developing mathematics patterns and ideas presented by sekender & shahjehan khan february 27,...
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Developing Developing MathematicsMathematics
Patterns and IdeasPatterns and Ideas
Presented
By
Sekender & Shahjehan Khan
February 27, 2005
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Cambridge College, Chesapeake, Va
Mat 603- Arithmetic to Algebra
Nancy E Wall Professor
Curtiss E Wall Professor
Patterns, Mathematics, Fibonacci, & Phyllotaxis
A Power Point Presentation
For The partial fulfillment
of the course
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Our Universe
Our universe, our Life, our living , our nature and everything around us is a pattern. Thus we see pattern in our physical, chemical, biological, mathematical and social construction of our daily lives.
Let us look at some patterns….
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The Solar System and How it Relates to an atom
MerVenE MarJu is SUN Proof
Here’s a Mnemonic device to memorize the planets in the universe
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Chemical Structures of Glucose and Benzene
Glucose Benzene DNA
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Everyday Needs
Houses Clothes Foods
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Food Industries
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Automobiles
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Architectures
Shalimar Garden Lahore, Pakistan.
Kutub minarTaj Mahal
White House The Tower of Pisa
The Forbidden Citypyramids in Egypt
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Traffic Patterns
Traffic Jam - ChinaBullock Carts - India Camels- Middle East
Traffic Jam
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Flight Patterns
Crop Circles
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Numerals
Arabic Numbers
Bengali Numbers
Hindi Numbers
Chinese Numbers
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Alphabet
Bengali
Arabic
Hindi
Greek
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Our Nature, objects in Nature and
Biological symmetry
Common Snail (Helix)Ovulate Cone (Pinus)Muscadine Grape Tendril (Vitis rotundifolia)
A type of symmetry in which an organism can be divided into 2 mirror images along a single
plane.
A packing arrangement in which the individual units are tightly packed regular hexagons. There is no more efficient use of packing space than this, and it occurred first in nature.
A symmetry based on the pentagon, a plane figure having 5 sides and 5 angles
Spirals Bilateral Symmetry
Hexagonal Packing Pentagonal Symmetry
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Math Patterns
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Pattern in Mathematics
Triangular Numbers Square Numbers
Pentagonal Hexagonal
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Pattern in Multiplication Table
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Sequences and SeriesA sequence is a function that computes an ordered list .The sum of the terms of a sequence is called a series.
Summation Rules
Sn= 1+2+3+ … + n = n(n+1) /2
Sn = 12 +22+32+… +n2 = n(n+1)(2n+1 )/6
Sn = 13 + 23+ 33+ …+n3 = n2 (n+1)2 /4
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Arithmetic Sequences and Series
Arithmetic sequences - A sequence in which each term after the first is obtained by adding a fixed number to the pervious term is Arithmetic Sequences (or Arithmetic Progression ) The fixed number that is added is the common differences.
In an Arithmetic Sequence with first term a, and common
differences d, the nth term an, is given by
an = a1 + (n-1)d Sum of the first n terms of an Arithmetic Sequence
Sn = n/2 (a1+an)
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Geometric Sequences and Series
A geometric sequence (geometric progration) is a sequences in which each term after the first is obtained by multiplying the preceding term by a fixed non zero real number, called the common ratio.
If a geometric sequence has first term a1 and common ratio r, then the first n term is given by
Sn = a1(1-rn)/ (1-r ), where r ≠ 1
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Pattern In Binomial Expansion
Pascal Triangle – The coefficient in the terms of the expansion of (x+y)n when written alone gives the following pattern.
And so on……..
To find the coefficients for (x+y)6, we need to include row six in Pascal’s triangle. Adding adjacent numbers we find row six as..
1 6 15 20 15 6 1
n- Factorial = n! , 0! = 1
For any positive integer n
n! = n(n-1) (n-2)…(3) (2)(1) and 0! =1
Factorial Pattern
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Permutations A permutation of n element taken r at a time is one of
the arrangements of r elements from a set of n elements, denoted by P(n,r) is
P(n,r) = n(n-1)(n-2) …(n-r+1)
= n(n-1)(n-2) …(n-r+1)(n-r)(n-r-1) …(2)(1)(n-r)(n-r-1)…(2)(1)
= n!(n-r)!
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Combinations of n Elements taken r at a time
C (n, r)
( )
represents the number of combination of n elements taken r at a time with r < n, then
If C (n, r) or
= ( ) = n!
(n-r)! r !
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Pattern for adding
consecutive odd numbers
series The formula is S=n2
Where S = sum
n = number of addends
Pattern for adding all even
number in series
S=n(n+1)Where S= Sum
n= Number of addends
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Pattern of Numbers from Triangle to Decagon
Table of squares and triangles of some naturals numbers
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Patterns and PolygonDefinition - A many -sided, closed –plane figure with three
or more angles and straight lines segment that do not intersect except at their end points.
Mathematicians use symbols to represent geometric numbers. Thus,
S4 = fourth square number = 16
T4 = Fourth triangle number = 10
n= numerals
So, we can derive
Sn = n2 for square
Tn =n(n+1)/2 for triangle
Pn= n(3n-1)/2 for pentagon
Hn= n(4n-2)/2 for Hexagon
HPn = n(5n-3)/2 for heptagon
On= n(6n-4)/2 for octagon
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Table of polygons Patterns and their
formula
Exploring Triangular and
Squares
numbers
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Pattern for adding all the natural
numbers in series S =n(n+1) /2
Where S= Sum
n= Number of addends
Pattern of adding cube of consecutive natural
numbersS= T n
2
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Pattern in square of consecutive natural
number with alternating negative
and positive signs
S=Tn when n is odd
S= - Tn when n is even
Pattern for adding consecutive odd
numbers with altering negative
and positive signsS = n for odd numbers addends
S = -n for even numbers addends
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Primes A Prime number is natural number that has exactly two factors, itself and 1. The pyramid below is called a prime pyramid . Each row in the pyramid begins with 1 and ends with the number that is the row number. In each row, the consecutive numbers from 1 to the row number are arrange so that the sum of any two adjacent number is a prime.
Prime Pyramid
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The Sieve Of Eratosthenes(prime numbers)
The table below represents the complete sieve. The multiples of two are crossed out by \ ; the multiples of 3 are crossed out by /, multiples of 5 are crossed out by -- ; the multiples of 7 are crossed
out by
The positive integers that remain are: 2,3,5,7,11,13,17,19,23,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97, are all prime numbers less than 100 There are infinite number of primes.
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Palindrome Pattern
Palindrome is a number that read the same backwards as forwards (for example 373, 521125, racecar, are palindromes)
Any palindrome with even number of digits is divisible by 11 Pattern with 11
1*9+2 = 11 (2)
12*9+3 =111 (3)
123*9+4=1111 (4)
1234*9+5=11111 (5)
12345*9+6=111111 (6)
123456*9+7=1111111 (7)
1234567*9+8=? 11111111 (8)
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Fibonacci
Leonardo Pisano ( 1170- 1250? ) our Bigolllo is known better by his nickname Fibonacci . He is best remembered for the introduction of Fibonacci numbers and the Fibonacci sequence. The sequence is 1,1,2,3,5,8,13 …... This sequence in which each number is the sum of two preceding numbers is a very powerful tool and is used in many different areas of mathematics
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What is Phyllotaxis?
Don’t Forget to file your Taxes!!
The arrangement of leaves on the node. Three kinds of Phyllotaxes are as follows:
Whorled- more than two leaves at each
node
Opposite -Two leaves
at each node
Alternate- one leaf at each node
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Terminology
Genetic Spiral (An imaginary Spiral) -- When an imaginary spiral line be drawn form one particular leave to the successive leaves around the stem so that the line finally reaches a leaf which stands vertically above the starting leaf
Orthostichy (Orthos, straight, stichos – line) -- The vertical rows of leaves on the stem.
Phyllotaxy ½ -- When third leaf stands above the first one.
Phyllotaxy 1/3 -- When fourth leaf stands above the first one.
Phyllotaxy 2/5 -- When sixth leaf stands above the first one and genetic spiral completes two circles.
Phyllotaxy 3/8 - When ninth leaf stands above the first one and genetic spiral completes three circles.
Golden Mean = (√5+1)/2 = 1.6180 = t.
Fibonacci Ratio – The ratio of two consecutive Fibonacci number F k+1 / F k for example 34/21 = 1.619 which converges toward golden mean.
Fibonacci Angle = 360° t-2 = 137.5 approximately.
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Phyllotaxes of Different plants understudy
Justimodhu Phyllotaxy ½
Puisak Phyllotaxy 1/3
Kalmi Phllotaxy 2/5 Neem
Peepul Phyllotaxy 3/8
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Pattern of Florets in a Sunflower head
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Types of Inflorescene Structures
Determinate Indeterminate
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Thank You and
The End