Developing Developing MathematicsMathematics

Patterns and IdeasPatterns and Ideas

Presented

By

Sekender & Shahjehan Khan

February 27, 2005

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

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….

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

Chemical Structures of Glucose and Benzene

Glucose Benzene DNA

Everyday Needs

Houses Clothes Foods

Food Industries

Automobiles

Architectures

Shalimar Garden Lahore, Pakistan.

Kutub minarTaj Mahal

White House The Tower of Pisa

The Forbidden Citypyramids in Egypt

Traffic Patterns

Traffic Jam - ChinaBullock Carts - India Camels- Middle East

Traffic Jam

Flight Patterns

Crop Circles

Numerals

Arabic Numbers

Bengali Numbers

Hindi Numbers

Chinese Numbers

Alphabet

Bengali

Arabic

Hindi

Greek

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

Math Patterns

Pattern in Mathematics

Triangular Numbers Square Numbers

Pentagonal Hexagonal

Pattern in Multiplication Table

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

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)

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

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

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)!

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 !

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

Pattern of Numbers from Triangle to Decagon

Table of squares and triangles of some naturals numbers

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

Table of polygons Patterns and their

formula

Exploring Triangular and

Squares

numbers

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

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

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

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.

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)

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

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

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.

Phyllotaxes of Different plants understudy

Justimodhu Phyllotaxy ½

Puisak Phyllotaxy 1/3

Kalmi Phllotaxy 2/5 Neem

Peepul Phyllotaxy 3/8

Pattern of Florets in a Sunflower head

Types of Inflorescene Structures

Determinate Indeterminate

Thank You and

The End