Top Banner
Discovering Fibonacci By: William Page Pikes Peak Community College
64

Discovering Fibonacci

Feb 24, 2016

Download

Documents

toviel

Discovering Fibonacci. By: William Page Pikes Peak Community College. Who Was Fibonacci?. ~ Born in Pisa, Italy in 1175 AD ~ Full name was Leonardo of Pisa ~ means son of Bonaccio ~ Grew up with a North African education under the Moors - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Discovering Fibonacci

Discovering Fibonacci

By:William Page

Pikes Peak Community College

Page 2: Discovering Fibonacci

Who Was Fibonacci?

~ Born in Pisa, Italy in 1175 AD~ Full name was Leonardo of Pisa~ means son of Bonaccio~ Grew up with a North African education under the Moors~ Traveled extensively around the Mediterranean coast~ Met with many merchants and learned their systems of arithmetic~ Realized the advantages of the Hindu-Arabic system

Page 3: Discovering Fibonacci

Fibonacci’s Mathematical Contributions

~ Introduced the Hindu-Arabic number system into Europe~ Based on ten digits and a decimal point~ Europe previously used the Roman number system~ Consisted of Roman numerals~ Persuaded mathematicians to use the Hindu-Arabic number system

1 2 3 4 5 6 7 8 9 0 and .

I = 1

V = 5

X = 10

L = 50

C = 100

D = 500

M = 1000

Page 4: Discovering Fibonacci

Fibonacci’s Mathematical Contributions Continued

~ Wrote five mathematical works~ Four books and one preserved letter~ Liber Abbaci (The Book of Calculating) written in 1202~ Practica geometriae (Practical Geometry) written in 1220~ Flos written in 1225~ Liber quadratorum (The Book of Squares) written in 1225~ A letter to Master Theodorus written around 1225

Page 5: Discovering Fibonacci

Fibonacci’s Rabbits

Problem:Suppose a newly-born pair of rabbits (one male, one female) are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month, a female can produce another pair of rabbits. Suppose that the rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?

Page 6: Discovering Fibonacci

The Fibonacci Numbers

~ Were introduced in The Book of Calculating~ Series begins with 0 and 1~ Next number is found by adding the last two numbers together~ Number obtained is the next number in the series~ A linear recurrence formula

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

Fn = Fn - 1 + Fn-2

Page 7: Discovering Fibonacci

Fibonacci’s Rabbits Continued

~ End of the first month = 1 pair~ End of the second month = 2 pair~ End of the third month = 3 pair~ End of the fourth month = 5 pair~ 243 pairs of rabbits produced in one year

1, 1, 2, 3, 5, 8, 13, 21, 34, …

Page 8: Discovering Fibonacci

The Fibonacci Numbers in Pascal’s Triangle

~ Entry is sum of the two numbers either side of it, but in the row above~ Diagonal sums in Pascal’s Triangle are the Fibonacci numbers~ Fibonacci numbers can also be found using a formula

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

Fib(n) =n – k

k – 1

n

k =1

( )1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

Page 9: Discovering Fibonacci
Page 10: Discovering Fibonacci

The Fibonacci Numbers and Pythagorean Triangles

a b a + b a + 2b

1 2 3 5

~ The Fibonacci numbers can be used to generate Pythagorean triangles

~ First side of Pythagorean triangle = 12

~ Second side of Pythagorean triangle = 5

~ Third side of Pythagorean triangle = 13

~ Fibonacci numbers 1, 2, 3, 5 produce Pythagorean triangle 5, 12, 13

Page 11: Discovering Fibonacci

The Golden Section and The Fibonacci Numbers

~ The Fibonacci numbers arise from the golden section~ The graph shows a line whose gradient is Phi~ First point close to the line is (0, 1)~ Second point close to the line is (1, 2)~ Third point close to the line is (2, 3)~ Fourth point close to the line is (3, 5)~ The coordinates are successive Fibonacci numbers

Page 12: Discovering Fibonacci

The Fibonacci Numbers in Nature

Pinecones clearly show the Fibonacci spiral 5 and 8, 8 and 13

Page 13: Discovering Fibonacci
Page 14: Discovering Fibonacci

A pineapple may have 5, 8, or 13 segment spirals depending on direction

Page 15: Discovering Fibonacci
Page 16: Discovering Fibonacci

Daisies have 21 and 34 spirals

Page 17: Discovering Fibonacci

Coneflower 55 spirals

Page 18: Discovering Fibonacci

Sunflower 55 and 89 spirals

Page 19: Discovering Fibonacci

Pine Cones, Pineapples, etc.

How many spirals are there on a

Pine cone ? 5 and 8 , 8 and 13

Pineapple ? 8 and 13

Daisy ? 21 and 34

Sunflower ? 55 and 89

Page 20: Discovering Fibonacci

Put these numbers together and you get:

5…8…13…21…34…55…89

Is there a pattern here?

Page 21: Discovering Fibonacci

The Fibonacci Numbers in Nature Continued

~ Sneezewort (Achillea ptarmica) shows the Fibonacci numbers

Page 22: Discovering Fibonacci
Page 23: Discovering Fibonacci

The Fibonacci Numbers in Nature Number of petals in flowers

Lilies and irises = 3 petals

Black-eyed Susan’s = 21 petalsCorn marigolds = 13 petals

Buttercups and wild roses = 5 petals

Delphiniums 8 petalsAsters 21 petals

Very few plants show 4 petals which is not a Fibonacci number!

Page 24: Discovering Fibonacci

What are the ratios of consecutive Fibs?1/1 = 1 2/1 = 2 3/2 = 1.55/3 = 1.667 8/5 = 1.6 13/8 = 1.62521/13 = 1.615 34/21 = 1.619 55/34 = 1.61789/55 = 1.61818 144/89 = 1.61798233/144 = 1.61806 377/233 = 1.618026610/377 = 1.618037 987/610 = 1.6180328

Page 25: Discovering Fibonacci

The Golden Section and The Fibonacci Numbers Continued

~ The golden section arises from the Fibonacci numbers~ Obtained by taking the ratio of successive terms in the Fibonacci series~ Limit is the positive root of a quadratic equation and is called the golden section

Page 26: Discovering Fibonacci

The Golden Section

~ Represented by the Greek letter Phi ~ Phi equals ± 1.6180339887 … and ± 0.6180339887 …~ Ratio of Phi is 1 : 1.618 or 0.618 : 1~ Mathematical definition is Phi2 = Phi + 1~ Euclid showed how to find the golden section of a line

b a

b + a

Page 27: Discovering Fibonacci

b a

a + b

The Golden Rectangle

Page 28: Discovering Fibonacci

A METHOD TO CONSTRUCT A GOLDEN RECTANGLE

Construct a square 1 x 1 ( in red)

Draw a line from the midpoint to the upperopposite corner.

Use that line as the radius to draw an arc that defines the height of the rectangle.

The resulting dimensions are the Golden Rectangle.

Page 29: Discovering Fibonacci

The Fibonacci Numbers in Nature Continued

~ Plants show the Fibonacci numbers in the arrangements of their leaves~ Three clockwise rotations, passing five leaves~ Two counter-clockwise rotations

Page 30: Discovering Fibonacci

The Golden Section in Nature~ Arrangements of leaves are the same as for seeds and petals~ All are placed at 0.618 per turn~ Is 0.618 of 360o which is 222.5o

~ One sees the smaller angle of 137.5o

~ Plants seem to produce their leaves, petals, and seeds based upon the golden section

Page 31: Discovering Fibonacci

Placement of seeds and leaves

• A single fixed angle for leaf or seed placement can produce an optimum design

• Provide best possible exposure for light, rainfall, exposure for insects for pollination

• Fibonacci numbers occur when counting the number of turns around the stem from a leaf to the next one directly above it as well as counting leaves till we meet another one directly above the starting leaf.

• Phi 1.618 leaves per turn or 0.618 turns per leaf 0.618 x 360 = 222.5 degrees or (1-0.618) x 360 = 137.5 degrees

Page 32: Discovering Fibonacci

The Fibonacci Numbers in Nature Continued

~ The Fibonacci numbers can be found in the human hand and fingers~ Person has 2 hands, which contain 5 fingers~ Each finger has 3 parts separated by 2 knuckles

Page 33: Discovering Fibonacci

At one time it was thought that many people have a ratio between the largest to middle bone, and the middle to the shortest finger bone of 1.618.

This is actually the case for only 1 in 12 people.

Page 34: Discovering Fibonacci
Page 35: Discovering Fibonacci

Leonardo Da Vinci self portrait

Page 36: Discovering Fibonacci

The Vetruvian Man

“Man of Action”

Full of Golden Rectangles:Head, Torso, Legs

Page 37: Discovering Fibonacci

Ratio of Distance fromfeet – midtorso- headIs 1.618

Page 38: Discovering Fibonacci
Page 39: Discovering Fibonacci
Page 40: Discovering Fibonacci
Page 41: Discovering Fibonacci

Madonna of the Rocks

This artwork uses the “Golden Triangle”

A Golden triangle is an isosceles triangle In which the smaller side( base) is in golden ratio ( 1.618) with its adjacent side.

Page 42: Discovering Fibonacci

Notre Dame Cathedral

Paris

Page 43: Discovering Fibonacci

The Golden Section in Architecture

~ Golden section appears in many of the proportions of the Parthenon in Greece~ Front elevation is built on the golden section (0.618 times as wide as it is tall)

Page 44: Discovering Fibonacci
Page 45: Discovering Fibonacci

The Golden Section in Architecture Continued

~ Golden section can be found in the Great pyramid in Egypt~ Perimeter of the pyramid, divided by twice its vertical height is the value of Phi

Page 46: Discovering Fibonacci
Page 47: Discovering Fibonacci
Page 48: Discovering Fibonacci

A METHOD TO CONSTRUCT A GOLDEN RECTANGLE

Construct a square 1 x 1 ( in red)

Draw a line from the midpoint to the upperopposite corner.

Use that line as the radius to draw an arc that defines the height of the rectangle.

The resulting dimensions are the Golden Rectangle.

Page 49: Discovering Fibonacci

As the dimensions get larger and larger, the quotients converge to the Golden ration 1.61803

Page 50: Discovering Fibonacci

Golden Spiral

Page 51: Discovering Fibonacci

The Fibonacci Numbers in Nature

~ Fibonacci spiral found in both snail and sea shells

Page 52: Discovering Fibonacci
Page 53: Discovering Fibonacci
Page 54: Discovering Fibonacci
Page 55: Discovering Fibonacci
Page 56: Discovering Fibonacci
Page 57: Discovering Fibonacci
Page 58: Discovering Fibonacci
Page 59: Discovering Fibonacci
Page 60: Discovering Fibonacci
Page 61: Discovering Fibonacci

Phi as a continued fraction

So …………

Phi is a famous irrational number that can be defined in terms of itself. It is a continued fraction.

Continued fractions were developed or discovered in part as a response to a need to approximate irrational numbers.

Page 62: Discovering Fibonacci

Let’s list 10 Fibonacci numbers in a column

I wont watch

When you get to the 10ht one tell me

I will turn around and you will start to calculate the sum using a calculator

I will get the sum faster than you will!!

Page 63: Discovering Fibonacci

Quick calculation of sum of 10 Fibonacci Numbers

• a• b The sum of 10 Fibonacci numbers is • a + b 55a + 88b. The 7th sum is 5a + 8b. By• a + 2b multiplying the 7th sum by 11, you get the• 2a + 3b total sum for the column of 10 numbers.• 3a + 5b• 5a + 8b• 8a + 13b• 13a + 21b• 21a + 34b 55a + 88 b

Page 64: Discovering Fibonacci

THE END, THANKS!

“Without mathematics there is no art” Luca Pacioli

[email protected]