29. 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.
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 and Fibonacci Numbers . 5 th 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.
47. Freitag, Mark. Phi: That Golden Number [Online]. Available http://jwilson.coe.uga.edu/EMT669/Student.Folders/F rietag.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/E rbas/KURSATgeome trypro/golden%20spiral/llogspira-history.html
