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1. Mathematical Department Geometric Representations of Fibonacci Sequence And Their Relation with the Golden Spiral KarnUdomsangpetch MahidolWittayanusorn School Academic Year 2007

2. Project Mentors Mr. SuwatSriyotee Ms. RangsimaSairuttanatongkum

3. Introduction 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.

4. 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.

5. 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.

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

7. Methods 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.

8. Review Literature

9. Golden Ratio 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

10. Golden Ratio in Geometry Golden Rectangle Inflation of Golden Rectangle

11. Golden Ratio in Geometry Pentagon with 1 unit side length Golden Triangle

12. Golden Spiral in Golden Rectangle and Triangle Golden Spiral inscribed in golden rectangle and golden triangle

13. Fibonacci Sequence Fibonacci sequence has a recursive relation of the form when and The sequence is as follow 1, 1, 2, 3, 5, 8, 13, …

14. Ratio between Successive Terms

15. Ratio between Successive Terms

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

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

18. Result

19. Fibonacci Spiral on Polar Coordinate

20. Fibonacci Spiral on Polar Coordinate The coordinate of D is

21. Fibonacci Spiral on Polar Coordinate

22. Geometric Representation from Squares

23. Geometric Representation from Squares

24. Geometric Representation from Squares

25. Geometric Representation from Squares

26. Geometric Representation from Squares The coordinate of D is

27. Geometric Representation from Squares

28. Geometric Representation from Right Triangles 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.

29. Geometric Representation from Right Triangles

30. Geometric Representation from Right Triangles Calculating in the same manner as the first representation, the relation with the spiral is as seen.

31. Geometric Representation from Equilateral Triangles Each triangles’ size correspond to each terms of Fibonacci sequence. Starting at BCD and BCA the representation is constructed.

32. Geometric Representation from Equilateral Triangles

33. Geometric Representation from Equilateral Triangles No relation is found between the representation and the spiral.

34. 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.

35. Geometric Representation from Pentagons

36. Geometric Representation from Pentagons Calculate the coordinate of each reference points on the representation in the same manner as the former representations.

37. 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.

38. Geometric Representation from Hexagons

39. Geometric Representation from Hexagons

40. Conclusion and Discussion

41. 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.

42. Discussion 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.

43. Suggestion 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.

44. Suggestion

45. 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.

46. Reference Freitag, Mark. Phi: That Golden Number[Online]. Available http://jwilson.coe.uga.edu/EMT669/Student.Folders/F rietag.Mark/Homepage/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/E rbas/KURSATgeometrypro/golden%20spiral/llogspira-history.html