RepresentacióN De La Secuencia GeoméTrica De Fibonacci

Karn Udomsangpetch

Mahidol Wittayanusorn SchoolAcademic Year 2007

Geometric Representations of Fibonacci SequenceAnd Their Relation with the Golden Spiral

Mathematical Department

Project Mentors

Mr. Suwat Sriyotee

Ms. Rangsima Sairuttanatongkum

Fibonacci sequence has been the center of study for mathematicians worldwide for over the centuries. It possesses various properties, algebraically and geometrically.

This project aims at extending the knowledge regarding the geometric representation of the sequence by using other geometric shape; equilateral triangle, right triangle, square, pentagon, and hexagon.

Introduction

Moreover, the relationship between the newly created representations and the golden spiral, in which is related to the former representation, will also be studied.

The former representation has led to the discovery of new properties of Fibonacci sequence. It will be of highest honor should these new representations of this project pave a new route for others in unearthing new knowledge about the sequence.

Introduction

Objective

To construct new representations for Fibonacci sequence by using equilateral triangle, right triangle, square, pentagon, and hexagon respectively.

To study the relationship between the new representations and the golden spiral.

The Geometer’s Sketchpad Microsoft Office Excel 2003 Compass Try Square and Straight Edge Graphing paper sheet

Programs and Equipments

The representations are first designed and drawn on the graphing paper, using geometric method; translation, reflection, and rotation.

Calculate and search for the relationship with the golden spiral.

Construct the representations in The Geometer’s Sketchpad.

Conclude the result of the study.

Methods

The golden ratio is an irrational number of the form which is about 1.61803

It is also the answer to the quadratic equation

Leading to the following properties

Golden Ratio

1 5

2

1 1 1 1

1

2 1 0x x

Golden Ratio in GeometryC

A B

D

AB

BC

1

1

1

1 1

21

Golden Rectangle

Inflation of Golden Rectangle

1

1

11

1

1

1

11

1

Pentagon with 1 unit side length Golden Triangle

Golden Ratio in Geometry

Golden Spiral in Golden Rectangle and Triangle

Golden Spiral inscribed in golden rectangle and golden triangle

Fibonacci sequence has a recursive relation of the form

when and

The sequence is as follow

1, 1, 2, 3, 5, 8, 13, …

Fibonacci Sequence

2 1n n nF F F 1n 1 2 1F F

7 8 1.60008 13 1.62509 21 1.6153

10 34 1.619011 55 1.617612 89 1.618113 144 1.617914 233 1.618015 377 1.6180

Ratio between Successive Terms

n nF 1n nF F

1n nF F

n1110987654

1.7

1.6

1.5

1

lim n

nn

F

F

Ratio between Successive Terms

Whirling Rectangles Diagram

2

1

2

1

3

The diagram is constructed by using squares whose sizes correspond with each terms of Fibonacci sequence.

Whirling Rectangles Diagram

The diagram can be inscribed with a spiral. This spiral is called “Fibonacci Spiral”.

Fibonacci Spiral on Polar Coordinate

10 5 5

6

4

2

2

2

2

OE

D

B

C A

1,

2,

4

OA OB

AD

BE

2 2

2 21 2

5 D

OD OA AD

r

ˆtan 2

ˆ arctan 2

arctan 2D

ADAOD

AO

AOD

The coordinate of D is 5, arctan 2


Point Polar CoordinateA

B

C

D

E

F

G

,r

1,0

1, 2

1,

5, arctan 2

117,arctan 4

137,arctan( )5 2

285,arctan( )9


J

H

F

A

I G

E

DC

B

K

Geometric Representation from Squares

J

H

F

A

I G

E

DC

B

K

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2 1 13 2 24 3 35 5 46 8 5

nnF

2

2 2

3 2

5 2

8 2


edge length2,3 13,4 14,5 25,6 3

, 1n n 1n nF F

AE 2

FG 2

HI 2 2

JK 3 2

J

H

F

A

I G

E

DC

B

K


6

4

2

2

4

10 5 5

F

E

D

A

B

C

G


2

2

O

E

D

A

B

C

1OA

2AD

2 2

2 21 2

5 D

OD OA AD

r

2ˆtan1

ˆ arctan 2

arctan 2D

ADAOD

AO

AOD


The coordinate of D is 5, arctan 2


Geometric Representation from Right Triangles

H

F

E

D

A

B

C

G

IJ

K

Starting from AHB and CHB whose side lengths are 1 unit, other triangles are created on the basis of the two triangles constructed before them.

H A

B

CH

D

A

B

C

I

H

E

D

A

B

C

I

H

F

E

D

A

B

C

IJ

K

H

F

E

D

A

B

C

G


Calculating in the same manner as the first representation, the relation with the spiral is as seen.


I

H

F

G

E

JD

A

CB

L K

Each triangles’ size correspond to each terms of Fibonacci sequence. Starting at BCD and BCA the representation is constructed.

Geometric Representation from Equilateral Triangles

JD

A

CB

G

E

JD

A

CB

H

F

G

E

JD

A

CB

I

H

F

G

E

JD

A

CB

K

I

H

F

G

E

JD

A

CB

L K


No relation is found between the representation and the spiral.


Geometric Representation from Pentagons

The construction starts with two 2 one-unit-pentagons. The representation whirls off in an anticlockwise direction.

Each of the pentagons’ sizes correspond with each terms of the Fibonacci sequence.


Calculate the coordinate of each reference points on the representation in the same manner as the former representations.


Geometric Representation from Hexagons

The construction starts with two 2 one-unit-hexagons. The representation whirls off in an anticlockwise direction.

Each of the hexagons’ sizes correspond with each terms of the Fibonacci sequence.

1H

D

FA

I

G

B

C

E

1H

LD

FAK

JI

G

B

C

E

2

1H

M

N

LD

FAK

JI

G

B

C

E

2

1H

M

N

LD

FAK

JI

G

B

C

E

3

2

1H

M

N

LD

FAK

JI

G

B

C

E



Conclusion

From the experiment, it is found that squares, right triangles, equilateral triangles, pentagons, and hexagons can all be used to construct geometric representations of Fibonacci sequence with side lengths corresponding to each terms of the sequence. However, only the representations from squares and right triangles possess relationship with the golden spiral.

DiscussionAlthough all the representations can be

successfully constructed, the processes are far more complicated than that of the whirling rectangle diagram. Moreover the relationship with the golden spiral is far less obvious than the former diagram.

The reason for the representations which share no relation with the spiral is that their turning angles are not 90 degree, while that of the spiral is exactly 90.

SuggestionThis project can be extended in order to find a

generalized method in constructing the geometric representation of Fibonacci sequence for any n-gons shape. The representation from octagon has been constructed with slight error in the process as in the figure.

Suggestion

Reference

Dunlap, Richard A. (1997). The Golden Ratio and Fibonacci Numbers. 5th edition. Singapore: World Publishing Co. Pte. Ltd.Smith, Robert T. (2006). Calculus: Concepts & Connections. New York, NY. McGraw-Hill Publishing Companiess, Inc.Maxfield, J. E. & Maxfield, M. W. (1972). Discovering number theory. Philadelphia, PA: W. B. Saunders Co.Gardner, M. (1961). The second scientific American book of mathematical puzzles and diversions. New York, NY: Simon and Schuster.

Freitag, Mark. Phi: That Golden Number[Online]. Available http://jwilson.coe.uga.edu/EMT669/Student.Folders/Frietag.Mark/Homepag e/Goldenratio/

ggoldenrati.html. (2000)ERBAS, Ayhan K. Spira Mirabilis [Online]. Department of

Math Education: University of Georgia. http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeome trypro/golden%20spiral/llogspira-history.html

Reference