Text of Geometric Representation of Fibonacci Sequence
2. 3. Mr. Suwat Sriyotee Ms. Rangsima Sairuttanatongkum 4.
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
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
To study the relationship between the new representations and
the golden spiral.
The Geometers Sketchpad
Microsoft Office Excel 2003
Try Square and Straight Edge
Graphing paper sheet
The representations are first designed and drawn on the
graphing paper, using geometric method; translation, reflection,
Calculate and search for the relationship with the golden
Construct the representations in The Geometers Sketchpad.
Conclude the result of the study.
The golden ratio is an irrational number of the form which is
It is also the answer to the quadratic equation
Leading to the following properties
11. Golden Rectangle Inflation of Golden Rectangle 12. Pentagon
with 1 unit side length Golden Triangle 13. Golden Spiral inscribed
in golden rectangle and golden triangle 14.
Fibonacci sequence has a recursive relation of the form
The sequence is as follow
1, 1, 2, 3, 5, 8, 13,
15. 7 8 1.6000 8 13 1.6250 9 21 1.6153 10 34 1.6190 11 55 1.6176
12 89 1.6181 13 144 1.6179 14 233 1.6180 15 377 1.6180 16. 17. The
diagram is constructed by using squares whose sizes correspond with
each terms of Fibonacci sequence. 18. The diagram can be inscribed
with a spiral. This spiral is called Fibonacci Spiral. 19. 20. 21.
The coordinate of D is 22. Point Polar Coordinate A B C D E F G 23.
24. 2 1 1 3 2 2 4 3 3 5 5 4 6 8 5 25. edge length 2,3 1 3,4 1 4,5 2
5,6 3 26. 27. The coordinate of D is 28. 29. Starting from AHB and
CHB whose side lengths are1unit, other triangles are created on the
basis of the two triangles constructed before them. 30. 31.
Calculating in the same manner as the first representation, the
relation with the spiral is as seen. 32. Each triangles size
correspond to each terms of Fibonacci sequence. Starting at BCD and
BCA the representation is constructed. 33. 34. No relation is found
between the representation and the spiral. 35. 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. 36. 37.
Calculate the coordinate of each reference points on the
representation in the same manner as the former representations.
38. 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. 39. 40. 41. 42. 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. 43.
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. 44. 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. 45. 46. Dunlap, Richard A. (1997).The Golden Ratio
andFibonacci Numbers . 5 thedition .Singapore: WorldPublishing Co.
Pte. Ltd. Smith, Robert T. (2006).Calculus: Concepts
&Connections . New York, NY.McGraw-HillPublishing Companiess,
Inc. Maxfield, J. E. & Maxfield, M. W. (1972).Discoveringnumber
theory . Philadelphia, PA: W. B. Saunders Co. Gardner, M.
(1961).The second scientific American bookof mathematical
puzzlesand diversions . New York,NY: Simon and Schuster. 47.
Freitag, Mark.Phi: That Golden Number [Online].
rietag.Mark/Homepag e/Goldenratio/ggoldenrati.html. (2000) ERBAS,
Ayhan K.Spira Mirabilis[Online]. Department ofMath