Transcript Plants - Angelfire

Developing
Mathematics
Patterns and Ideas
Presented
By
Sekender &
Shahjehan Khan
February 27, 2005
Mat 603Arithmetic to Algebra
Nancy E Wall
Professor
Curtiss E Wall
Professor
Patterns, Mathematics,
Fibonacci, & Phyllotaxis
A Power Point
Presentation
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
Here’s a Mnemonic device to memorize the planets
in the universe
MerVenE MarJu is SUN Proof
Chemical Structures of Glucose
and Benzene
Glucose
Benzene
DNA
Everyday Needs
Houses
Clothes
Foods
Food Industries
Automobiles
Architectures
Kutub minar
Taj Mahal
White House
pyramids in
Egypt
The Tower
of Pisa
Shalimar Garden
Lahore, Pakistan.
The Forbidden City
Traffic Patterns
Traffic Jam - China
Bullock Carts - India
Traffic Jam
Camels- Middle East
Flight Patterns
Crop Circles
Numerals
Chinese Numbers
Arabic Numbers
Hindi Numbers
Bengali Numbers
Alphabet
Arabic
Hindi
Bengali
Greek
Our Nature, objects in Nature and
Biological symmetry
Spirals
Common Snail
(Helix)
Ovulate Cone
(Pinus)
Muscadine
Grape Tendril
(Vitis
rotundifolia)
Hexagonal Packing
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.
Bilateral Symmetry
A type of symmetry in
which an organism can
be divided into 2 mirror
images along a single
plane.
Pentagonal
Symmetry
A symmetry based on
the pentagon, a
plane figure having 5
sides and 5 angles
Math Patterns
Pattern in Mathematics
Triangular Numbers
Pentagonal
Square Numbers
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
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.
Factorial Pattern
n- Factorial = n! , 0! = 1
For any positive integer n
n! = n(n-1) (n-2)…(3) (2)(1) and 0! =1
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
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
If C (n, r) or
(
)
represents the number of combination of n
elements taken r at a time with r < n, then
C (n, r)
=
(
)
=
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 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,
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
numbers
S= T n2
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 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, 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?
The arrangement of leaves on the node.
Three kinds of Phyllotaxes are as follows:
Alternateone leaf at
each node
Opposite Two leaves
at each
node
Whorled- more than
two leaves at each
node
Don’t Forget to file your Taxes!!
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
example 34/21 = 1.619 which converges toward golden mean.
Fibonacci Angle = 360° t-2 = 137.5 approximately.
k+1 /
F k for
Phyllotaxes of Different plants understudy
Justimodhu
Phyllotaxy ½
Puisak
Phyllotaxy 1/3
Kalmi
Phllotaxy 2/5
Peepul Phyllotaxy 3/8
Neem
Pattern of Florets in a Sunflower head
Types of Inflorescene Structures
Determinate
Indeterminate
Thank You
and
The End