Developing Mathematics Patterns 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 HousesClothesFoods

Food Industries

Automobiles

Architectures Shalimar Garden Lahore, Pakistan. Kutub minar Taj Mahal White HouseThe Tower of Pisa The Forbidden Citypyramids in Egypt

Traffic Patterns Traffic Jam - China Bullock 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 SpiralsBilateral Symmetry Hexagonal Packing Pentagonal Symmetry

Math Patterns

Pattern in Mathematics Triangular Numbers Square Numbers Pentagonal Hexagonal

Pattern in Multiplication Table

Sequences and Series A sequence is a function that computes an ordered list. The sum of the terms of a sequence is called a series. Summation Rules S n = … + n = n(n+1) /2 S n = … +n 2 = n(n+1)(2n+1 )/6 S n = …+n 3 = n 2 (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 n th term a n, is given by a n = a 1 + (n-1)d Sum of the first n terms of an Arithmetic Sequence S n = n/2 (a 1 +a n )

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 a 1 and common ratio r, then the first n term is given by S n = a 1 (1-r n )/ (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 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=n 2 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 Polygon Definition - 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, S 4 = fourth square number = 16 T 4 = Fourth triangle number = 10 n= numerals So, we can derive S n = n 2 for square T n =n(n+1)/2 for triangle P n = n(3n-1)/2 for pentagon H n = n(4n-2)/2 for Hexagon HP n = n(5n-3)/2 for heptagon O n = 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 numbers S= T n 2

Pattern in square of consecutive natural number with alternating negative and positive signs S=T n when n is odd S= - T n when n is even Pattern for adding consecutive odd numbers with altering negative and positive signs S = 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, , 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= (6) *9+7= (7) *9+8=? (8)

Fibonacci Leonardo Pisano ( ? ) 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 = = t. Fibonacci Ratio – The ratio of two consecutive Fibonacci number F k+1 / F k for example 34/21 = which converges toward golden mean. Fibonacci Angle = 360° t -2 = 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