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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.
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.
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
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.
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
1 5
2
1 1 1 1
1
2 1 0x x
C
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 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, …
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
n nF 1n nF F
1n nF F
n1110987654
1.7
1.6
1.5
1
lim n
nn
F
F
2
1
2
1
3
The diagram is constructed by using squares whose sizes correspond with each terms of Fibonacci sequence.
The diagram can be inscribed with a spiral. This spiral is called “Fibonacci Spiral”.
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
J
H
F
A
I G
E
DC
B
K
รู�ปที่�� ความยาวด้ านสี่��เหลี่��ยมจั�ตุ�รู�สี่
2 1 1
3 2 2
4 3 3
5 5 4
6 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
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.
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.
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.
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
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.
Although 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.
This 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.
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