse from time to time, which causes minor earthquakes and also subsidence

of the ground above. The subsidence stresses the rock layers, which then fissure, or break.

Evidence of this can be seen by the fissures visible inside and outside the Pyramid. The Figure

below shows known fissures, in yellow, in and around the Pyramid labeled GH through VW.

Two or more of these must have existed before the construction of the Pyramid since the original

builders blocked them up. Davidson, who was a structural engineer, concludes that the designer

foresaw the possibility of the fissured strata being further impacted by the added weight of the

Pyramid and it ʺ..is as perfectly designed to meet, and adjust itself to, the conditions of

subsidence as it well could be;ʺ (Davidson 186).

Further possible evidence of subsidence is provided by Petrie (P24) who observes that the

courses of the Great Pyramid have dips of 0.5° to 1° inwards, which is taken to mean that the

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outside stones of a course dip toward the center of the Pyramid. These dips are likely a deliberate

design feature to make the Pyramid more stable when built. In this case, stones would tend to

move toward the center, rather than away from it during subsidence. We will see shortly to what

extent this may have occurred.

The sloping passages also provide evidence of subsidence. Petrie states (P36) that the mean axis

of the whole length of the Descending Passage is 26° 31ʹ 23ʺ. He further states (P39) that the

mean axis of the whole length of the Ascending Passage and Grand Gallery is 26° 12ʹ 50ʺ. The

total angle between them is thus 52° 44ʹ 13ʺ. This system can be viewed as two passage

subsystems with angles of 26° 22ʹ 6.5ʺ about a line which dips 0° 9ʹ 16ʺ below horizontal.

Inductive metrology suggests that the mean was intended to be horizontal, i.e., 0° 0ʹ 0ʺ, then 0° 9ʹ

16ʺ represents the mean subsidence in degrees.

Please note that Inductive Metrology would assume that the design intent was that the angle of

all up-sloping and down-sloping passages is the same, which is the same as assuming that the

pitch of a gabled roof is intended to be the same on both slopes.

The Kingʹs Chamber, Queens Chamber, and end of the Descending Passage are approximately in

a vertical line at the central axis of the Great Pyramid. The distance from the Entrance to the

central axis horizontally is approximately 4000ʺ. Therefore the

Approximate Predicted Subsidence = 4000ʺ x TAN(0° 9ʹ 16ʺ) = 11ʺ

Petrieʹs measure of the floor level of the Queenʹs Chamber is 834.4ʺ, and the level predicted by

the theory is 846.7ʺ, which is a difference of 12.3ʺ, and that compares well with the 11ʺ predicted

above.

It can also be seen, by induction, that the passage to the Queenʹs Chamber was intended to be

horizontal. According to the table in Petrieʹs measurements (P40), the Queenʹs Chamber passage

changes level as follows:

On flat floor 858.4ʺ at 52ʺ along passage

Minus step in floor 19.7ʺ at1307.0ʺ along passage

Minus niche N side 834.4ʺ at 1620.7ʺ along passage

For a level change due to subsidence of 4.3ʺ over 1568.7ʺ

The slope of this change = tan-1

(4.3/1568.7) = 0.1571° 0° 09ʹ 25ʺ

Smyth measured this slope as 0° 11ʹ (S2p152) by the roof.

It can be concluded from these witnesses that the subsidence along the passage to the Queens

Chamber is from 0° 09ʹ 25ʺ to 0° 11ʹ. This is very much of the same order as that shown from

comparing the difference in the slopes of the descending and ascending passages above, which is

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© Copyright 2019 M J Cooper In Accordance With Title Page – Oregon USA 3

0° 9ʹ 16ʺ. This provides visibility into the manner and extent to which the interior of the Pyramid

has subsided, i.e., it has caused the Pyramid to sink close to the Queen’s Chamber by 11ʺ.

Most of the passages from which subsidence data is obtained are in the north half of the Pyramid,

but there is some data from the south half. Here the subterranean chamber and passages do show

that once the center is reached, the slopes start to tilt upward as one moves further south, which

indicates that the maximum subsidence occurred at about mid-Pyramid. For example, from P37,

the north side of the Subterranean Chamber roof is 124.3ʺ above the end of the Descending

Passage, whereas it is 125.4 to 127.6ʺ above at the south side; this is 1.1 to 3.3ʺ higher. The roof

at the south end of the South Subterranean Passage is 1184ʺ below the Pavement level, which is

4ʺ above the roof of this passage at the Subterranean Chamber, where it is 1188ʺ below

Pavement level.

Passage and Chamber Floor Angles

The following is a complete list of passages and chambers that were intended to be horizontal:

Passage or Chamber Acronym Measured Angle ° ʹ ʺ Intended Angle °

1st Northern Subterranean Passage 1NSP 0° 0ʹ 0ʺ

Recess Rx 0° 0ʹ 0ʺ

2nd

Northern Subterranean Passage 2NSP 0° 0ʹ 0ʺ

Subterranean Chamber SC 0° 19ʹ 33ʺ 0° 0ʹ 0ʺ

Southern Subterranean Passage SSP 0° 21ʹ 17ʺ 0° 0ʹ 0ʺ

Queens Passage Low QPL 0° 09ʹ 25ʺ 0° 0ʹ 0ʺ

Queenʹs Passage High QPH 0° 11ʹ 00ʺ 0° 0ʹ 0ʺ

Queenʹs Chamber QC 0° 0ʹ 0ʺ

First Low Passage 1LP Distorted 0° 0ʹ 0ʺ

Antechamber AC Distorted 0° 0ʹ 0ʺ

Second Low Passage 2LP Distorted 0° 0ʹ 0ʺ

Kings Chamber KC Distorted 0° 0ʹ 0ʺ

There are four sloping passages, and each one has a measured angle. The slope of these four

individual passages can only be determined from Petrieʹs table in P6, which gives the coordinates

of the endpoints from which their angles can be derived. As Davidson points out, Petrie has an

unfortunate error in his calculation of the top of the Grand Gallery, which has been corrected in

the data below. The sloping passages are also divided by Petrie into two sets of passages, one set

for the up-sloping passages, AP + GG, and the other for the down-sloping passages EP + DP.

The angles shown below are those that Petrie set his theodolite to when taking measurements

along the entire length of both sets of passages. The table below summarizes all these

measurements and provides the mean and theoretical angles.

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Sloping Passage Acronym Absolute Angle

° ʹ ʺ

Measured or Theoretical

Angle °

EP and DP as one length EP/DP 26° 31ʹ 23ʺ 26.5231

AP and GG as one length AP/GG 26° 16ʹ 40ʺ 26.2778

Entrance Passage EP 26° 29ʹ 13ʺ 26.4869

Descending Passage DP 26° 33ʹ 05ʺ 26.5514

Ascending Passage AP 26° 03ʹ 37ʺ 26.0602

Grand Gallery GG 26° 17ʹ 02ʺ 26.2839

Mean of all four angles above Mean 26° 20ʹ 44ʺ 26.3456

Pyramidology Theoretical Theory 26° 18ʹ 10ʺ 26.3027

1 vertical for 2 horizontal 2:1 26° 33ʹ 54ʺ 26.5651

37 vertical for 79 horizontal 37:79 26° 16ʹ 27ʺ 26.2742

The combined passage angles EP and DP and also AP and GG in the second and third rows were

used above to determine the extent of the subsidence within the Pyramid and will not be taken

into consideration for the passage angle. To choose the intended passage angle from the above

data, we need to consider the following:

From Inductive Metrology, the passage angle for all the up and down sloping passages

should be the same magnitude but opposite in direction.

Like the base angle of the Pyramid, the passage angle will be computed mathematically

rather than some randomly chosen angle, which has no formula from which to calculate

it. Three candidate angles are included at the bottom of the above Table, which fit this

criterion.

Compare the measured and averaged passage angles, which are the fourth through eighth

rows of the table with the three candidate angles at the bottom of the table. The 1:2 angle

is outside the range of the measured and average values, and we will see that the

Pyramidology theoretical angle is the closest of the two remaining candidates to the mean

angle. The Pyramidology Theoretical angle should be chosen on this basis.

The base angle was chosen in Paper 4 to be the Pyramidology π angle. The pi angle is

expressed as tan-1

4/π, and the selection of another, very similar, π angle, sin-1

√π/4, for the

passage angle satisfies Inductive Metrology.

The arc length theory demonstrates that if the base angle is the π/4 angle, then the only

angle in the arc length table that would not be definable by π would be the passage angle

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unless the √π/4 angle is assumed to be the intended angle.

The Figure below is repeated from Paper 4 and shows the exterior of the Pyramid with

dimensions based on the Sacred Cubit. The equal-area square was rotated to the position

shown, and it subtended an angle of Sin-1

√π/4° with the vertical axis of the Pyramid.

Selecting the √π/4 angle as the base angle leads to a much more consistent “intended design”

than the 37:79 angle would, and the 1:2 angle can be excluded because it is out of range. The

consistency of the intended design selected to date can be shown by the small subset of

measurement systems and values as follows:

Sacred Cubit

Royal Cubit

1

1.1

2

3

π

365.25 Number of days in Julian Year

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It is concluded that the √π/4 angle is the intended passage angle.

How the Passages and Pathways are Measured

Journeying through the passages of the Pyramid, as with any building, would be via the floors, at

least where they exist. In places, the floor disappears and then reappears again, such as when

transitioning from the Entrance Passage to the Ascending Passage through the Granite plugs and

the Great Step. In other places, there are steps such as in the Subterranean Chamber, Queenʹs

Chamber Passage, and the 2nd Low Passage. In general, the Pathways predicted by the M Circle

Arc Length theory mimic how the surveyors have used reference points to measure the passages,

and this is shown in the Figure below.

The theory assumes that all sloping Passages were intended to be at the same angle, 26° 18ʹ 10ʺ,

up or down. The dimensions shown above are those predicted by the Theory rounded to the

nearest inch.

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Theoretically, the Entrance Passage ends 1111ʺ from the junction of its floor with the north face

of the Pyramid to the intersection of the downward extension of the floor of the Ascending

Passage with the Entrance Passage.

The floor of the Descending Passage starts at the floor at the south end of the Entrance Passage,

and its length is 3037ʺ down to its end.

The Subterranean Passages and Chamber is called the Subterranean System for convenience. It

starts at the end of the Descending Passage, and its length is 1318ʺ to the end of the 2nd

Subterranean Passage. Differences in floor levels of the Passages and Chamber are ignored.

The Ascending Passage commences at the south end of the Entrance Passage and slopes up

1547ʺ to its end, which is its intersection with the north wall of the Grand Gallery. The first 60ʺ

of this passage is free air to permit the Descending Passage to continue down to the Subterranean

System. There is then 15ʺ of stone floor until the beginning of the Granite Plug. The line

continues between the bottom of the Granite Plugs and the Ascending Passage floor for 179ʺ

where it re-emerges as the floor of the Ascending Passage.

The Queenʹs Chamber System commences at the end of the Ascending Passage, which is the

North Wall of the Grand Gallery. It proceeds 1627ʺ horizontally to the center of the Queenʹs

Chamber, directly under the apex of its sloping roof. The downward step in the passage is

ignored.

The Grand Gallery also starts at its north wall and slopes upwards 1885ʺ until it reaches its south

wall. As the floor line transitions through the vertical axis of the Pyramid, it passes through the

Great Step for about 69ʺ. It terminates at the south of the Grand Gallery about 6ʺ below the floor

line of the Kingʹs System.

The Kingʹs System starts at the south wall of the Grand Gallery, and its length is 475ʺ to the

south wall of the Kingʹs Chamber. A small upward step of about 0.75ʺ at the south end of the 2nd

Low Passage, which is also the north wall of the Kingʹs Chamber, is ignored.

With one exception, the calculation of the Pathways follows how Petrie, and others, have defined

them. The exception is at the start of the Kingʹs System, which has a short slope and is described

later.

Reconstructing the Designed Measures of the Passage System of the Great Pyramid

We have two resources that will enable us to mathematically reconstruct the intended design of

the Passages of the Pyramid. The first is the M Circle Table that was created in Paper 4, which is

repeated below with the theoretical value of the circumference of the M Circle, 25793.03ʺ. The

second resource is a reasonably accurate understanding of the impacts of subsidence on the

Passage System.

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M Circle Table – Showing All Known Arc Lengths

AL# Exterior

Angle

Degs ° Arc

Length

Measured

Value Bʺ

Diff

Possible Relationship

1 RCS Base

Angle

51.854 3715.2 Discuss

below

- Level of Subterranean Chamber Roof

2 RCS Apex

Angle

76.292 5466.1 5465.6 0.6 The path length of Entrance,

Descending, Subterranean Passages,

and Subterranean Chamber

3 FCS Base

Angle

58.298 4176.9 4175.3 1.6 The path length of Ascending

Passage, Grand Gallery and Kingʹs

Chamber (See Figures below)

4 FCS Apex

Angle

63.405 4542.8 4541.0 1.7 Path Length of Entrance and

Ascending Passages and Grand

Gallery

5 DCS Base

Angle

41.997 3009.0 Discuss

below

Queenʹs Chamber Floor Level

6 DCS Apex

Angle

96.006 6878.5 When divided by four this gives a

path length from the North wall of the

Grand Gallery to the midpoint in the

Queenʹs Chamber (See Figures

below)

/4 1719.6 1719.5 0.2

7 BCS Base

Corner

Angle

90.000 6448.5 Discuss

below

- Confirms that the Pyramid Base

Angle = tan-1

(4/π)

8 Passage

Angle

26.303 1884.5 1883.6 0.9 Length of Grand Gallery

9 Queenʹs

Chamber

Roof

Angle

30.459 2182.3 Discuss

Below

- Aids in defining the sloping length of

Petrieʹs virtual pathway in the 1st Low

Passage. Discussed later in this paper.

Reconstructing the King’s Chamber and Grand Gallery

The Table above shows that the FCS Base Angle defines the combined length of the Ascending

Passage (AP), Grand Gallery (GG), and Kingʹs Chamber and its Passages (KC). The Table also

shows that the FCS Apex Angle defines the combined lengths of the Entrance Passage (EP),

Ascending Passage (AP), and Grand Gallery (GG). These two pathways overlap each other

because both of them include the Ascending Passage and Grand Gallery. The combined pathway,

EP, AP, GG, and KC, is referred to as the Primary Pathway. The abbreviations defined above

represent the lengths of the sloping or horizontal passages; the following two equations can be

stated:

AP + GG + KC = 4176.9ʺ

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EP + AP + GG = 4542.8ʺ

Also, arc length eight defines the length of the GG = 1884.5ʺ

These three equations have four unknowns, AP, GG, KC, and EP, so it is not possible to solve

them unless we know one more of the variables. The length of the Kingʹ Chamber system can be

calculated using inductive metrology, and once we have that result, all four of these variables

will be known mathematically. However, other dimensions need to be theorized before it is

possible to define the Kingʹs Chamber and its passages mathematically.

It should be noted that although arc length two defines the EP, DP, and the Subterranean System

Passages and Chambers, it adds two more unknowns for just one more equation, so it is no help

in solving the primary path. Arc length six can only be used to reconstruct the Queen’s Chamber

and its passages, so it is no help either.

Reconstruction of the King’s Chamber and its Passages

The Kingʹs Chamber has been subject to subsidence, as shown by the exaggerated Davidson’s

Figure below. The subsiding GG generated the force that pulverized and displaced the Great

Step, which pushed the passage floors toward the south wall of the Kingʹs Chamber, where the

Pyramid core resisted it. One effect of this was to push the south end of the GG and the north end

of the Kingʹs passages upward. You can see this effect by placing your fingertips together in

front of you and pushing them together. Your fingertips will rise, which simulates the relative

movement of the Great Step and the KC passages.

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The floors of the Passages, Antechamber, and Kingʹs Chamber have buckled. Petrieʹs

measurement (P47) shows the point N, in the above figure, has been pushed into the Kingʹs

Chamber by 0.20ʺ. All of the nine beams in the Kingʹs Chamber have cracked toward their south

side. Everything is askew, and regarding the Kingʹs Chamber Petrie says (P51) ʺOn every side,

the joints of the stones have separated, and the whole chamber is shaken larger. ʺ

Evaluation of Theoretical Length of Grand Gallery and Kingʹs Chamber Passages

The following determines the measured and theoretical lengths of the GG and the passages to the

Kingʹs Chamber.

The last three entries in Petrieʹs series of measurements (P48), from the south wall of the GG to

the Kingʹs Chamber, are as follows with my estimate in red. This estimate is necessary because a

gap is indicated between the end of the passage floor at 269.04ʺ and the start of the raised floor

in the King’s Chamber, 269.17ʺ.

The base of Kingʹs Chamber wall (Assumed end of the passage) 268.90ʺ

End of passage floor 269.04ʺ

Estimated length of passage floor from the south wall of GG 269.10ʺ

(Half distance between the end of passage floor and the start of Kingʹs floor)

Raised floor, Kingʹs Chamber 269.17ʺ

From the doorway of the 1st Low Passage to the doorway to the Kingʹs Chamber, the theory

predicts a total length as follows:

LP1 length (365.25/7) 52.17ʺ

Antechamber Length (365.25/π) 116.26ʺ

LP2 Length (365.25 - 365.25/(2π) – 365.25/√π) 101.05ʺ

Total Theoretical Length 269.49ʺ

The theoretical equations shown above are derived later. The theory predicts that the length of

this passage has been reduced from 269.49ʺ to 269.10ʺ, which is 0.39ʺ. Please note that the

passage floor has also been displaced from 268.90ʺ to 269.10ʺ, which is 0.20ʺ into the Kingʹs

Chamber.

The combined measured length of the GG and the Kingʹs Chamber passages is 1883.60ʺ plus

269.10ʺ, which is 2152.70ʺ. Theoretically, this length is arc length eight, 1884.52ʺ, for the GG,

plus 269.49ʺ for the passages, which is 2154.01ʺ. The difference between the theoretical and

measured lengths is, therefore, 2154.01ʺ minus 2152.70ʺ or 1.31ʺ.

Is it possible that the Grand Gallery and Kingʹs Chamber passages could have been intended to

be 1.31ʺ longer than measured by Petrie? The answer is yes because there are three ways in

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which the length of the Great Step and the passages to the King’s Chamber could have been

shortened.

All three ways would be caused by the need for the passages to increase their horizontal length

as they subsided during an earthquake. As the passage ends subside, a force will be generated

along their length because the core masonry of the Pyramid will resist the increase in horizontal

length. The force will be transmitted through whatever surfaces are in touch with each other, i.e.,

floors, walls, or ceilings. Let us look at what happens to a stone floor being subjected to the

forces caused by subsidence.

Before the subsidence, the floor slabs lie on their bed, square with each other. The length shown

is similar to the length of the Kingʹs Chamber passages and Antechamber.

As the force is applied, the stones will buckle with respect to each other. If the slabs could reach

the state shown below, they will have increased their apparent total length of the floor, and

intuitively this cannot happen as the force being applied is trying to shorten the length. The

buckling is exaggerated so that the effect can be seen more clearly.

As the stones rise or fall with respect to each other, the force is initially transmitted via just the

edges in contact. Edges are just lines, which have zero areas, and the force per unit area becomes

infinite. Stone is not very compressible, and the edges in contact will pulverize, crumble or

break, which will continue until the area in contact between the stones increases to such an

extent that the force per unit area reduces to a point where pulverizing ceases. The end effect will

be something similar to that shown below. Small gaps may be left between the stones at the

apexes of the buckling because the force was not sufficient to pulverize the entire face. If they

could have lifted the stones, surveyors would have found this in the Kingʹs Chamber passages. It

can be seen below that, as expected, the length of the passage will decrease because of the loss of

material due to pulverizing.

240.00

246.77

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The second way in which the length of the GG and KC passages can have been reduced is by

ʺdisplacementʺ of the subfloor of the 1st and 2nd Low Passages and the Antechamber to the south

between the walls of these passages. From Davidsonʹs KC figure above, it can be seen that,

except for the first two stones in these passages, which are limestone rather than granite, the

others sit between the walls of the passage and Antechamber. Because of the force due to

subsidence, the stone of the Great Step appears to have been displaced south relative to the walls

that sit on it. Petrie provides evidence of this occurring because the entire passage floor

terminates inside the Kingʹs Chamber by 0.20ʺ rather than in line with the North wall of the

chamber.

It can be seen from the second entry in the above table that the top surface of the passage floor

extends into the Kingʹs Chamber 0.14ʺ followed by a gap of 0.13ʺ, 269.04ʺ-269.17ʺ, before the

actual Kingʹs Chamber floor begins. Given the example of buckling above about half, this gap

should be assigned to the passage floor and the other half to the Kingʹs Chamber floor, which is

about 0.06ʺ each. Therefore, the observed displacement through the passages would be 0.14ʺ

plus 0.06ʺ = 0.20ʺ.

The third way in which the GG and KC passages could have been shortened is because the Great

Step has ʺfracturedʺ. The Edgars provided the Figure on the left below, and Smyth provided the

center and right Figures.

237.26

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The left-hand Figure shows the general dilapidation of the Great Step in 1910. There is a loss of

material in the middle of the Step, affecting both the horizontal and vertical faces. The center

Figure clearly shows that the Great Step has fractured along two lines on its top surface. The

larger fracture extends down the vertical face of the Great Step, as shown by the right-hand

Figure. The loss of material in the center of the Great Step, in the form of a V, would result from

this fracture. It is easy to see that the sides of the Great Step could have been pushed to the sides

of the GG along these fracture lines, and its length would have been shortened. The Great Step

has since been restructured, as shown by the photo below.

Some structures are designed to withstand seismic activity, as evidenced by the photo below,

which shows how the 1989 Loma Prieta earthquake affected the San Francisco Bay Bridge. A

portion of the upper roadway collapsed onto the lower roadway.

If a structure fails, it will be at its weakest point, and the media later explained that this was, in

fact, a feature of the design. Damage to a small part of the bridge, which in this case, was

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designed to occur over a primary support, avoided loss of the complete structure. The Great Step

was designed to be a similar weak link so that when it fractured during subsidence, the Grand

Gallery was protected from major damage. Engineers repaired the Bay Bridge quite quickly.

The question now is, what would the original length of the Great Step have been? The measured

length of the step from north to south is 61.0ʺ on the east and 61.5ʺ on the west, P46. Given the

evidence from Smyth and the Edgars concerning fracturing, this length should be longer.

Considering that the Great Step was designed as the weak link in the upper Pyramid passages,

there should be a simple way to determine what its original length was. Now one royal cubit

(RC) is 20.607ʺ, and three RCs are therefore 61.82ʺ. It can be seen from a plan view of the Great

Step, Grand Gallery and First Low Passage, as shown below, that all the dimensions can be

reduced to whole numbers of royal cubits (RCs), in the range 1 to 5, if the north to south length

of the Great Step is taken to be 3 RCs. I am sure you remember the (3,4,5) Pythagorean triangle

from school, and this leads to the realization that the top surface of the step is intended to be

nothing more than two of these triangles joined together as a rectangle. It could not be any

simpler to understand how to determine the correct length of the Great Step. Any length other

than 3 RC would have been crass. Like the Bay Bridge, reconstruction happens quickly.

Someone once asked me why there are not any dimensions with whole numbers in the Pyramid.

As you can see, there are and right where they are needed! The GG is the place for dimensions

that are either whole numbers of royal cubits or simple fractions thereof. Many of Petrieʹs

measurements in the GG, P46, are in the range 20ʺ to 21ʺ or multiples thereof. Petrie and others

conclude that these were originally intended to be whole number multiples of royal cubits of

20.6ʺwhich further supports the case for the step length being 3 RC. Other whole and fractional

RCs are as follows:

Width of GG floor 2 RC

Width of each Ramp 1 RC

Height of Ramps perpendicular to floor of GG 1 RC

Width of GG above ramps 4 RC

Width of 7 corbels on all walls 1 RC

Width of GG ceiling 2 RC

1 RC

2 RC

1 RC

3 RC

Floor of Grand Gallery

Ramp

Great

Step

Part of Ramp

4 RC5 RC2 RC1st Low Passage

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Horizontal Length of Long Holes in Ramps 1 RC

Sloping Length Short Holes in Ramps 1 RC

Horizontal north to south length of Great Step 3 RC

Diagonal horizontal length of Great Step 5 RC

And simple fractions of Royal Cubits

HorizontalWidth of each Corbel 1/7 RC (4 Digits)

Height of Groove above 3rd

Corbel Above Edge 1/4 RC (7 Digits)

Perpendicular Height of Groove 2/7 RC (8 Digits)

Depth of Groove 1/28 RC (1 Digit)

Width of Ramp Holes (East to West) 2/7 RC (8 Digits)

Continuing, we see that the Grand Galleryʹs south end traverses what Petrie calls a ʺvirtual floorʺ

which passes through the Great Step. Having determined that the horizontal length of the Great

Step should be 3 RC and also that the passage angle is the √π/4 angle, then the theoretical length

of the virtual sloping floor can be calculated to be = 68.961ʺ. Subtracting this length from the

theoretical length of the GG, 1884.518ʺ, = 1815.557ʺ, which is the theoretical length of the GG,

from its north wall to the Great Step. This is just 0.057ʺ longer than Petrieʹs measurement of

1815.5ʺ, and this is the key to understanding that fracturing of the Great Step is by far the major

contributor to the shortening of the GG.

The length of the Antechamber and its passages have been reduced by pulverization 0.39ʺ or

more compared to the theoretical length. We can see from the above analysis that the concept of

pulverization and the fracturing of the GG can account for this reduction and a little more as

follows:

Corrected length of GG – measured length of GG = 1884.52 - 1883.6 = 0.92ʺ. All of this could

have been absorbed within the Grand Gallery, or part could have been pushed into the passages

and increased the amount of pulverization by a commensurate amount.

Also, the King’s Chamber passages have been displaced 0.2ʺ into the Chamber.

So it can be seen how the Grand Gallery and King’s Chamber passages were shortened by 1.31ʺ.

Based on the above analysis, we can conclude that the Grand Gallery has been shortened

by subsidence in length at the Great Step and that its theoretical length is 1884.518ʺ based

on arc length eight. It can also be concluded that the intended horizontal length of the

Great Step is 3 RC or 61.821ʺ, which is 68.961ʺ on the slope. The length from the north

wall to the Great Step is therefore intended to be 1815.557ʺ.

Passage and chamber dimensions herein are expressed up to 3 decimal places. The length of a

year is slightly less than an inch, and three decimal places represent about one-third of a day.

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However, the data is based upon spreadsheet calculations, which are always maintained to 15

digits. Once in a while, a rounding difference of ± 1 digit may be seen, but this is of no

consequence.

At this stage, it is possible to fit the EP, AP, and GG into the RCS cross-section of the Pyramid.

Theoretically, arc length four equals the combined length of these three passages along their

slope. Since they are all sloping at the same angle, we can calculate the horizontal length that

they would occupy, which lets us see how they mathematically fit into the Pyramid.

Fitting the GG, AP, and EP into the Pyramid

The northern half of the RCS is the orange triangle and comprises the Vertical Axis, Pavement,

and North Face. The Great Step is shown in light blue, and its vertical face is aligned with the

vertical axis of the Pyramid, as concluded by Petrie, which places the south end of the GG three

royal cubits, 61.821ʺ, south of the vertical axis. The north end of the Grand Gallery is, therefore,

1815.557ʺ, on the slope, from the Vertical Axis along the dark blue line. This dimension is not

shown in the Figure for clarity. The theoretical lengths of the floors of the AP, shown in red, and

4542.768

GG

APEP

APE

4010.613

667.138

61.821

4072.435

Great Step

Vertical

Axis North

Face

Pavement

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of the EP, shown as a solid green line, are not known at this time. Their collective theoretical

length, however, is arc length four minus the GG length, which is 2658.250ʺ. To draw arc length

four, 4542.768ʺ the EP component is drawn along the same line as the GG and AP floors as the

dashed green line.

As an aside, this introduces a new concept which is called the Ascending Passage Extension

(APE), which is the combination of the green and magenta dashed lines. It will be seen later that

this is an intentional, but virtual, feature necessary to generate the chronological timeline. The

APE is constructed by extending the north end of the AP northward and downward until it

intersects the extension of the north face also northward and downward. This intersect occurs

about 942ʺ below ground.

The combined horizontal length of the GG, AP, and EP, which includes the 3 RC south of the

vertical axis, is shown above as 4542.768ʺ × cos P = 4072.435ʺ where P is the passage angle.

Subtracting the 3 RCs to the south of the Vertical Axis results in a horizontal length 4010.613ʺ to

the north of the Vertical Axis. This theoretical value is 0.613ʺ longer than Petrieʹs measured,

P64, and is due to the theory predicting longer lengths for the AP and EP. This difference will be

revisited later. The level above the pavement, which corresponds to this width from the Vertical

Axis to the North Face, is 667.138ʺ. Petrie’s average measurement of the bottom of the 19th

course was 666.95ʺ, based on the NE and SW corners of the Pyramid, which is 0.188ʺ below the

theoretical level, which is considered an acceptable difference. However, it is about 1.1ʺ lower

than Petrieʹs measurement of the floor level of the EP at the bottom of the 19th

course at the

Entrance, which is 668.2ʺ.

Although not shown, the horizontal distance from the floor at the entrance to the north base of

the Pyramid is 523.969ʺ compared with Petrieʹs measured value of 524.1ʺ. The difference is

0.131ʺ, which is acceptable due to measurement tolerances.

The above evaluation fixes the theoretical location of the junction of the floor of the entrance

passage at 667.138ʺ above the pavement and 4010.613ʺ north of the vertical axis of the Pyramid

which assumes a Pyramid with a base length of 9069.165ʺ and base angle tan-1

(4/π). The south

end of the GG lies on a line parallel to the vertical axis and exactly three royal cubits, 61.821ʺ,

south of it. The exact level of the south end of the GG has not been calculated at this point in the

evaluation and depends upon the theoretical lengths of the EP and the AP. There is a 0.613ʺ

difference between the measured and theoretical combined lengths of the EP and AP, which

needs to be evaluated.

It is necessary to return to the subject of the King’s Chamber Dimensions later.

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Determination of the Subterranean Chamber and Passage Dimensions

Arc length one provides the equation for the dimensions of the Subterranean Chamber and

passages as follows:

EP + DP + SS = 5466.120ʺ

Which is one equation with three unknowns so it cannot be solved until two of these theoretical

unknowns become known. The EP length will become known when the upper passage equations

are solved. The DP is a mostly featureless passage, the theoretical length of which cannot be

directly discovered. Fortunately, the theoretical dimensions of the Subterranean System (SS) can

be determined by employing Inductive Metrology.

Determination of the Theoretical Dimensions of

the Subterranean System (SS)

First off, we need to recognize and correct for the

fact that in this area, some of Petrieʹs measurements

are vague.

On his admission, Petrie did not take his usual care

when measuring the DP:

ʺFor the total length of the entrance passage, down

to the subterranean rock-cut part, only a rough

measurement by the 140-inch poles was made,

owing to the encumbered condition of it. The poles

were laid on the rubbish over the floor, and where

any great difference of position was required, the

ends were plumbed one over the other, and the

result is probably only true within two or three

inches.ʺ

Petrie did not find an important feature that was discovered by the Edgars. He established a level

datum at the bottom end of the DP from which his vertical measurements were made. He

mentions and provides measurements of three margins around the north Subterranean Passage.

These three margins can be seen in the photo above, which was taken by the Edgars. The left

margin is 4.5ʺ, the right margin is 3.2ʺ, and the top margin is 5.4ʺ to 6.0ʺ. The Edgars also found

a bottom margin, the height of which was 1.25ʺ and of which Petrie made no mention. Looking

at the Edgar photos below of the east side of this margin, it can be seen, from their pencil marks,

that the margin rapidly reduces in height to nothing as it crosses the junction of the DP and the

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north Subterranean Passage. If Petrie had cleaned the loose dust and stones out of the way, he

would have found the floor of both passages was at the same level.

The Edgars measure in the nSP is 37.25ʺ above the lower extremity of the inclined floor of the

DP, whereas Petrieʹs measure is 38.3ʺ slightly closer to the junction of the two passages. The

following figure shows these two measures.

Petrieʹs measurements are shown in gray in the figure above. It can be seen that, given the lower

30ʺ of the DP remain at the perpendicular height of 48.50ʺ, the only way Petrieʹs dimensions for

the top margin, 5.4ʺ to 6ʺ, and north Subterranean Passage height, 38.3ʺ, can hold is that his

reference level must have been the lower extremity of the DP which is shown by the red cross

which is the junction of the floors of the nMP and DP. The height of the nMP, as given by the

5.78

48.50

38.30

20.00

10.00

37.25

30.00

DP

nSP

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Edgars, is 37.25ʺ, which is 1.05ʺ less than Petrieʹs height. The Edgars express doubt over this

measurement in their 1913 edition of ʺGreat Pyramid Passages Vol IIʺ in the footnote at the

bottom of pages 201-202, so precedence is given to Petrieʹs measures in this area.

However, precedence is given to the Edgars measurements of the length of the lower portion of

the DP and the length of the nMP. The Edgars cleared out both passages to improve the accuracy

of their measurements. Also, they used 600ʺ tape measures, similar to Petrieʹs, to minimize the

issues of slippage, etc. that plagued Smyth. They also measured both the east and west walls of

the DP and nMP and determined that the east and west junctions varied from 1ʺ to 2ʺ along the

north to south axis. The west side of the DP is longer than its east side, whereas the east side of

the nMP is shorter than the west side. Ibid pages 8-9.

More Inductive Metrology

In ʺInductive Metrologyʺ, Petrie shows how to derive the base unit of measure used in ancient

monuments by analyzing individual measurements to find integer or fractional relationships with

another measurement. Any designer will likely use simple ratios in his design and having found

them the underlying basic unit of measurement can be determined. As an example, consider the

following from pages 13 and 14 retyped from the referenced book.

ʺAs an example we may take the porch of Chideock Church, Dorsetshire; the measures are –

Inches. Differences. Units.

Radius of Sundial 6.9 6.9 1

Side of door molding to side of porch 40.7 32.8 6

Side of doorway to side of porch 47.7 7.0 7

Door wide 54.6 6.9 8

Stone arching blocks over door, wide 74.6 20.0 11

Whole width of porch 150.0 75.4 22

Here we see the same amount (6.9 or 7.0 inches) repeated three times over in the differences;

and taking this hint, by adopting this as a base, we find that the measures agree pretty closely to

it; the multiples being given in the last column.

Of course the small internal errors will appear far larger in proportion on the difference than on

the whole; and sometimes a double error (i.e., a + error in one measure and a – error in the

next) will thus be thrown on a difference of only a few inches, and appear monstrous, when it

would be quite unobtrusive on the whole measures of several feet in length. This method, though

quick and easily applied without a slide rule, is not to be worked with such care as the others;

and if in any case the differences are not seen to group well, it may be discarded as unsuitable in

that instance.ʺ

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As an aside, we can see that Petrie was conversant with slide rules, and his use of one may

account for some of the errors previously noted in his calculations. Imagine using a slide rule by

candlelight in a tomb, which was Petrieʹs lodging near the Pyramid.

Notice here also that Petrie recognizes that an error can sometimes appear ʺmonstrousʺ when

compared with a measurement of a few inches, but when compared with a larger measurement it

is ʺunobtrusiveʺ.

Reconstruction of the Subterranean System

In inductive metrology, Petrie looked for a ʺhintʺ on which to base his reconstruction. In the

church porch, it was the repetition of the 7ʺ dimension. The two photos below, of the Recess,

reveal the ʺhintʺ applicable to the Subterranean System.

It can be seen that it is not possible to determine the exact height of the Recess because the roof

is quite unfinished. The same is true of the top of the walls, so only their bases have reasonably

well-defined dimensions. Therefore the only things to be learned from the Recess are the

dimensions of its floor. If we take the intended dimensions of the Recess to be about 72ʺ to 73ʺ

square, then the diagonals are about 102.5ʺ. We can round this to 5 royal cubits, which are

103.04ʺ. The diagonals of the floors form a cross, and, as it was shown earlier that the Pyramid

was intended to be interpreted in this day and age, we can, in the modern idiom, take this cross to

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say that ʺRoyal Cubits are not used hereʺ. The mystery of why the roof of the Recess is

unfinished is because it draws attention to the floor so that its measurement system can be found.

According to Dieter Lelgemann (Recovery of the Ancient System of Foot/Cubit/Stadion –

Length Units):

ʺThe development of those units is closely connected to the Egyptian method to mark off a

square such as the ground-plan of the pyramid of Cheops; (Petrie 1934) found a description

of this method in an old papyrus.

Based on the Remen (R = (20/28) NC) a new length unit was defined, the

Old royal cubit = √2 R = 523,75 mm.ʺ

It can be concluded that the theoretical length and width of the Recess is 5 Remens.

Therefore by Inductive Metrology the measurement system of the Subterranean System is

based on Remens.

According to Lelgemann, one Remen is equal to 1 Royal Cubit divided by √2, which equals

14.571ʺ (370.11 mm). The Remen is related to the Nippur Cubit (NC) by the ratio of 20/28. An

NC is, therefore, 20.399ʺ (518.16 mm) based on the length of the RC defined herein (20.607ʺ or

523.42 mm). For interest, please note that the Megalithic Yard, which is not used in the Pyramid,

is defined as 2.72 ft, which is 32.64ʺ, which is 1.600 NC.

The following Figure shows a top view of the three components of the North Subterranean

Passages. Note that there are three sets of data per dimension. The number in black is Petrieʹs

measurement in inches; the number in red is the theoretical dimension calculated from the

equation in blue.

218

218.571

R*15

73

72.857

R*5

55

54.643

R*15/4

346

346.070

R*19*5/4

72

72.857

R*5

102.5

103.04

Ys

2nSP 1nSP

Recess

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The three parts of the passage are the long first passage, 218ʺ (P37), a Recess, which it appears,

was intended to be 72ʺ-73ʺ square, and a second short passage measured as 55ʺ long. Petrie only

measured the width of the Recess as 71ʺ, but the left-hand photo above shows that the Edgars

could place a 72ʺ measuring rod across it.

Using Petrieʹs Inductive Metrology, the ratio of the lengths of the three components of the 1st

Subterranean Passage can be determined as follows:

Length ̋ Dividing by the length Multiply by 3 to bring

of second passage (55ʺ) close to a whole number

First Passage 218 3.96 11.89

Recess 73 1.327 (≈ 4/3) 3.98

Second Passage 55 1.0 3.00

If these values are then rounded to the nearest whole number, we see their lengths are in the

ratios 12 to 4 to 3, which when added equal 19. So the first passage is 12 units, the Recess is four

units, the second passage is three units, and the total length of the North Subterranean Passage is

19 units. The total length of the two passages and the Recess is 346ʺ, which divided by 19 equals

the length of the unit. 18.21ʺ. Since a Remen equals 14.571ʺ, then the unit equals 18.21/14.571

Remens, which equals 5/4 R (Remen). So the total theoretical length of the north

Subterranean Chamber Passage is defined by the equation 19 * R * 5/4 = 346.070ʺ.

The remaining theoretical horizontal dimensions of the SS can then be computed by dividing

Petrieʹs measurements by 19 Remens. These are shown in the Table below.

The Table below summarizes the lengths of the three components discussed above and also adds

the remaining theoretical horizontal dimensions. Petrie does not provide tolerances for his

measurements in the Subterranean System, so they have been computed as shown below. The

resolution of Petrieʹs measurements (P37) in the Subterranean System appears to be 1ʺ so we can

set a resolution of ±0.5ʺ for that and add a tolerance of ±0.2ʺ for measurement accuracy. The

total tolerance range is, therefore, ±0.7ʺ. The Edgars appear to measure with a resolution of

0.25ʺ, so we can set a resolution of 0.125ʺ plus the 0.2ʺ measurement tolerance, also assigned to

Petrie, for a total tolerance of 0.325ʺ.

The remainder of the horizontal dimensions of the Subterranean System can be derived, as

shown in the Table below.

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Theoretical Lengths of Subterranean System ʺ (SS)

R = 1 Remen Equation Theoretical

Dimension

Lower

Limit

Nominal

(Petrie)

Upper

Limit

Length 1nSP R*15 218.571 217.3 218 218.7

Length & Depth

Recess

R*5 72.857 71.8 72.5 73.2

Length 2nSP R*15/4 54.643 54.3 55 55.7

Total Length nSP R*(15+5+15/4) = R*19*5/4 346.070 345.3 346 346.70

Length SC R*19*3/2/TanA 326.163 325.3 326 326.7

Length sSP R*19*7/3 645.997 645.3 646 646.7

Total Length SS R*19*(7/3+3/2/TanA+5/4) 1318.230 1317.3 1318 1318.7

Depth SC R*19*2 553.712 553.1 553.8 554.5

Green numbers in the third column indicate that the theoretical value fits within the tolerance

range of the measurements, while red shows it does not. As can be seen, all theoretical values fit

within their range, so the theory holds good.

Also, please note that the theoretical length of the Subterranean Chamber, from north to south, is

expressed by the equation R*19*3/2/TanA, where A is the base angle of the Pyramid. As

mentioned during the passage offset evaluation, some of the equations pertaining to the length or

height of a chamber contain a trig function. This equation is the second example. It is horizontal,

so it points in the direction of the south Subterranean Passage, which comes to a dead end. Dead

ends are not good, so symbolically, TanA can be taken as indicating ʺbadnessʺ.

There is a second possibility for the length of the SC, which is defined by the equation

R*19*37/40*TanA = 326.066ʺ, which is also within the tolerance range. It, too, contains a trig

function. The equation for the easterly passage offset at the entrance to the Pyramid is

365.25/TanA, where the division by TanA is matched by the equation in the table above. The

alternate equation multiplies by Tan A, which is not as consistent as the Entrance offset equation.

The equation in the table has, therefore, been selected based on consistency.

The values tabulated above were used to draw the Figure below. Also, the Descending Passage

and the north and south Subterranean Passage theoretical widths are shown. The black, red, and

blue colors are the same for all similar drawings in this study. The final part of the Subterranean

System pathway is shown in green. Although the depth of the Subterranean Passages and

Chamber have been provided, as Remen based equations, they do not appear to be relevant to the

theory.

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From the above figure, we can see that when the SS theoretical length is subtracted from arc

length two, the result is 4147.890ʺ, which is the theoretical length of the EP and DP combined.

The theoretical vertical height of these two passages is this value times SinP, where P is the

passage angle, which equals 1837.986ʺ. Subtracting this height from the height of the floor at the

entrance, 667.138ʺ, shows the depth of the bottom of the DP below the pavement, = -1170.848ʺ

which Petrie measures as 1181.0ʺ below the pavement. There is approximately a 10ʺ difference

between the theoretical and measured values, which has been caused by subsidence.

The theoretical horizontal distance of the south wall of the Recess from the vertical axis of the

Pyramid is significant, as discussed in Paper 7. It equals the base length of the Pyramid/2

(9069.165/2ʺ) – horizontal distance from the north base to the entrance (523.969ʺ) – horizontal

length of the EP and DP (4147.890ʺ*cosP = 3718.440ʺ) – length of 1nSP (218.571ʺ) – length of

Recess (72.857ʺ) = 0.746ʺ.

Petrie has the north wall of the Subterranean Chamber 40ʺ south of the vertical axis of the

Pyramid, and so the south wall of the Recess is 15ʺ north of the vertical axis (P37), which puts it

nearly 14ʺ closer to the north base. As observed above, the end of the passage has sunk 10ʺ and

moved northward by 14ʺ, but there was no appreciable change in passage length. This factor

must be taken into consideration if we are to define an accurate theoretical model of the Pyramid

and its passages. Fortunately, the model puts the passages and chambers back in their correct

locations to ensure that symbolisms are interpreted correctly. The south edge of the Recess does

have significant symbolism, as will be seen later.

41.8

41.214

2*RC

32.5

32.786

R*9/4

218

218.571

R*15

73

72.857

R*5

55

54.643

R*15/4

346

346.070

R*19*5/4

72

72.857

R*5

102.5

103.035

RC*5

326

326.163

R*19*3/2/TanA

646

645.997

R*19*7/3

553.8

553.712

R*19*2

28.8

29.143

R*2

Subterranean Chamber

sSP 2nSP 1nSP

Recess

Shaft not Part of

Original Design

646

1318.230

R*19*(7/3+3/2/TanA+5/4)

DP

Well

Shaft

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There is a second level that is associated with the SS which is derived from arc length one which

is 3715.198ʺ which can be converted to a level within the Pyramid by the technique shown by

the light blue lines in the Figure below:

The technique is to draw a horizontal line, light blue above, from the center of the Pyramid Base

to the left with its length equal to the value being converted. That length is 3715.198ʺ. A vertical

line is drawn, downward in this case, until it intersects the line representing the mirror image of

the Casing of the Pyramid. A horizontal line is then drawn back toward the center of the

Pyramid, and its interaction with any Pyramid feature is observed. In this case, the lower

horizontal blue line passes either at or just above the reconstructed level of the roof of the

Subterranean Chamber and about the level of the floor of the lower mouth of the Well Shaft. The

symbolism of this line is discussed later. This theoretical line indicates a level of 1043.273ʺ

below the Pyramid Base. This depth can also be calculated by the equation:

Level = -(Height of Pyramid)*(1- Arc length one/(Pyramid Base Length/2))

Subtracting the level of the south end of the Descending Passage, -1170.848ʺ, from the level of

the above blue line, -1043.273ʺ results in a difference of 127.575ʺ which is shown in the

following figure, which also shows the remaining dimensions of the Subterranean System which

are the vertical heights.

3715

Base Angle

51° 51' 14.31"

Entrance

Passage

1111

Descending Passage

3038

Ascending Passage

1547

Grand Gallery

1885

Passage Angle

26° 18' 9.73"

Queen’s System

1627

King’s System

475

North Face

Pyramid Base

Granite Plugs

1043

Mirror Image

of Casing

Subterranean System

1318

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Petrieʹs range for the height of the roof of the Subterranean Chamber above the end of the

Descending Passage floor is 124.3ʺ to 127.6ʺ. The upper end of this range, 127.6ʺ, occurs at the

SE corner of the chamber above the doorway to the south Subterranean Passage. The difference

between the 127.6ʺ level and the level of the arc length one blue line, 127.575ʺ, is 0.025ʺ.

It is concluded that arc length one, converted to a level, represents the intended roof level

of the Subterranean Chamber and the floor level of the lower Well Shaft opening.

The conclusion explains the mystery of why the Subterranean Chamber roof is flat, but the floor

is not. The roof confirms the location of and the need for arc length one. In turn, arc length one

provides the level of the SC roof. This scenario is quite the opposite of the Recess, which has a

flat floor and undefined roof to draw attention to the fact that the measurement system of the

Subterranean System is in Remens.

The passage widths from the above drawings are summarized in the following Table:

Theoretical Lengths of Subterranean System ʺ (SS)

R = 1 Remen Equation Theoretical

Dimension

Lower

Limit

Nominal

(Petrie)

Upper

Limit

nSP Depth R*9/4 32.786 31.6 32.5 33.3

nSP Height R*11/4 40.071 38.9

sSP Depth R*9/4 32.786 26.7 28.7 30.1

sSP Height R*9/4 32.786 26.0 32.5 32.5

RC = 1 Royal Cubit

DP Depth RC*2 41.214 Discussed Later

DP Height Numerically computed 47.666 Discussed Later

The depths, which are the east to west horizontal dimensions and the heights shown above for

the SS, are not crucial to the lengths of the passage floors, and so are not crucial to the theory.

However, an estimate has been made where possible.

The widths and heights selected in the table above are chosen to be one Remen multiplied by a

real fraction with a denominator of 4. This combination is a common factor in the three

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components of the nSP. The nominator of the fractional part is chosen so that the final dimension

is larger than Petrieʹs upper limit in the table above so that if the Subterranean System were ever

to be finished, this would allow for removal of stone to accomplish that finishing. However, this

is not quite true for the nSP passage width, which violates these criteria by 0.5ʺ but only at the

north door of the SC.

Note that both Petrie and the Edgars measured the height of the nSP as 36ʺ. However, to allow

for material to be removed for finishing and squaring up and leveling the existing passage, the

floor has been chosen as Petrieʹs reference level, and since the top of the passage is 38.9ʺ above

that, then the next fractional increment is R*11/4 which is 40.071ʺ.

Final Considerations Relating to the Subterranean System

The entrance to the EP occupies the entire 19th

course, according to Petrie (P32). The length of

the North SP is R*19*5/4, the length of the SC is R*19*3/2/TanA, the length of the South SP is

R*19*7/3, and the depth of the SC is R*19*2. Therefore all major dimensions of the SS are a

function of 19 Remens. The association with the number 19 at both ends of the EP and DP

combination is not coincidental as it verifies the fact that this study is authorized by Isaiah 19:19.

It is not possible to complete the theoretical reconstruction of the descending passages at this

time; however, we can carry out a ʺsanityʺ check. Arc length two (AL2) is defined to equal the

length of the EP, DP, and SS as follows:

EP + DP + SS = 5466.120

The theoretical length of the SS in the table above is 1318.230ʺ

The measurement of the DP that the Edgars made:

2 Roof, west side, with steel tape 3035.6ʺ

3 Floor, west side, with steel tape 3037.7ʺ

4 Floor, west side, with steel tape 3037.3ʺ

6 Floor, east side, with steel tape 3035.5ʺ (+ 1.4ʺ adjustment)

7 Floor, east side, with steel tape 3035.7ʺ (+1.4ʺ adjustment)

The data above shows that at the floor, the east side of the DP is, on average, 1.9ʺ shorter along

the slope than the west side. The Edgars also show that the west side of the north Subterranean

Passage, which at 347.00ʺ is 1.75ʺ longer than the east side, 345.75ʺ, horizontally. The Edgars

and Petrie both report that the interface between the nSP and the DP is slightly skewed; the

[To clarify Morton Edgar states in the 1913 edition of ʺGreat Pyramid Passagesʺ footnote page 8:

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ʺBy subsequent (in March 1912) square measuring I found that the top-west to bottom-east

diagonal of the flat end of the Descending Passage, is nearly at right-angles to the incline, and

that the other diagona1 from the top-east to bottom-west corners, is also nearly at right-angles

to the incline; but the first mentioned diagonal is about 1.25 inches further out or more to the

north than the other. Consequently, any measuring along the west roof-line, and east floor-line,

is bound to give a less result than along the east roof-line, and west floor-line.ʺ

which is shown in the figure below.

The figure shows the Edgars measurements for the west and east sides of the Descending

Passage and what they call the Small Horizontal Passage leading to the Pit, herein identified as

north Subterranean Passage or nSP. The dimensions can be seen to be partially compensating

when comparing west to east. The DP, 3037.5 average from AP to the nSP, plus the length of the

nSP converted to a slope, 345.75/cosP, equals 3423.181 compared with 3035.6, 347.0/cosP and

3422.676 respectively on the east side. It seems likely that the excess length here, on both sides,

was left to permit the end of the DP to be squared up, but it looks like this was not done. In this

West

East

3037.500

3035.600

347.000

345.750

Great Pyramid Passages

Vol II 1913 pp 7-10

3423.181

3422.676

Descending Passage

Descending Passage

nSP

nSP

AP

AP

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case, the excess length should be assigned to the DP rather than the nSP. The average of the total

sloping length of the west and east sides is 3423.181 plus 3422.676 divided by 2 equals

3422.929. If we then subtract the sloping length of the nSP, the result will be an estimate of the

average length of the DP. To ensure sufficient material remains after any squaring is done, the

theoretical sloping length of the nSP should be used. This is 346.070 horizontally and 386.038

along the slope. The result is in an estimated measured DP length of 3036.89ʺ.

When the SS and DP values are substituted into the equation above the estimated theoretical

length of the EP is 1110.999ʺ, which is within 0.4ʺ of Petrieʹs measurement of 1110.64ʺ. It can

be concluded that the sanity check is valid.

Note that Petrie measured the east margin as 4.5ʺ wide and the west as 3.2ʺ. Hence the 1st

Subterranean Passage appears to be offset to the west, relative to the axis of the bottom of the

Descending Passage, by approximately 0.65ʺ. This offset is probably intentional rather than

being an allowance that would have been dressed down later.

Petrie measured the level of the roof at the south end of the 1st Subterranean Passage as being

38.9ʺ above his reference, which is 0.6ʺ higher than the north end at 38.3ʺ. The question is

whether the passages and chambers are now starting to slope, due to subsidence, in the opposite

direction, i.e., upwards from north to south? Why? Because Petrie measures the level of the north

wall of the Subterranean Chamber as 124.3ʺ above his reference and the south wall from 125.4ʺ

to 127.6ʺ above his reference.

The following photo, taken by the Edgars, of the entire length of the east wall of the chamber,

indicates that this difference is more likely to have been caused by the fact that the whole

chamber is unfinished. The thin red line, which has been added to the photo, indicates the

fluctuations of the roofline across the chamber appear to match Petrieʹs stated 3ʺ variation.

The roof of the SE corner, to the right, can be seen to be higher by a few inches compared to the

NE corner, which is to the left. The roof is also higher just to the left of the center of the image.

These deviations are caused by the unfinished and rough nature of the roof rather than the fact

that it slopes upward to the right. It can be concluded that if the Subterranean Chamber is not

sloping upward, then there is no reason for the north Subterranean Passage to slope upward

either.

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Note that the Edgars took the black and white photos over 100 years ago. They are very detailed,

well composed, well exposed, well-focused, and very useful for evaluating the Subterranean

System. They used magnesium wire to illuminate the scene, which, being extremely bright,

probably accounts for why the people in the image are not looking at the camera. The only

problem is some Moiré-like patterns, which were probably caused by a printing or scanning

process.

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The above photo shows the rounded nature of the original rock cutting. It also shows Stanley

looking over a thin ridge of rock at a small recess in the west wall of the Chamber. The recess is

at the roof of the chamber, and the Edgars report this as follows:

ʺ300 At the north end of the west wall at the roof, we disclosed in our clearing operations a

small and roughly squared recess-Plate X. In appearance it is as if a small westward passage

had been contemplated but had been abandoned shortly after work on it had commenced, as it is

only from six to eighteen inches deep, the inner end being very irregular. Adjoining the wall to

the north of this recess, there is a peculiar upright ridge of rock reaching from the floor to within

13 inches of the roof. It runs parallel with and about three feet from the north wall of the

chamber ; the long narrow space between the two in not unlike a horse-stall. After getting

Stanley to creep into this space and look out over the ridge in the direction of the recess, we

photographed the corner-Plate LIV.ʺ

The following Figure is cut from the Edgars Plate X, which shows, on the right-hand side, the

location of this small recess and ʺstallʺ on the west wall. If there are some other unknown

chambers or passages within the Pyramid, then this recess would be the place to dig to find them

since, from the above photo, its quality appears consistent with the rest of the original Chamber

and would, therefore, be part of the original design.

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The photo below is taken more recently and shows the Subterranean Chamber cleared of all

debris so that the form of the unfinished western half of the Chamber can be seen. To Petrie, it is

an example of quarrying, but then he did not see it as shown in the photo. Some think this looks

like a cave or a tomb, but to this author, it looks like two cities divided against each other.

However, it may have other symbolism.

A Modern Day View of the Subterranean Chamber

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The End of the South Subterranean Passage and the End of Time (For Some)

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Reconstruction of the Dimensions of the Queenʹs Chamber and Its Passage

The Queenʹs Chamber, Niche and Entrance

The section that determined the measurement standards of the Pyramid showed that there is a

Year Line (YL), Year Square (Ys), and Year Circle (Yc), which are all based on the Bʺ, and

which apply to the interior of the Pyramid. However, in the case of the Subterranean System, we

saw there was a modifier in that the Recess that told us not to use 5 RCs (= 1 Ys) but rather to

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use Remens, which is RC/√2. It appears that there is nothing in the path of the Queenʹs Chamber,

which modifies the measurement system, so we should expect only to use Bʺ, Ys, or RCs. It

should be pointed out that YC and YL are transitional measurement systems between Bʺ and Ys

and are not used for passage lengths except where previously noted in the Kingʹs Chamber

system. Ys is the Year Square.

For defining dimensional equations, the Subterranean System equations were of the general

form:

(Measurement System)*(integer)*(simple fraction)*(Trig function) = Dimension

Where the measurement system = Remens, RC or Bʺ, etc.

Integer = e.g. 1 or 19 in the SS

Simple fraction = e.g. 5/4 or 7/3

Trig function = e.g. TanA in SS

For consistency, it is most reasonable to expect that the designer used the same form for the

theoretical equations for the Queenʹs Chamber and its passage.

This Queenʹs pathway traverses south along the center of the Queenʹs Chamber Passage, which

comprises a low part and a high part. At the entrance to the Queenʹs Chamber, the pathway is 1

RC from the east wall. It continues on 5RC along the same path and directly under the apex turns

right and continues another 4.5 RC to the center of the Chamber. See the green line in the Figure

below.

In Paper 4, which defined the internal measurement system of the Pyramid, it was determined

that the theoretical length and depth of the Queenʹs Chamber were ten and eleven RC,

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respectively. These dimensions translate to 20.607*10ʺ = 206.070ʺ and 20.607*11ʺ = 226.677ʺ

respectively. The equation above can be rewritten as:

(Arc length six)/4 = 6878.539/4 = 1719.635 – 5RC – 4.5RC - QCPLow - QCPHigh

QCPLow + QCPHigh = 1719.635 – 103.035 – 92.732 = 1523.868ʺ

Petrie measures these two lengths combined as 1523.9ʺ (P40), which is an exceptionally close

match. For the record, Petrie considers the cubit to be 20.62ʺ, so ten cubits are 206.2ʺ to him.

Since the average length of the Queenʹs Chamber is 205.85ʺ, which Petrie accepts was intended

to be ten cubits in P41, then a difference of 206.2ʺ – 205.85ʺ is acceptable to Petrie and is ±0.35ʺ,

and which is, therefore, the tolerance selected for the Queenʹs Chamber system.

A spreadsheet search was conducted to help find equations that would accurately fit the above

dimensions and also the heights of the walls and the apex of the roof of the Queen’s Chamber.

The Table below provides a set of consistent equations for three of the six parameters relative to

the Chamber since they contain the same fraction n/77, and a fourth equation contains n/11.

Pyramid Feature Equation Theoretical

Dimension "

Measured Dimension " Difference "

Length QP Low Ys*77/3*tanP 1307.182 1307.00 -0.182

Length QP High Residual 216.686 216.90 0.214

Length QC Ys*2 206.070 205.85 -0.220

Depth QC Ys*11/5 226.677 226.47 -0.207

Height QC Walls Ys*77/43 184.505 184.47 -0.035

Height QC Apex Ys*77/16*tanP 245.097 245.10 0.003

All the equations fit Petrieʹs measured values, P40, within the range of ±0.35ʺ. A consistent set

of equations for the theoretical length of 6 dimensions relating to the Queenʹs Chamber and its

passage have been derived, and their impact is shown in the following two Figures. In the

dimension values, black is the measurement, and red is the theoretical value based on the

equation in blue. Residual indicates that the length of the high portion of the Queen’s Chamber

Passage was assigned the remaining value after the 9.5 cubits in the Chamber, and the theoretical

length of the low portion of the passage was subtracted from arc length six.

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* Note that by the theory the ratio of QCP Low to QC Apex is 16:3 exactly

During the derivation of these theoretical lengths, we came across three dimensions that were of

a similar form to those found in the Subterranean System. From the Table above the form is:

Ys*77/a*b

Where a is an integer and b is 1 or TanP (Passage Angle).

The value 77 is pertinent to the Queenʹs Chamber because it contains seven surfaces, i.e., one

floor, four walls, and two roof surfaces. The depth of the Queenʹs Chamber is 11 RC. So the

multiplication of these two numbers is 77. Is it coincidental that the height of the Pyramid is

based on the number 17.6, which is 8/5*11?

Niche

The dimensions of the Niche in the Queenʹs Chamber are shown in the following Figure

113.339

5.5 RC

92.732

4.5 RC

226.47

226.677

11 RC

103.035

5 RC

205.85

206.070

10 RC

QCP

High

216.9

216.686

Residual

QCP Low

1307

1307.182

Ys*77/3*TanP

Step

41.214

2 RC

North Wall of

Grand Gallery

20.607

1 RC

Niche

Queen’s Passage

1523.9

1523.868

AL-6 – 9.5 RC

Queen’s Chamber

Apex of Queen’s Chamber Roof

North Wall of

Grand Gallery245.1

245.097

Ys*77/16*TanP

184.4

184.505

Ys*77/43

Queen’s Chamber

High

67.21

66.958

Low

46.5

47.042

Step

19.7

19.916

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None of these dimensions affect the length of the pathway through the Queenʹs Chamber, so they

are as Petrie measured them as shown in black. However, Petrie states that he believes that the

horizontal dimensions of the Niche are based on cubits. He also states that the depth, which here

is into the page, is RC*2, which makes the base 2*3 RC2 = 6 RC

2 in this area.

Note that the Niche is divided into five narrowing sub-niches the higher it rises. Petrie measured

its height as 183.8ʺ, which was the top of the seventh course and which is the top of the walls in

other places. It has been shown that the height of the seventh course, as Petrie measured it, at the

Niche is 183.8ʺ, but it was probably intended to be the same level as the top of the walls which

Petrie measured as 184.47ʺ. The theoretical height of the walls, Ys*77/43 or 184.5ʺ, where Ys =

5RC.

The numbers 5 and 6 are critical in understanding the symbolism of the Niche and the Queenʹs

Chamber.

Airshafts

The following Figure shows dimensions related to the airshafts in the Queenʹs Chamber.

The green line represents the Queenʹs Chamber pathway, which enters the chamber at the bottom

right corner from the high part of the QCP and continues on the same straight line till it turns

Queen’s Chamber

67.14

31.79

28.23

28.94

28.03

20.3 20.61 RC*1

30.43 30.91 RC*1.5

41.83 41.21 RC*2

52.74 51.52 RC*2.5

61.74 61.82 RC*3

183.8

184.505

Ys*77/43

25.11

113.386

2.88m113.339

114.173

2.90m

77.165

1.96m

75.984

1.93m

8.268

21cm

8.268

21cm

113.339226.677

20.607

Queen’s Chamber

South

Airshaft

North

Airshaft

Step

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right just under the apex. It then proceeds to the exact center of the QC. This point is 113.39ʺ

from the east wall of the chamber, which is at the bottom of the Figure. It can be seen that the

bore of the north and south Air Shafts, which are both 21 cm in width and also in height, do not

commence until after the chamber center in the westerly direction. The southern airshaft

commences 113.386ʺ (2.88 m), which is 0.047ʺ west of the chamber center. The north Air Shaft

commences after 114.173ʺ (2.90 m), which is 0.834ʺ west of the chamber center. The dimensions

here are from Rudolph Gantenbrink, who, in 1993, sent robots as far as he could up the Air

Shafts in the Kingʹs and Queenʹs Chambers. Please visit cheops.org.

Also, the above Figure shows that the horizontal lengths of the Air Shafts, as they leave the

Queenʹs Chamber, are different lengths with the south shaft being 77.165ʺ (1.96 m) and the north

being 75.984ʺ (1.93 m). Petrie also measured these lengths, P(44), as 80ʺ for the south, and 76ʺ

for the north shaft. The following image shows how Gantenbrink and Petrie arrived at different

lengths. These two images are cropped from Gantenbrinkʹs cyber drawings and show the

southern horizontal shaft in the left image and the northern horizontal shaft in the right image.

The white dotted line represents the wall blocks of the Queenʹs Chamber.

The Air-shafts are those parts of the above image which contain a 2 meter scale within them.

Solid stone blocks are ʺhatchedʺ. It can be seen from the two scales that Gantenbrink measured

just the floor of the wall block of the Air-shaft because the meter scales indicate what looks like

1.96 m to the south and 1.93 m to the north as reported above. It seems like Petrie slid a

measuring rod in until it was stopped by the upward sloping floor of the shafts. An approximate

measurement was made from the above images, and Petrie would have measured 80ʺ to the

south, as he reported, but 77.4ʺ to the north, which is 1.4ʺ longer than reported. Gantenbrinkʹs

results will be used as being more appropriate and accurate in the following discussion.

The discussion centers around Petrieʹs interpretation that the center of the Queenʹs Chamber, i.e.,

the apex of its roof, is aligned with the Great Step in the Grand Gallery and both are aligned to

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the Vertical Axis of the Pyramid in the north to south direction. The arc-length theory herein

does not predict this result. The reason is as follows:

The theoretical sloping length of the Grand Gallery from its north wall to the Great Step has been

stated as 1815.557ʺ, which is a very close match to Petrieʹs measure of 1815.5ʺ, (P45). The

horizontal length is determined by multiplying the sloping length by cosP, where P is the passage

angle. The horizontal length of the GG is, therefore, 1627.584ʺ. The horizontal length of the

QCP, from the north wall of the GG to the apex of the QC, is shown as 1626.903 in the fourth

figure above. Theoretically, the apex of the QC roof is therefore 1627.584 – 1626.903 = 0.681ʺ

north of the Great Step. Normally this would require the theory to be reworked or abandoned, but

the theory is reporting precisely what has been measured. Petrie, P46, reports the Great Step is

located 0.4 south of the Pyramid center, and the apex of the Queenʹs Chamber roof is 0.3 north of

the Pyramid center. The difference is, therefore, 0.7ʺ toward the north, according to Petrie, which

virtually matches the 0.68ʺ to the north required by the theory.

The difference in length of the horizontal parts of the Queenʹs Chamber Air Shafts comes into

play as a second witness to this difference. The horizontal lengths from above are south 77.165ʺ

and north 75.984ʺ, which is a difference of 1.181ʺ. Based on the data we can calculate that:-

The north face of Queenʹs Chamber wall = distance of apex from center + distance of south face

of QC wall + width of wall = 0.681 + 103.035 +75.984 = 179.700ʺ north of Pyramid center

So similarly, the south face of the south wall of the Queenʹs Chamber can be shown to be located

179.520ʺ south of Pyramid center.

In other words, the Pyramid was designed so that even with the Queenʹs Chamber being

deliberately offset 0.68ʺ to the north, the start of the sloping portion of its Air Shafts are

symmetrical about the center of the Pyramid.

It can be concluded that the apex of the Queenʹs Chamber roof does not lie on the plane,

which marks the north to south transition of the Pyramid. It does not align with the face of

the Great Step.

The ability of the theory to reveal this error, which is nearly 140 years old, is quite remarkable

and adds to its robustness.

The Theoretical Floor Level of the Queenʹs Chamber

The arc-length table shows that when arc length five (DCS Base Angle) is subtracted from arc

length six (DCS Apex Angle), the resulting value can be used to determine the floor level of the

Queenʹs Chamber. This value = 3869.551ʺ which needs to be converted to a level above the

pavement in this case, by the same method as used to obtain the roof level of the Subterranean

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Chamber. The following equation was used, which has been modified to use it for a level above

the pavement:

Theoretical Queenʹs Chamber Floor Level

= (Height of Pyramid)*(1- (Arc length six - Arc length five)/(Pyramid Base Length/2))

= 5773.610 * (1-3869.551/(9069.165/2)

= 846.744ʺ above the pavement

The Theoretical Position of the Queenʹs Chamber within the Pyramid

The theoretical position of the Queenʹs Chamber is therefore fixed within the Pyramid. The

following three facts are theorized:

Height of QC Floor = 846.744ʺ above the pavement

= 834.4ʺ P40

= 12.344ʺ difference

Height of QC Roof Apex = 245.097ʺ above QC floor

= 1091.841ʺ above the pavement

QC Apex North of Pyramid Vertical Axis = 0.681ʺ

Also, it is theorized that the north wall of the Grand Gallery is 1523.868ʺ north of the north wall

of the Queenʹs Chamber, which in turn is 103.035ʺ north of the apex of the QC roof.

Adding the three previous values shows that the north wall of the Grand Gallery is 1627.584ʺ

north of the Pyramid vertical axis.

At this stage, the relationship of the level of the north floor of the Grand Gallery has not been

theorized.

The 12.344ʺ difference between Petrieʹs QC floor level and the theorized level is mostly

attributable to subsidence, but a small part of the difference is due to theoretical versus

measurement differences, which will be defined in a later section.

Small Dimensions

It should also be noted that the south wall and passage of the Recess is theoretically 0.746ʺ north

of the centerline of the Pyramid, just 0.065ʺ further north than the apex of the Queenʹs Chamber.

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Additionally, the theoretical height of the small step into the Kingʹs Chamber from the second

Low Passage is 0.756ʺ, just 0.01ʺ different in dimension from the south wall of the recess from

the center of the Pyramid. These three dimensions are symbolically linked, as will be shown in a

later section.

Theoretical Angle of Queenʹs Chamber Gable Roof

One final detail that can be gleaned from the Queenʹs Chamber that is required to complete the

Pyramid passage reconstruction accurately is the theoretical angle of its gable roof. Please see

the Figure below:

The height of the apex from the floor of the QC = Ys*77/16*TanP

Where TanP = tangent of the passage angle = √ (π/(16-π))

The height of the top of the walls from the floor of the QC = Ys*77/43

Horizontal length from wall to apex = 1 Ys

Theoretical QC Apex Angle Z° = Tan-1

((Apex height –Wall height)/(Half chamber width))

= Tan-1

((Ys*77/16*TanP-Ys*77/43))/Ys

= Tan-1

(77*(√(π/(16-π))/16– 1/43))

= 30.459° = 30° 27ʹ 31ʺ

Petrie measures this angle in 4 places, (P42), and the average measure is 30.438°, which is

0.021° or 1.3ʹ less than the theoretical angle. This difference is considered acceptable since it is

within the ±2ʹ tolerance allocated to the Pyramid base angle, and also it is close to the center of

the range measured by Petrie which is 30° 10ʹ to 30° 48ʹ, i.e., 30° 29ʹ.

245.1

245.097

77*Ys/16*TanP

184.4

184.505

77*Ys/43

Queen’s Chamber

205.85

103.035

5 RC (1Ys)

Z = 30.459°

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Since this angle, Z is defined by an equation that contains π it is included in the arc-length Table,

which is the final angle required to complete the arc-length theory.

Reconstruction of the Dimensions of the Kingʹs Chamber and Its Passages

The diagram below shows the top view of the Kingʹs System.

Part of the Grand Gallery is shown in this diagram outlined in dark blue to the right-hand side.

The east and west ramps are 1 RC wide, and the floor between them is 2 RC wide. The Great

Step occupies the final portion of the Grand Gallery. It is 3 RC long, 4RC wide, and its diagonal

is 5 RC, as previously described. The plane of the face of the Great Step, is aligned with the

Vertical Axis of the Pyramid, which is shown by the fiduciary near the top right corner.

The Kingʹs System begins at the left-hand side of the Great Step, which is also its south side. The

first subsystem is the Antechamber and its passages outline in red.

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Plan View Kingʹs Chamber Great Pyramid

The Antechamber system comprises the 1st Low Passage (1LP), the Antechamber, and the 2

nd

Low Passage (2LP). The dimensions are shown with the black, red, and blue notation to show

the measured, theoretical, and equation of the dimensions. The theoretical length of the 1LP is

deemed to be Y/7 where Y is 365.25, and the reason for this is made clear in the discussion of

the front view below which makes the length of the 1LP 52.1786ʺ long compared with Petrieʹs

measurement of 52.02ʺ, P47, and which is a difference of 0.1586ʺ. Given the effects of

subsidence in this area, this is an acceptable difference, and the theoretical length of the

1LP can be concluded to be Y/7.

Passage Offset

287

286.8667

365.25/TanA

1 RC

2 RC

1 RC

3 RC

1st

Low

Passage

King’s

Chamber

Antechamber

Part of Floor of

Grand Gallery

2nd Low

Passage

Midpoint of King’s

Chamber

Part of Ramp

Original Position of Coffer

N

W

N

Vertical Axis

of Pyramid

116.30

116.263

Yc

100.8

101.048

Y–2*Ys–Yc/2

52.02

52.18

Y/7

Great

Step

Part of Ramp

4 RC

103.15

103.035

5 RC

412.11

412.1405

20 RC

206.29

206.070

10 RC

205.97

206.070

10 RC

412.40

412.1405

20 RC

5 RC

Pathway for

Arc Length 3

65

65.256

19/6*RC

48

48.083

14/6*RC

L

E

A

F

365.24

365.25

Y 58.15

58.131

Yc/2

58.15

58.131

Yc/2

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In the earlier section discussing the internal measurement standard of the Pyramid, the

theoretical lengths of the Antechamber and 2nd

Low Passage were concluded to be Yc =

116.2627ʺ, and Y – 2*Ys – Yc/2 = 101.0484ʺ respectively.

Following the Antechamber system is the Kingʹs Chamber. In the earlier section discussing the

internal measurement standard of the Pyramid, the theoretical width of the Kingʹs

Chamber was concluded to be 2*Ys = 206.0702ʺ.

The final dimension that can be obtained from the above diagram is the east to west length of the

Kingʹs Chamber. Petrie, P52, shows the mean lengths to be 412.4ʺ and 412.11ʺ at the north and

south walls, respectively. The mean of these is 412.255ʺ. The theoretical length can, therefore, be

deemed to be 4*Ys = 412.14ʺ, which differs by 0.114ʺ from Petrieʹs measure.

The following are theoretical lengths of the Kingʹs System from the face of the Great Step

to the south wall of the Kingʹs Chamber and points in between.

Length Equation of

Theoretical

Length

Theoretical

Length ʺ

Petrie

Length ʺ

Difference

ʺ

Face of Great Step to north

wall 1LP

3RC 61.82 61.32 0.50

Length of 1LP Y/7 = 365.25/7 52.18 52.02 0.16

North wall 1LP to KC

north wall

1LP + AC + 2LP 269.49 269.10 0.39

North wall 1LP to mid KC 1LP + AC + 2LP +

Ys

372.52 372.27 0.25

North wall 1LP to KC

south wall

1LP + AC + 2LP +

2*Ys

475.56 475.41 0.15

North wall 1LP to mid KC

to mid Coffer

(Arc length three ?)

1LP + AC + 2LP +

23/5*Ys

743.45 743.29 0.16

The distance from the north wall of the 1LP to the north wall of the Kingʹs Chamber was

discussed in the subsidence section earlier. It was shown that the 0.39ʺ difference between the

theoretical and measured lengths could be attributed to pulverization and displacement. As the

measurement transitions across the Kingʹs Chamber, it can be seen that at the middle of the

chamber the difference has reduced by 0.14ʺ to 0.25ʺ and at the south wall, it is reduced to 0.15ʺ.

The reduction in these differences can be mainly attributed to pulverization. The average

measured length of the Kingʹs Chamber from east to west is the average of 412.11ʹ and 412.40ʺ,

which is 412.26ʺ, from P52. The theoretical length is 4*Ys = 412.14ʺ, which is 0.12ʺ less than

measured. Though no direct contribution can be seen, this difference is minimal and does not

sway the selection of the theoretical length.

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On the above basis, it is acceptable to conclude that all the theoretical lengths in the

pathway of the Kingʹs Chamber, which is shown in green in the above diagram, have been

identified correctly.

The following diagram is the front view of the Kingʹs System and part of the south end of the

Grand Gallery.

Front View, from the East, of the Kingʹs System and part of the South End of the Grand Gallery

At the right is the south end of the Grand Gallery. Seven corbels can be seen above the south end

of the Great Step. The horizontal dimension of all seven corbels is one cubit, so their lengths are

RC/7. According to Petrie, the vertical heights of these corbels are challenging to measure, but

the following analysis is submitted for consideration.r4

Petrie measures the vertical heights of the top 5 corbels as 33.6ʺ, 33.7ʺ, 33ʺ, 34ʺ, and 33.8ʺ but

does not provide the height of the bottom two corbels. Smyth provides the vertical heights of the

235.2

235.212

Yc/TanP

43.6

43.590

Great

Step

1st Low

Passage

Antechamber

149.35

149.481

9/7*Yc

Yc

235.212

Yc/TanP

2nd Low

Passage

7 Corbels

33.6017

Yc/(7*TanP)

43.7

43.590

Num

Vertical Axis

of Pyramid

0.8

0.756

Num

King’s Chamber

42.835

Num

West

Wall

52.02

52.18

Y/7

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bottom three corbels as 35.9ʺ, 34.8ʺ, and 34.6ʺ. The one common corbel between them is the last

in each series, which is 33.8ʺ for Petrie and 34.6ʺ for Smyth. Adding the top five corbels from

Petrie and the bottom two from Smyth results in a total height of 238.8ʺ. Adding just Petrieʹs five

measures results in 168.1ʺ which when averaged, by dividing by 5, and then multiplied by seven

results in 235.34ʺ. Petrieʹs height of the five courses of the Kingʹs Chamber walls, which is

Petrieʹs ʺpride and joyʺ measurement equals 235.2ʺ ±0.06ʺ, Plate 13. It is likely, using Inductive

Metrology, that the height of the seven corbels is intended to be the same as the height of the

walls of the Kingʹs Chamber.

In all of the Pyramid, this is Petrieʹs most accurately determined measure hence the reason I call

it his ʺpride and joyʺ. He states that it is accurate to ±0.06ʺ in 235.2ʺ, which is an uncertainty of

only 0.026%. A theoretical value of Yc/TanP is assigned to it, which is equal to 235.212ʺ, which

is within the range 235.2 ± 0.06ʺ. Note also that this applies to a vertical measurement at the end

of a pathway that contains a trig function in the equation. Since it is upward and related to the

passage angle, it symbolizes “goodness”.

From where the green pathway enters the face of the Great Step to where it emerges again in the

1LP Petrie refers to as a virtual path, which is so because the path cannot be seen, but it can be

defined mathematically. Similarly, extra ʺvirtualʺ corbels can be created at the north end of the

1LP, going southward, starting at the point where the real corbels terminate. Adding two such

corbels adds clarity to the interpretation of the historical path length through the Pyramid

passages. So at this point, it is considered that the designer wants us to recognize seven real and

two virtual corbels for a total of nine.

On the other side of the 1LP, we have the Antechamber, and the height of its roof, above the

floor, is 149.35ʺ, P48. Considering the seven real and nine total corbels then the theoretical

height of the AC can be set to Yc*9/7 = 149.48ʺ which is 0.13ʺ longer than Petrieʹs measurement

of 149.35ʺ, P48, and this difference is negligible.

So to the north of the 1LP, we have seven corbels arranged vertically, and the measurement

standard is Royal Cubits (Ys/5) for the Great Step, the GG, and the length of the seven corbels.

To the south of the 1LP, there is a vertical height of 9/7 Year Circles (Yc), and the AC is one Yc

in length. So using Inductive Metrology, we can see that the length of the 1LP should be defined

in 1/7th of something related to the Year Line, Circle, or Square. The Year Line (Y)/7 fits the bill

as it is 52.1786ʺ, which is just 0.16ʺ longer than Petrieʹs measurement of 52.02ʺ, P47, and this

difference is negligible.

In the above figure, the remaining dimensions define the height of the two Low Passages in

conjunction with the height of the small step at the entrance to the Kingʹs Chamber. The

theoretical dimensions of these features are discussed next.

Length of Pathways

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At this point we can determine a theoretical length for the EP and AP by substituting the Kingʹs

System length that we calculated in the last row of the table above which is 743.45ʺ in the

equation for Arc length 3 as follows:

AP + GG +KS = 4176.87ʺ

Rearranging results in

AP = 4176.87 – GG – KS = 4176.87 - 1884.52 – 743.45 = 1548.9ʺ (From P38 = 1546.8ʺ)

We can also determine the length of the Entrance Passage from Arc length four

EP + AP + GG = 4542.8ʺ

Rearranging results in

EP = 4542.8 –GG – AP = 4542.8 – 1548.9 – 1884.52 = 1109.35ʺ (From P38 = 1110.64ʺ)

This tells us that if we ignore the length of the sloping part of the Kingʹs System pathway, where

the green path enters from the Grand Gallery, then the theory will add 2.1ʺ, (1548.9ʺ - 1546.8ʺ)

to the length of the AP and subtract 1.29ʺ, (1109.35ʺ - 1110.64ʺ) from the length of the Entrance

Passage. Therefore we now need to determine the designerʹs intent regarding the length of the

sloping pathway in the Kingʹs System.

Length of Sloping Part of Kingʹs System Pathway

The figure below shows the sloping portion of the KCS pathway, which is part of the arc length

three pathway, which commences at the north end of the AP and finishes one RC from the west

wall of the Kings Chamber. Here the pathway, shown in green, enters at the bottom right and

follows the floor of the GG until it becomes hidden, or virtual, as it enters the Great Step. It

remains hidden until it exits the floor of the 1st Low Passage. Here it transitions from sloping to

horizontal and continues to the end of the Kingʹs Chamber with one small upward step at the

entrance to the King’s Chamber.

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The green line shows the sloping part of the KCS as it exits the Great Step to where it intersects

the floor of the 1st Low Passage. The red vertical line shows its height.

Petrie, table P47, defines this as 4.2ʺ on the east side, which could be calculated as 5.1ʺ on the

west side, based on levels of the face of the Great Step in the same table. The average is 4.65ʺ.

From Smythʹs Great Step data, we can calculate this average height as about 5.9ʺ, from ʺLife &

Work. Vol 2, p74ʺ. This average was calculated from the data for the horizontal length and

height of the Great Step, as shown below.

Since there is no close consensus between Petrie and Smyth, we need to find another approach,

and one is shown in the diagram below. Petrie does not provide all the data we need, so we have

to turn to Smyth in the Grand Gallery section of his volume 2. Smyth published all the data in

black, or it was mathematically deduced from his data. The one dimension in red shows the

West East

6.0483

36.2000

61.00

5.7472

35.8000

60.80

Great

Step

Great

Step

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calculated height of the sloping part of the KCS path. The dotted lines indicate that the GG has

been shortened to fit the view, and an expanded view of a corbel is provided for clarity.

The sloping pathway based on arc length three is the green line along the floors of the AP, GG,

and 1LP. The AP enters the GG via the north wall of the Lower GG, and it is 53.2ʺ vertical.

Above that, up to the first corbel is a vertical wall of 41.2ʺ. By inductive metrology, this would,

theoretically, be 2 RC since there is a similarly dimensioned area at the south wall above the

door to the 1LP. The combined height of these two features is 94.4ʺ. There is no corbel on the

north wall at this point, but there is a corbel on the sidewall, above the Lower GG, which slopes

upward to the south wall. It is shown as the upper sloping line in red.

Usually, buildings are designed to be symmetrical, and that should apply to the area outlined in

red, which is the lower part of the GG. This area is from the floor of the GG up to the first corbel

on the sidewall, both of which should slope upward at the passage angle and which should,

therefore, be parallel. The north and south walls should both be vertical, and so they too should

be parallel. The red outlined area is, therefore, a parallelogram.

In this case, the combined height of the dimensions shown on the right of the figure should equal

those on the left. Looking at the bottom right corner of the GG, it can be seen that the

parallelogram contains an acute angle. The expanded view of the corbel shows that there should

AP

1LP

N Wall

S Wall

Lower

GG

Great

Step

Ramp

Expanded View

of Corbel

Sloping Part

of KCS

1.5

41.3

43.6

84.994.4

41.2

53.2

94.48.0

1.5

2.9

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be a similar angle at the top left of the red area for symmetry, and there is. It is a virtual area

since it is embedded in a stone block, shown in dark blue outlined by red, and it represents the

edges of the corbel on the south wall. The height of this triangle is 1.5ʺ and should be included in

the total dimensions on the left to maintain the symmetry.

The next area below this is a blank wall of 41.3ʺ, which is 2 RC by inductive metrology, as

mentioned above. Below this are the 1LP, which is 43.6ʺ, and then the sloping portion of the

green pathway in the KCS. If this height is set to 8.0ʺ, as shown in red, then the dimensions on

the left add up to 94.4ʺ, which is the same as on the right.

However, as shown above, the height is measured as 4.65ʺ by Petrie and 5.9ʺ by Smyth. The

difference is 3.35ʺ for Petrie and 2.1ʺ for Smyth compared with the parallelogram approach.

These significant differences show that there is severe distortion around the Great Step, which

we already know since we have seen that it fractured at some point in time. How then do we

determine an approach that will recover the dimensions of the parallelogram of the lower part of

the GG and the sloping part of the KCS?

Theoretical Sloping Length of Kingʹs System

Fortunately, the Pyramid provides data which helps us theorize the sloping length of the Kingʹs

System. Passage lengths have been derived from the M circle arc-lengths, and they have all been

used except arc length nine from which, fortunately, the sloping length can be determined. This

process is quite complex, and it also reveals the theoretical height of the passages, the height of

the small step at the entrance to the Kingʹs Chamber, and the position of the ʺScored Linesʺ in

the Entrance Passage. It also explains why there are only six corbels on the north wall of the GG.

Once we have all these theoretical dimensions, the reconstruction of the Pyramid passages and

chambers will be complete.

To determine the interaction, if any, of arc length nine with the Pyramid features, it is best to

draw a circle with this length as its radius. The circumference of this circle will then define all

the points which are arc length nine distant from its center. Since arc length nine is defined by

the angle of the Queenʹs Chamber gabled roof, we can draw three circles; two centered on the

eaves and one centered at its apex. These are shown by the dotted lines in the following Figure.

The three circles do not interact with many of the Pyramid features. All three cut across all four

Air Shafts, the Descending Passage, and the Subterranean Passage, but there is nothing out of the

ordinary at these intersections. The unique interaction is that the circle, centered on the apex of

the QC roof, passes through the lower entrance to the Well Shaft.

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Based on this unique interaction, a Z Circle centered on the lower mouth of the Well Shaft

should be drawn to see how it interacts with other features. Its center will be somewhere close to

the Well Shaft entrance, and it will at least pass through the QC roof apex, which is shown in the

figure below. So far, the Well Shaft has not been a factor in the mathematics of reconstructing

the passages, but the following Figure shows further interactions between it and the M Circle.

Firstly we can see that the Z Circle, now centered on the lower Well Shaft entrance at X2, Y2,

passes through or close to a few more features. Secondly, as required, the circle passes through

the apex of the QC roof at X1, Y1, which is a known point. Secondly, it also passes close to the

lower end of the GG and the upper end of the Well Shaft, at a to be determined point. Thirdly, at

the bottom of the figure, it intersects the intersections of two pairs of circles at X3, Y3, which is

also an unknown point. If it is possible to fit these three points on the circumference of the Z

Circle, then the theoretical position of the circle and the passages will have been finalized.

What are these two new pairs of circles? Two double circles have been added to the former

figure. The smaller circle of each pair is an M Circle, while the larger is the M Circle divided by

the Time Scale, 0.9932 Inches/Year, which is the M/T Circle. These circle pairs were created

during a search phase of this study using ʺwhat ifʺ scenarios to eke out relationships between

features and what follows is the result of such a search. In this case, the scenario is ʺwhatʺ will be

seen ʺifʺ we pair an M circle with an M/T circle and center them at major points of the Pyramid.

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.

The blue pair is centered at the intersection of the downward extension of the Ascending Passage

(APE) with the downward extension of the line of the north casing of the Pyramid (CE). This

point is X4, Y4, but its exact theoretical position is unknown at this point in the study. The red

pair is centered at the intersection of the north casing of the Pyramid with the Pavement at X5,

Y5, which is a known point.

Starting at the top and following the red circles counter-clockwise, we see that they pass either

side of, or bracket, the lower entrance of the Well Shaft, as shown below. This point is X2, Y2,

which is to be determined at this time. This point is defined by the blue and white fiducial,

which is the center of the dark green Z Circle in the above figure. The center of the fiducial is

where the tips of the two white and two blue quadrants meet, which identifies the referenced

point. This bracketing of the lower Well Shaft entrance can only mean that the Well Shaft is

mathematically part of the Pyramid. We see this is corroborated in the above figure by the blue

pair of M circles bracketing the upper entrance of the Well Shaft at the lower end of the GG.

These two witnesses substantiate that the Well Shaft is an integral and original part of the

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Pyramid. This conclusion is backed up by it being mathematically identified by the centering of

M Circle pairs at significant parts of the Pyramid. The Well Shaft was not added later or as an

inspection or escape shaft. A complete reconstruction of the Great Pyramid should, therefore,

include the Well Shaft, which should lead to the interpretation of its symbolism.

In the above figure, the beige rectangle is the lower entrance of the Well Shaft, and the dotted

blue line is the roof level of the Subterranean Chamber, which will be discussed later. The

sloping green lines represent the Descending Passage, and the red horizontal lines represent the

north Subterranean Passage. The red circles then continue until they intersect the blue pair of

circles at the bottom of the primary figure above and as shown below:

As can be seen, the two pairs of M and M/T circles intersect at four places. The Z Circle is

centered at the blue and white fiducial in the second figure above. This center can be adjusted so

that the Z Circle intersects the top intersection of the top red M Circle and top blue M/T Circle.

However, the other three intersections are too far from the potential center of the Z Circle for a

similar interaction with them. X3, Y3 represents the point of intersection of the red M Circle, the

blue M/T Circle, and the Z Circle, which is centered on the lower entrance of the Well Shaft.

To lock the Z circle in place, we need to define three points on its circumference, so we need to

find a third point which, as shown above, is at the upper entrance of the Well Shaft. The

following figure shows the area of the upper end of the Well Shaft:

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As mentioned above, the blue pair of M Circles bracket the top entrance of the Well Shaft, or

close enough to make the connection, and then continue down until they intersect the red pair of

M circles. The blue pair also continues upwards and brackets the two lower corbels on the north

wall. These are the only two corbels different from all the other corbels. There are four points

shown in the above figure, which could be used as the third point to fix the location of the Z

Circle. These points were chosen because they involve the Z Circle, which provides the final

answers to the theoretical dimensions of the Pyramid passages.

Before evaluating these points to see if any of them are useable to lock the Z Circle, it is

necessary to consider the implications of the above sentence ʺThe blue pair also continues

upwards and brackets the two lower corbels on the north wall. These are the only two corbels

different from all the other corbels. ʺ The red pair of circles bracket the lower end of the Well

Shaft, and the blues circles bracket the upper end, which tells us that when the M and M/T

Circles bracket something it is an indication of something we did not know and a positive

statement is being made. The red circles also bracket the 17th

pair of ramp holes in the GG, but

no significance for this has been found. However, why are the lower two corbels in the GG

bracketed by the blue circles? We know from Petrieʹs measurements that they are different from

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all other corbels and that all other corbels are the same as each other as shown in the figure of the

Grand Gallery below:

The two lowest corbels on the north wall are circled in red to indicate they are different. Their

widths are 1.5 palms and 0 palms, top to bottom. The width of all other corbels, some of which

are circled in green, are theoretically one palm. A palm is one-seventh of a Royal Cubit or about

3ʺ.

Since it is clear that these two corbels are different from all the others, there is no point in

bringing that to our attention a second time except for emphasis. Like the upper and lower ends

of the Well Shaft are brought to our attention, in a way that tells us they are an original part of

the design, then it can be concluded that the two unique corbels were originally something

different. Dimensionally there is an infinite number of ways that they could have been different,

but of these, the one rational way is to assume that originally they were the same as all other

corbels but have been and were intended to be changed by subsidence. So it can be concluded

that the original dimensions of these two corbels were each one palm and that somewhere along

the line, they were changed to the values shown in the figure above. For now, the original lengths

will be used, and the reason for the change in length to what is seen today will be explained later.

For convenience, the second figure above is repeated below: There are four points in the Figure

which could be used as the third point to fix the location of the Z Circle. The first is where the

dark green Z Circle is tangential to the ramp in the Grand Gallery, which is shown by the orange

arrow and the corresponding orange line which is the radius of the Z Circle drawn from its center

at the angle 90 – P degrees, where P is the Passage angle. The second is shown by the blue and

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white fiducial, which is the point where the south side of the upper entrance of the Well Shaft

intersects the top of the west ramp in the Grand Gallery. The Z Circle and blue M/T circles might

intersect at this point. The light blue arrow shows the third at the point where the dark blue M

circle intersects the top of the ramp and the Z Circle. The fourth, shown by the magenta arrow, is

where the dark green Z Circle might intersect the north wall of the Grand Gallery, maybe at the

same height as the top of the ramp.

For reference, the small green boat shapes are the holes along the top of the ramps.

The position of the junction of the APE and CE and also the center of the Z Circle can be

mathematically adjusted, using Excels ʺSolverʺ feature, to solve for all four points identified

above. ʺSolvingʺ requires the points to meet the criteria which identify the points as described

above. A few characteristics of each of the four points is evaluated to determine if any, can be

rejected to leave one clear point, as shown in the Table below:

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1LP Slope

Height ʺ

EP Length ʺ

(1110.64, P38)

AP Length ʺ

(1546.8, P38)

EP + AP

Length ʺ

SWS from

N Wall ʺ

(49.87, P46)

NWS from

N Wall ʺ

(21.83, P46)

Tngt 5.4696 1110.6229 1547.6269 2658.2498 Adjustable Adjustable

M/T 6.6701 1110.9034 1547.3464 2658.2498 49.16 Adjustable

M 7.9210 1111.1957 1547.0541 2658.2498 Adjustable 23.63

GGAP 9.4536 1111.5538 1546.6960 2658.2498 Adjustable Adjustable

The table compares the lengths of the EP, AP, and south and north distances of the Well Shaft

edges from the GG north wall. The equations, derived from the arc lengths, require that the sum

of the length of the EP and AP is always the same. If one increases in length, the other must

decrease by the same amount. The fifth column in the above table shows the sum of the EP and

AP lengths are the same, thereby validating the mathematics.

The tangential method in the second row leads to an AP length that is 0.8ʺ too long compared

with Petrie The method which uses the intersection of the Z Circle with the junction of the GG,

AP, and top of the ramps, in the fifth row leads to an EP that is 0.9ʺ too long compared with

Petrie. The M Circle defining the north edge of the Well Shaft, in the fourth row, leads to a

distance from the north wall of the GG to the north edge of the WS is 1.8ʺ too long. For these

reasons, these three points can be excluded from selection.

Although not perfect, the point where the Z Circle is locked to the south edge of the Well Shaft,

in the third row, is the closest solution to Petrieʹs measurements. Also, it is associated with the

Well Shaft, and since this Z and M Circle analysis revolves around validation of the Well Shaft,

this is why it was assigned the fiducial to pair with the fiducial at the lower entrance. The center

of the Z Circle identifies the lower entrance, and a possible doorway can be drawn with the

fiducial on the right-hand side. No way can be seen to define the width of the Well Shaft

theoretically, and so, arbitrarily, it is set to the distance separating the M and M/T Circles, which

is 28.1ʺ. This fiducial is labeled X6, Y6, and its exact location can now be found at this time.

The following describes the mathematical solution to lock the Z Circle in position. It turns out

that trying to define equations for this is extremely difficult or even impossible as there are four

unknown points that lead to the need to solve fourth-order equations. However, it is possible to

use the Excel ʺSolverʺ feature to solve the problem numerically, and the following is a copy of

the Excel worksheet that was used for this purpose. The solutions for the unknown coordinates

described above are the red values shown in the third column.

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Radius M/T 4133.190616

Radius M 4105.088327

Radius Z 2182.271128

Apex QC X1 0.680931 Initial Initial 0.000000

Y1 1091.840707 Conditions 1 Conditions 2

Center of Z Circle X2 554.071606 500.000000 650.000000 0.000000

at lower Wellshaft Y2 -1019.098903 -1000.000000 -1050.000000

Intersect M and M/T Circles X3 1640.721710 1500.000000 1800.000000 0.000000

and Z circle from X2 Y3 -2911.583731 -3000.000000 -2800.000000

Center M circle at end APE X4 5274.510272 5000.000000 5500.000000 0.000000

Y4 -942.105284 -1000.000000 -900.000000

Center of M/T Circle X5 4534.582513 4534.582513 4534.582513

Y5 0.000000 0.000000 0.000000

Intersect M/T Circle and top X6 1578.424629 1500.000000 1700.000000 0.000000

of ramp Y6 907.817846 1000.000000 800.000000 0.000000

Pyramid Base Angle tanα 1.273240 0.000000

Angle from APE/CE to X6, Y6 tanB 0.500509 26.588365 0.000000

The description of the points to be ʺsolvedʺ by Excel is shown in the first column. Their

coordinates, e.g., X1, Y1, are shown in the second column. The first three rows of the third

column provide the known radii of the M/T, M, and Z Circles that ʺSolverʺ needs. The remaining

rows of the third column, except for the last two, provide the calculated values of the desired

coordinates. Some points, in blue, are already known, and so there is no need to calculate them,

but 4 points, consisting of one X and Y coordinate each, are unknown and are shown in red. The

fourth and fifth columns provide two sets of initial conditions from which ʺSolverʺ computes the

final coordinates in the third column. Both sets of initial conditions lead to the same result, which

is good since multiple solutions are not discovered by the spreadsheet. The last two rows are

angles that are required by the spreadsheet equations to satisfy the constraints.

The sixth column contains the constraints that ʺSolverʺ must meet to arrive at a solution. The

equations here have been designed to produce a zero result when the constraints of the distances

between certain points are met. The conditional formatting feature of Excel was used to indicate

when these constraints are exactly zero by setting the result to green. All are green, so all the

constraints were met.

The problem was solved using the GRG Nonlinear method. Constraint Precision was set to 1E-

20 in options. Microsoft Office Home and Business 2010, version 14.0.7208.5000 (32-bit), was

used.

The following figure shows the constraints which make it possible to arrive at a solution.

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The first three constraints are that the distances from the center, X2, Y2, of the Z Circle, shown in

dark green, to the apex, X1, Y1, of the QC roof and the intersection of the two pairs of M Circles,

X3, Y3, and the intersection of the top of the ramp in the GG with the left hand side of the Well

Shaft, X6, Y6, should all be equal to the radius of the Z Circle, from arc length nine, which is

2182.271128ʺ.

The next two constraints are that the distance from the intersection of the APE and the Casing

Extension (CE) to the intersection of the top of the ramp with the south side of the Well Shaft,

X6, Y6, and the intersection of the two pairs of M Circles, X3, Y3, should both be equal to the

radius of the M/T Circle which is 4133.190616ʺ.

There are three more constraints. The first of these is that the distance from the north edge of the

Pyramid face, X5, Y5, to the intersection of the two pairs of M Circles should be equal to the

length of the radius of the M Circle which is 4105.088327ʺ. The second is that the angle between

the lower end of the APE, X4, Y4, and the edge of the north face of the casing should be the same

as the base angle of the Pyramid, which equals 4/π. The third and last constraint is that the angle

between the end of the APE, X4, Y4 to the intersection of the top of the ramp in the GG with the

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left hand side of the Well Shaft, X6, Y6, should be sin-1

(1RC/radius M/T Circle) + P which is the

height of the ramp, 1 RC, divided by 4133.190616ʺ + Passage angle P.

The most significant result is the coordinates of the position of the intersection of the APE and

CE. Given the following triangle:

The Sine Rule says

a/sinA = b/sinB = c/sinC

Moreover, we can calculate all the major theoretical dimensions from the arc-length equations

we have previously discovered. The figure below shows a triangle similar to the one above. Point

A above is the junction of the APE and AP shown below. Point B above is the Entrance to the

Pyramid at point X0, Y0. Point C is the junction of the CE with the lower end of the APE and its

coordinates are X4, Y4. It can be seen from the figure below that the angle A = 2*P, the angle B

= 180 – α – P, and the angle C = α – P. The length a above is the distance from X0, Y0 to X4, Y4

shown below. All these coordinates are now known.

b

c

a

A

B

C

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Please note in the above figure that the features shown as dotted lines are what Petrie would call

virtual features. Their magnitude, position, and orientation can be calculated even though they

cannot be seen.

The Entrance Passage is extended upwards from the Entrance until its floor crosses the upward

vertical line from the lower end of the floor of the APE. This extension is the Entrance Passage

extended, EPE.

The result is the creation of two virtual features with the same length. The first is the total of the

length floors of the EP and EPE, which is oriented along the axis of the Descending Passage. The

second is the floor of the APE, which is oriented along the axis of the Ascending Passage. These

features are both used when interpreting the symbolism of the passages.

We are almost in a position to complete the reconstruction of the passages, but we need to

recover two more dimensions. The first is the theoretical height of the passages, and the second

is the height of the small step at the entrance to the Kingʹs Chamber. These both use the concept

of the M Circle pair introduced above. The following figure shows how M Circle pairs are used

to find the theoretical passage height.

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The figure shows just one of eight possible solutions to determine the theoretical height of the

sloping passages. This problem is also solved numerically as follows:

A pair of M Circles are drawn, in blue, centered on the intersection of the APE and the CE. A

second pair of M Circles are drawn in purple, such that the M/T circle passes through the

intersection of the floor of the APE and the CE. The circumference of the M Circle of this

second pair should pass through the intersection of the roof of the APE and the CE and also the

intersection of the roof of the EP and Casing. There are eight possible ways that the M Circles

can be adjusted to pass through the actual and virtual roofs and floors.

The constraints are that the center of the second pair, X8, Y8 will lie on the circumference of the

M/T Circle of the first pair. For this to happen, the distance from the intersection of the floor of

the APE and the CE to the center of the second pair of M Circles will be the radius of the M/T

Circle, which is 4133.1906ʺ. The distance from the intersection of the roof of the APE with the

CE to the center of the second pair of M Circles will be the radius of the M Circle, which is

4105.0883ʺ. Finally, the distance from the intersection of the roof of the Entrance Passage to the

center of the second pair of M Circles will also be the M Circle radius, which is 4105.0883ʺ. The

following Table provides the results of the numerical solution from ʺSolverʺ:

Solver Parameters Passage Height

H = Passage Height 47.665515

Length along casing for EP = H/sinP 48.702266

Length along Casing for APE = H/sin(α - P) 110.511146

x y

EP/Casing Floor 4010.613490 667.138080

EP/Casing Roof 3980.531667 705.439446

APE/Casing Floor 5274.510272 -942.105284

APE/Casing Roof 5206.251093 -855.194997

Center of 1st M Circle Pair 5274.510272 -942.105284

Center of 2nd M Circle Pair X8, Y8 1460.711328 -2535.278935

4105.088327 0.000000

4105.088327 0.000000

4133.190616 0.000000

The cells which solver can vary are in red, and the constraints are in green. Known values are in

blue. The initial conditions for the three red cells can be 0. Solver generates a value for the

Passage Height in the 2nd

row and the 2nd

column and the lengths that the EP and APE subtend

on the Casing and CE are calculated, in the 3rd

and 4th

row respectively, so that the x and y

coordinates for the roofs and floors can be calculated in the 2nd

and 3rd

columns, rows 6 through

9. The coordinates of the 2nd

pair of M Circles are then calculated, columns 2 and 3, rows 10

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through 11. The three constraints, as described above, are calculated, and ʺSolverʺ terminates

when the last three cells in the 3rd

column are all zero and conditionally set to green.

The last three values in the second column, in black, represent the values of the radii of the M or

M/T Circles. Each entry can be set to either the M or the M/T radius for a total of 8

combinations. Of these eight solutions, four are zero, and two are negative, and these six are

unusable. There are two positive values; one is 50.410381ʺ, which is outside Petrieʹs range of

46.2ʺ to 48.6ʺ, P150. The remaining value is 47.665515ʺ, which is within Petrieʹs range, and

it is therefore taken as the theoretical height of the sloping passages.

The following table shows a sequence by which most theoretical lengths of the Pyramid passages

and chambers can be theoretically reconstructed:

Theoretical Parameter Equation Inches

Royal Cubit (RC) 365.25/(10 × √π) 20.607025

Remen (Rm) RC/√2 14.571367

Entrance Floor, EPNx, X0 AL4 × cos(P) - 3 × RC 4010.613490

Entrance Floor, EPNy, Y0 H - EPNx × tan(α) 667.138080

APENx Numerical Solution 5274.510272

APENy Numerical Solution -942.105284

a, Entrance Floor to APE/CE Intersect ((X0 - X4)2 + (Y0 - Y4)2))0.5 2046.240280

a/sin(A), (A = 2P) a/sin(2P) 2575.598065

APE a × sin(180-α-P) 2520.769967

EP sin(α-P) × a/sin(2P) 1110.903407

EPSx X4 - APE × cos(P) 3014.726870

EPSy Y4 + APE × sin(P) 174.881825

EPE APE - EP 1409.866560

EPENx X0 - EPE × cos(P) 5274.510272

EPENy Y0 + EPE × sin(P) 1291.868934

AL4 EP+AP+GG 4542.767563

AP AL4 - EP - GG 1547.346403

APSx EPSx+ AP × cos(P) 1627.584146

APSy EPSy + AP × sin(P) 860.531848

AL8 GGP/360 × M 1884.517753

GGSx APSx + GG × cos(180 - P) -61.821074

GGSy APSy + GG × sin(180 - P) 1695.587035

AL3 AP+GG+KCS 4176.873883

KCS with sloping portion AL3 - AP - GG 745.009727

KCSh - horizontal no sloping portion Y * (8/7 + 0.5/π + 1.3/√π) 743.451233

Sloping part of KCS (KCSsl) KCS-KCsh 1.558493

Horizontal part of KCSsl (hKCSsl) KCSsl × cos(P)/(1 - cos(P)) 13.494401

Sloping part of KCSsl (slKCSsl) KCSsl/(1 - cos(P)) 15.052894

Vertical part of KCSsl (vKCSsl) slKCSsl × tan(P) 6.670140

AC Floor Level Above Pavement (FLAP) GGSy + vKCSsl 1702.257175

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Theoretical Parameter Equation Inches

Sloping Passage Perpendicular Height

(SPph)

Numerical Solution 47.665515

Sloping Passage Vertical Height (SPvh) SPRH/cos(P) 53.170494

GGNh SPVH + 2 × RC 94.384543

Height of Kingʹs Chamber Step (KCstep) Numerical Solution 0.755538

Kingʹs Chamber FLAP ACFLAP + KCStep 1703.012713

The above tables show how the sequence in which the main theoretical dimensions of the

Pyramid passages and chambers are calculated to arrive at the dimension of the sloping part of

the KCS. Note that the table is color-coordinated to help group the different types of dimensions.

A red value is a coordinate that has a mnemonic intended to identify the location of that

coordinate. For example, EPNx is the x coordinate of the north end (N) of the Entrance Passage

(EP). Passage and chamber floors, rather than roofs, are usually assumed, and so ʺFʺ is not used

in the identification. A blue value is one that is already known at this stage. Green values are

ones that are being derived for the first time in this analysis. The second column provides the

equation used to calculate the theoretical dimension, except where numerical analysis was used.

The third column provides the theoretical values of the dimension, which are all in British

inches.

The first two entries in the table define the Royal Cubit, and Remen whose values have already

been determined. The sequence, but not the timeline, starts at the entrance to the Pyramid, which

is already known, and the equations for the X and Y coordinates have already been calculated.

These are EPNx and EPNy. They are coded red, rather than blue, because they define a

theoretical point corresponding to one that Petrie has measured and which was used earlier to

determine the impacts of subsidence on the location of the ends of the passages and chambers.

The next two entries are APENx and APENy, which define the point at which the APE intersects

the CE. These were computed by numerical analysis earlier. The distance between the entrance

and APEN, which is defined from the sine rule as ʺaʺ, is then calculated as is the ratio a/sin(2P).

These two intermediate values are required so that the length of the AP extension (APE) can be

calculated in the next row. EP, one of the major passage dimensions of the Pyramid, is then

calculated from ʺaʺ and the sine rule. The coordinates of the south end of the EP, EPSx, and

EPSy, are then calculated for subsidence evaluation.

Since we now know the theoretical length of APE and EP, the length of EPE can be calculated

and also its northern coordinates EPENx and EPENy. This point and also APEN define the start

of the timeline as APE = EP + EPE.

To compute the theoretical length of the AP arc length four (AL4) and the length of the GG,

which is arc length eight (AL8), need to be known. Once AP is calculated it is possible to

compute its southern point, APS, and the southern point of the GG, GGS.

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Knowing the length of the AP, in conjunction with arc length three, allows the length of KCS to

be computed. This length includes the sloping part of KCS. Earlier, the horizontal length of KCS

was computed, which did not include the sloping portion of KCS. So finally, after computing all

major passage lengths, it comes down to the sloping portion of the KCS is 1.558493ʺ longer than

KCSh. The following figure shows the relevance of the three parameters hKCSsl, slKCSsl, and

vKCSsl. The sloping portion of the KCS is 1.558493ʺ, which is the difference between the length

of the sloping portion, which is slKCSsl, 15.052894ʺ and the horizontal part of that slope,

hKCSsl, which is 13.494401ʺ. vKCSsl is the vertical height required, which at the passage angle

P is 6.670140ʺ, which is shown as the vertical red line in the figure below. This dimension is the

difference in height between the bottom south end of the GG and the floor level of the passages

leading to the Kingʹs Chamber entrance.

By adding vKCSsl to the y coordinate of the south end of the GG, GGSy, the theoretical floor

level above the pavement of the passages, ACflap, is calculated as 1702.257175ʺ.

Having calculated a theoretical height for the sloping portion of the KC, it can be fed back to the

previous analysis to see if it fits, which is shown in the figure below.

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When compared with the previous analysis, it can be seen that the lowest corbel on the north

wall has been added because the M and M/T circles indicated this, centered at the north end of

the APE, bracketing the two lowest corbels. Looking at the right-hand side of the above figure,

we can see the vertical edge of a parallelogram in red. The distance from the floor of the GG to

the underside of the corbel comprises two other dimensions. This first is the height of the AP,

SPvh, which is derived from Sloping passage height, SPrh. These two heights are 53.170494ʺ

and 47.665515ʺ, respectively.

Smyth (v2,p70) measured the vertical height of the AP as 53.2ʺ, which is remarkable as it is just

0.03ʺ taller than the theoretical value. To this is added 2 RC which is 41.214ʺ for a total height,

GGNh, of 94.384543ʺ which is just 0.015ʺ less than the 94.4ʺ measured by Smyth (v2, p89). The

theory, therefore, matches the measurements at the GG north wall quite accurately.

From this measurement, the height of one corbel should be subtracted, because the slope of the

lowest side corbel touches the southmost corner of the corbel before continuing down at the

passage angle, P until it intersects the north wall. Since the length of a corbel is 1 RC divided by

7 then its height is 1.455ʺ on the slope. The north end of the parallelogram is therefore 94.384 –

1.455 = 92.929ʺ and the south end of the parallelogram should match this.

The other vertical of the parallelogram is the south wall, shown in red, and there are four

measurements shown vertically above each other. The top dimension is the height of one corbel,

which should be added because the slope of the lowest corbel on the sidewall passes through the

1.455

1LP

N Wall

S Wall

AP

Lower

GG

Great

Step

Ramp

Expanded Views

Sloping Part

of KCS

41.214

53.170

94.384 92.929

1.455

1.455

43.59092.929

41.214

6.670

2.944

1.4552.944

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corner of the corbel on the south wall before continuing up and intersecting the line of the north

wall. Below that, there is a height of 2 RC, 41.214ʺ, which matches the corresponding height

above the door but below the first corbel on the north wall. The lowest of these four dimensions

is the height of the sloping portion of the KCS, 6.670ʺ. When these three dimensions are

summed, 49.339ʺ, and then subtracted from the height of the parallelogram at the north wall,

92.929ʺ the result is 43.590ʺ, which is the theoretical height of 1LP. Smyth, (v2, p74), measured

the average height of 1 LP as 43.55ʺ, which is just 0.04ʺ less than the theoretical height.

Smythʹs measures have been used here for consistency since Petrie did not provide all those

required for the analysis. The close correlation between the theoretical values and the measured

values is quite remarkable and supports the conclusion that using the M and M/T circle pairs, for

both the KCS sloping length and the sloping passage height, is the correct theoretical approach.

The following analysis shows where the projected rooflines of the AP and 1LP intersect in the

lower part of the GG. Mathematically they intersect exactly two horizontal corbels to the north of

the south wall, which sets a precedent for the QC passage. The following figure is a variation of

several similar figures found above:

The north and south walls have common characteristics that include a vertical doorway at the

floor and then a wall between the top of the door to the overhang of the first corbel above, with a

theoretical height of 2 RC between A and C at the north wall.

The top sloping red line of the parallelogram is the lowest side corbel. It is drawn sloping up at

the passage angle between the noses of the first corbels on the north and south walls,

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respectively. This corbel slopes down from the nose of the first north corbel, at the passage

angle, to meet the north wall at B and up at the other end to meet the south wall at D. This

continuation of the slope decreases the height of the parallelogram and the 2 RC part of the north

wall by RC/7 × TanP. Therefore at the north wall, the height from the top of the doorway to the

top sloping line of the parallelogram is 2RC - RC/7 × TanP.

The roofline of the AP is projected up at the passage angle until it intersects the south wall. The

height below the top of the parallelogram, point D, and this intersection, point F, has to be the

same as the height at the north wall, which has just been calculated to be 2RC - RC/7 × tanP.

However, the top of 1LP, point G, is 2 RC below the bottom of the corbel at point E. The height

from point D to point E equals RC/7 × TanP, and therefore, the height from point D to point G

equals this value plus 2 RC, i.e., 2RC + RC/7 × tanP. By subtracting the height of the projection

of the AP roofline at point F from this value, we will arrive at the height between points G and F,

which is 2RC + RC/7 × TanP – (2RC - RC/7 × TanP) = 2RC/7 × tanP.

The remainder of the analysis can be seen in the lower expanded view. FG equals 2RC/7 × TanP,

and so GX, where X is the intersection of the projections of the AP and 1LP roofs, equals 2RC/7

since FX is at the passage angle P, which is the theoretical width of two corbels. This

mathematical fact has significance with regard to the roof of the QC passage and also in defining

the chronology.

EP Length Theoretical v Measured

In the table above, two other, new theoretical passage lengths have been calculated, which

should be compared against Petrieʹs measures. These are for the EP and AP. The theoretical EP

length is 1110.903ʺ. Petrieʹs mean EP length is 1110.77ʺ, which is the average of 1110.64ʺ and

1110.90ʺ from P38.

The figure below shows Petrieʹs assumptions for the start of the EP, which are derived from P32.

Bear in mind that there is a small difference between how Petrie defined the start of the EP

compared with the theoretical start. It is difficult to draw an accurate image since Petrieʹs tools

for calculating angles and lengths were not as accurate 150 years ago as our computer-based

calculations today, and there are differences because of this. For example, the horizontal length

from the casing edge to Petrieʹs station mark is 638.58ʺ using todayʹs tools versus 638.4ʺ using

Petrieʹs tools.

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Petrie calls this the ʺoriginal positionʺ in P35, but it reflects the positions of the components in

1881, a much later time. The essence of the figure shows that there is 127.9ʺ from the casing

edge to the station mark. It is then 982.74ʺ from there to the junction of the EP and AP which

reflects Petrieʹs 1110.64ʺ inches for the length of the EP and which is based on a half-length of

4534.1ʺ from the center of the Pyramid to the center of the north edge and a casing angle of 51°

53ʹ 29ʺ.

The following figure shows the same parameters for the theoretical lengths, which are based on a

pyramid half-length of 4534.583ʺ. It is also based on the height of the entrance being at the

average height of the bottom of the 19th

course and a casing angle of 51° 51ʹ 14ʺ. As can be seen,

the theoretical length of the EP is 1110.93ʺ.

So any difference between Petrieʹs measured values and the theoretical values can be ascribed to

the difference between the current configuration of the entrance versus the theoretical.

668.20in.

524.10in.

638.58in.

611.16in.

1110.64in.

982.74in.

Floor of Entrance Passage

Petrie’s Station Mark

Surface of Casing

Pyramid Base

127.90in.

26° 29' 0"

51° 53' 29"

667.14in.

638.88in.

26° 18' 10"

51° 51' 14"

128.19in.

Pyramid Base

Floor of Entrance Passage

Petrie’s Station Mark

Surface of Casing

1110.93in.

610.34in.

523.97in.

982.74in.

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The theoretical length of the AP from the table above is 1547.346ʺ compared with Petrieʹs

measured value of 1546.8ʺ, P38. The difference is 0.546ʺ. Like the EP, we can evaluate how

Petrie calculated the initial start length of this passage, which is based on his calculated AP floor

angle. In turn, this is based on his footnote in P38, page 61, which reads as follows:

*The elements in question are

(1) Prof Smythʹs plumb-line 48.5 on slope below his zero in Ascending passage; and

(2) 180.5 on slope of entrance passage, below beginning of Ascending roof.

(3) My level in A. P., 71.3 on slope above C.P.S.ʹs zero in A.P.

(4) My level in E.P. 1015.0 on slope below C.P.S.ʹs E.P. zero.

(5) Difference of my A.P. and E.P. level marks 156.2 vertically.

(6) My plumb-line on E.P. floor 1027.3 on slope below C.P.S.ʹs E.P. zero.

(7) Height on my plumb to floor of A.P. 37.0.

(8) Height of plug-blocks 47.3, and angle of end 26º 7ʹ,

(9) Angle of E.P. at junction 26º 21ʹ. From these measures we get 125.1 tan. θ +142.9 sin. θ =

124.7; ∴ θ = 26º 12½ʹʺ

When reconstructing what Petrie is trying to convey here, there are a couple of problems. In

comment (4), Petrie states that his level in the EP is 1015ʺ below Smythʹs EP zero, and in (6), his

plumb-line is 1027.3 inches below Smythʹs EP zero. Smythʹs EP zero is the end of the ʺBasement

Sheetʺ (SV2, p11) which Petrie measures as 124.2ʺ from the entrance, P35. So these two points

are 1015 + 124.2 = 1139.2ʺ and 1027.3 + 124.2 = 1151.5ʺ respectively from the entrance. The

junction of the floor of the AP with the EP is 1110.64ʺ below the entrance, and by the time it

crosses Petrieʹs plumb-line, it is 37.0ʺ above the floor of the EP from comment (7). These

measures are shown in the following figure. 1000ʺ has been cut off the three dimensions along

the EP floor from the entrance for brevity. The brown line is the EP floor at an angle of 26° 21ʹ

0ʺ from comment (9), and the red line is the AP floor.

The first problem is that the angle of the floor of the AP is 27° 15ʹ 28ʺ compared with Petrieʹs

computed value of 26° 12ʹ 30ʺ. This angle does not permit the remainder of Petrieʹs dimensions

in this area to be calculated with any reasonable degree of fidelity.

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Looking at comment (9) it can be seen that Petrie has derived the following equation from which

the AP floor angle can be calculated:-

ʺFrom these measures we get 125.1 tan. θ +142.9 sin. θ = 124.7

∴ θ = 26º 12½ʹʺ

The following figure shows a complete restoration of Petrieʹs comments but with what I believe

are necessary corrections for the problem outlined above and others to be outlined below.

The elements of the equation above are highlighted in magenta in the figure above, but there are

some differences. The heart of the equation is the top part of Petrieʹs plumb-line CEF, where F is

the floor of the AP. Petrieʹs equation says this is 124.7ʺ long. The top of the plumb-line, CE, is

the vertical height of the triangle ABD which can be calculated from the hypotenuse of the

27° 15' 28"

110.64

139.20

151.50

37.0

Petrie’s Plumb Line

26° 21' 0"

Petrie’s

EP Level

Al Mamoun’s

Hole

ϴ

181.74

(180.5)

324.69

143.0

71.3

48.5

156.2

37.0

143.0 sinϴ

125.6 tanϴ

140.20

152.50

23.2

26° 21' 0"74.19

47.3

50.76in.

110.90

110.64

125.6

124.7

Start of E

P 1000

Petrie’s

EP Level

Petrie’s AP Level

C

Petrie’s Plumb Line

CPS’s

Plumb Lines

A

B

D E

CPS’s AP Zero

F26° 9' 15"

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triangle, AD which equals 143.0ʺ times the tangent of the angle of the floor of the AP which

Petrie says is 26° 12ʹ 30ʺ. Petrie says AD equals 142.9ʺ. The midpoint of the plumb-line, EF, is

the length DE multiplied by the sine of the angle of the floor of the AP. Petrie says this length is

125.1ʺ, but after applying corrections to the figure, it turns out to be 125.6ʺ.

When the 124.7ʺ of the plumb-line CF is added to the 37ʺ of the plumb-line below the AP floor,

the total is 161.7ʺ. The EP level mark is 1027.3 – 1015.0 = 12.3ʺ higher up the EP floor from the

plumb-line. As can be seen from the figure the dark green line has been drawn vertically from

Petrieʹs EP level with a height of 156.2ʺ, and this ends at the same level as Petrieʹs AP level mark

which represents the difference in Petrieʹs EP and AP levels. It is 156.24ʺ, but the 0.04ʺ

difference has been ignored. These dimensions support the fact that the distance along the

plumb-line from the AP to the EP floor should be 37ʺ.

Petrieʹs comments regarding the length of the plumb-line and EP mark from Smythʹs EP zero are

short by 1ʺ which could have happened by means of a computational or typographical error. If

the dimensions are increased to 1140.20ʺ and 1152.50ʺ respectively, then the angle of the AP

floor becomes 26° 9ʹ 15ʺ. So a correction of 1ʺ has been made to both these dimensions in the

Figure and are shown as 140.20ʺ and 152.50ʺ in red at the bottom right-hand corner, which may

or may not be right, but it is required in mathematical terms. The distance between the plumb-

line and EP level remains at 12.3ʺ, so the relationship between the height of the plumb-line and

the difference between the EP and AP levels remains the same. Please recognize that the

magenta plumb-line shown above the AP roof is only a virtual line.

Petrie computed the length AD from three separate dimensions. 71.3ʺ is the difference between

his AP level and Smythʹs EP zero. 48.5ʺ is the length of the base of the first granite plug block.

The 23.2ʺ is the overhang of Smythʹs plumb-line from point B to point D. These values sum to

143.0ʺ versus Petrieʹs 142.9ʺ. The minor difference, 0.1ʺ, is due to the availability of more

accurate calculation tools. Since these are well-established dimensions, any changes to Petrieʹs

equation can only be made to the value of 125.1ʺ. The dimensions in the figure were all

computed by the drawing program Vizio, rather than using a spreadsheet, so the value for this

length, 125.6ʺ, is somewhat more accurate than Petrieʹs value. It also reflects the increase in the

length of Petrieʹs plumb-line and EP level from Smythʹs EP zero.

In comment (2), Petrie introduced Smythʹs dimension from the junction of the EP and AP roofs

to Smythʹs plumb-line in the EP. This dimension is 180.5ʺ, and it and Smythʹs plumb-line are

shown in light green in the figure. However, the dimension has increased to 181.74ʺ, in part

because of the corrections. Possibly Petrie introduced Smythʹs dimension because he found later,

when he was back home, that he had not measured the horizontal distance from his EP level

mark to his AP level mark. However, there appears to be no way to use Smythʹs dimension of

180.5ʺ to arrive at a dimension of 125.1ʺ in Petrieʹs equation, with or without the 1ʺ correction.

The use of the 180.5ʺ dimension is ignored in this analysis.

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With the corrections and ignoring the 180.5ʺ, the figure above represents the essence of what

Petrie was trying to say in his comments. It should be borne in mind that small variations in the

height of Petrieʹs plumb-line below the AP floor will result in different lengths from the junction

of the EP and AP to Petrieʹs AP level mark which is shown by the following table:

AP Floor Angle Petrieʹs Plumb-line Height ʺ EP Junction to AP Level Mark ʺ

26° 12ʹ 30ʺ 37.044 324.068

26° 10ʹ 35ʺ 37.018 324.437

26° 09ʹ 15ʺ 37.000 324.692

26° 07ʹ 00ʺ 36.970 325.125

26° 02ʹ 30ʺ 36.909 325.996

It can be seen that just a small difference in the angle of the floor of the AP, say the 1ʹ 55ʺ

between the first and second entry in the table above will lead to a change in the calculated

length from the EP Junction to the AP Level Mark of 0.369ʺ, which changes the length of the AP

by the same amount.

The purpose of this analysis was to determine if the theoretical length of the AP is acceptable as

it is 1547.346ʺ, which is 0.546ʺ longer than Petrieʹs value of 1546.8ʺ. The analysis has shown

that given Petrieʹs range of AP floor angles, 26° 02ʹ 30ʺ to 26° 12ʹ 30ʺ, his AP length will change

from 1546.8ʺ to 1548.8ʺ. The theoretical value, 1547.346ʺ does fit within this range, but this is

highly susceptible to the AP floor angle. A new survey of the length of the AP would be

beneficial to arrive at a more reliable range for the length of the AP.

Kingʹs Chamber Step Height and Position of Scored Lines in Entrance Passage

There is a small step at the junction of the 2nd

Low Passage and the Kingʹs Chamber, which

Petrie measured as 0.8ʺ, at the end of the first table in P47. I recall stumbling somewhere along

2LP when I visited the Pyramid in 1973. There are also scored lines in the EP 481.59ʺ from the

entrance. These values of these two dimensions are related and can be determined as follows:

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There are four pairs of M and M/T Circles shown in the figure above. The red pair is centered at

the north edge of the casing and the blue pair on the floor of the entrance. The M Circles of this

pair intersect at X9, Y9. The lower green pair of M Circles is centered at X9, Y9 and the M/T

Circle passes through the Kingʹs Chamber as shown below:

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The top green arc is the M/T Circle, and the point where it intersects the vertical junction

between the Kingʹs Chamber and 2LP represents the top of the step, which is identified as point

X10, Y10. If this junction seems to be a little off, this is because the drawing program draws

circles with an offset when converting to screen coordinates. It computes the coordinates of any

feature accurately. The coordinates of the circles are computed by the spreadsheet program using

solver as with the Z Circle and passage height, as shown in the table below.

KC Small Step Height & Scored Lines

Small Step Height x y

North Base 4534.58251284 0.00000000

EP/Casing Floor 4010.61348972 667.13808046

X9, Y9 1061.47499353 -2188.44107250

Top of Step, X10, Y10 -331.31074262 1703.01271286

Step Height 0.75553799

0.00000000 4105.08832652

0.00000000 4105.08832652

0.00000000 4133.19061627

Scored Lines y = mx + c Tan P

m 0.49428946 0.49428946

c -1315.26587999

x y 3579.10293014 453.84696067

0.00000000 4105.08832652

Make Step height = 0ʺ

Distance from Entrance 481.35 481.03

Petrie (P55) 481.59 481.59

Difference 0.24 0.56

The table shows the numerical solutions for both the KC Step and the Scored Lines. The

constraints for the step are the three green zeros, which, top to bottom, represent the distance

from the north base to the point X9, Y9, from the entrance to point X9, Y9, and from the intersect

to the top of the step point X10, Y10. The first two distances should be the radius of the M Circle,

and the third should be the radius of the M/T Circle. The numerical solution shows that the

height of the step is theoretically 0.7555ʺ, which compares well with Petrieʹs 0.8ʺ, P47. The level

of the KC floor should, therefore, be 1703.0127ʺ.

For the position of the Scored Lines in the EP, please refer to the second figure above. The upper

pair of green M Circles is centered at the top of the step in the KC, which is point X10, Y10. The

point at which the M Circle of this pair intersects the EP is the location of the Scored Lines,

which are at right angles to the slope of the passage.

The numerical solution for this point is shown in the bottom part of the above table. There is

only one constraint, and that is the distance from the nose of the step to the floor of the EP is the

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radius of the M Circle. The numerical solution shows that this intersection is 481.35ʺ below the

entrance, which is just 0.24ʺ less than Petrieʹs measure of 481.59ʺ. This difference is acceptable.

To be sure that the Scored Lines relate to the KC Step, the distance between the two was

computed in the last four rows of the last column of the table above with the step height set to 0ʺ.

In this case, the position of the Scored Lines is 481.03ʺ below the entrance, which is 0.56ʺ less

than Petrieʹs measurement and cannot be justified so easily.

Some believe that the Scored Lines align with some major astronomical event occurring at the

time indicated by the position of the lines within the EP. Jumping ahead to the chronology, we

will see that the date represented by the Scored Lines is Julian Day 926775.1428 (TT), which is

03:35, Thursday, 16th May 2176 BC (UT+2:20:56) ± 1.25 hours. No such event could be found.

Therefore, it is concluded that the Scored Lines validate the existence of the KC Step and

to help identify its precise location.

Queenʹs Chamber Connection to the AP and GG

The Queenʹs Chamber has already been fixed in the Pyramid by the level of its floor and by the

level of the floor of the GG at its north wall, which is also the junction of the AP and GG. The

riser of the Great Step that is its vertical face is assumed to be aligned with the vertical axis of

the Pyramid. That is a vertical line that passes through the center of the base of the Pyramid and

its apex. The Great Step has been shown to have a tread of exactly 3 RC, which extends

horizontally from the vertical axis to the south by that amount. The north wall of the GG has

therefore been fixed in the horizontal axis, but until now, its height above the pavement was not

theoretically known. This theoretical horizontal point is APsx 1627.584ʺ, and now it is known

that the theoretical vertical coordinate of this point, APsy, equals 860.532ʺ.

The method that determined the level of the QC floor, which is derived from Arc length five

(AL5), was shown earlier in ʺThe Theoretical Floor Level of the Queenʹs Chamberʺ to be

846.744ʺ.

The following figure shows how the QC is connected to the GG and AP. Dotted lines are used to

shorten the lengths to add clarity to the figures. The connection is via the Queenʹs Chamber

Passage, which comprises a high and low part. The lengths of these two passages have been

determined from arc length six (AL6), but their theoretical heights are determined in the figure.

Three other coordinates are shown in the figure. The first is the reference fiducial at the top of

the south side of the Well Shaft 1578.425, 907.818, which is included just as a reminder. The

second is the top of the doorway of the AP 1627.584, 860.532, which is included to show the

level of the junction of the AP and QCP roofs. The third coordinate is the center of the floor of

the QC 0.681, 846.744.

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When defining the intersection of the AP and 1CP roof lines above, it could be seen that the

point was mathematically defined and was a multiple of the width of a corbel, i.e., it was more

than just a random point. Regarding the intersect of AP and QCP roof lines, it can be seen that

the top of the AP doorway, at the north wall of the GG, can be projected horizontally to form the

roof of the QCP which is also more than just a random line. It, therefore, starts in line with the

corbels which start at the north wall.

When the passage roof reaches the north wall of the QC, it will be at a height above the

Pavement of 913.702ʺ, which is the level of the roof of the AP, minus the level of the QC floor

which is 846.744, i.e., 66.958ʺ as shown in the figure. Petrie measured the height of the top of

the second course of stones in the QC as 67.44ʺ (Table P42) minus 0.23ʺ, i.e., 67.21ʺ, which is

where the QCP enters the QC at the NE corner. It is, therefore, 0.25ʺ higher than the theoretical

value.

In pages 59 to 61 of volume 2, Smyth recognizes that the measurements in this passage are

reduced by an amount equal to the thickness of the encrustation. Smyth measured the height of

the low portion of the passage as 46.36,ʺ but he believes that it is not improbably 47.0ʺ

originally. Petrieʹs average measure of height in this passage is 46.3ʺ. He believes that in general,

the passage heights ought to be related to the height of the courses in the Kingʹs Chamber, which

are, by his measure 47.042ʺ, tall, P150. Since both these surveyors agree, it is concluded that the

height of the low part of the QCP equals the height of the courses in the KC, which theoretically

are 47.024ʺ which is the value shown in the figure and which has no significant difference from

the surveyors ʺtheoreticalʺ values.

19.7

19.916

46.5

47.04267.21

66.958

Well Shaft

AP

x 1627.584

y 860.532

x 1578.425

y 907.818

x 1627.584

y 913.702

x 0.681

y 846.744

Queen’s Chamber

Queen’s Passage

GG

7 Corbels

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© Copyright 2019 M J Cooper In Accordance With Title Page – Oregon USA 80

This height, when subtracted from the height of the high portion at the QC, results in the height

of the step in this passage being 19.916ʺ. Petrie measured the height of this step as 19.7ʺ, P40, so

the theoretical value is 0.22ʺ taller, which is an acceptable difference given the fact that treasure

hunters dug under this step, which most probably created a variation from the original height.

All-in-all the differences between the theoretical and measured values of the height of the

connection between the AP and GG and the QC, via the QCP, is quite remarkable for its

accuracy given the evaluation above.

Final Theoretical Length of the Descending Passage

The parameters in the table below are used to determine the final theoretical length of the

Descending Passage, which turns out to be 3036.986215ʺ.

Theoretical

Parameter

Equation Inches Measured

ʺ

Difference

ʺ

AL2 EP+DP+SS 5466.120099

EP sin(α-P) × a/sin(2P) 1110.903407 1110.64 -0.26

SS Rm × 19 × (43/12 + 3/2 × tan(α)) 1318.230477 1318.0 -0.23

DP AL2 - EP - SS 3036.986215 3036.92 -0.07

DPSx EPSx - DP × cos(P) 292.173392 306.0 13.83

DPSy EPSy - DP × sin(P) -1170.847653 -1181.0 -10.15

In ʺFinal Considerations Relating to the Subterranean Systemʺ the estimate for the length of

the Descending Passage was shown to be 3036.89ʺ which is 0.1ʺ less than the theoretical value

calculated above which is an acceptable difference especially considering the difficulty of

measuring it.

M Circle Table – Showing All Known Arc Lengths

At this point, we have been able to reconstruct the exterior and interior dimensions of the

Pyramid. Since we are not looking for any more clues to help us in this respect, a final update to

the M Circle Table can be made, which started the whole process of looking for clues. The

revised Table is shown below.

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AL

#

Exterior

Angle

Degs ° Arc

Length

Measured

Value Bʺ

Diff

Relationship

1 RCS Base

Angle

51.854 3715.2 -1043.3/-

1056

- Level of Subterranean Chamber

Roof

2 RCS Apex

Angle

76.292 5466.1 5465.6 0.6 The path length of Entrance,

Descending, Subterranean

Passages, and Subterranean

Chamber

3 FCS Base

Angle

58.298 4176.9 4175.3 1.6 The path length of Ascending

Passage, Grand Gallery and Kingʹs

Chamber (See Figures below)

4 FCS Apex

Angle

63.405 4542.8 4541.0 1.7 Path Length of Entrance and

Ascending Passages and Grand

Gallery

5 DCS Base

Angle

41.997 3009.0 846.7/834.4 Queenʹs Chamber Floor Level

6 DCS

Apex

Angle

96.006 6878.5 When divided by four this gives a

path length from the North wall of

the Grand Gallery to the midpoint

in the Queenʹs Chamber /4 1719.6 1719.5 0.2

7 BCS Base

Corner

Angle

90.000 6448.3 - - Confirms that the Pyramid Base

Angle = tan-1

(4/π)

8 Passage

Angle

26.303 1884.5 1883.6 0.9 Length of Grand Gallery

9 Queenʹs

Chamber

Roof

Angle

30.459 2182.3 - - Assists in defining the sloping

length of Petrieʹs virtual pathway in

the 1st Low Passage.

As recognized earlier, it can be seen that similar definitions group the M Circle relationships by

cross-section, which adds to its value as a clue as it indicates that the clues were created by an

intelligent being and are not just a random occurrence.