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Fibonacci Spiral by Luis Miguel López Herranz

Fibonacci Spiral by Luis Miguel López Herranz
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2016-04-22 03:08
FunctionSequenceGeometric Construction
In collections Regularity
This example demonstrates the generation of a Fibonacci spiral, which is based on the Fibonacci sequence. Key mathematical concepts include sequences, the golden ratio, and the golden spiral. Construction highlights include the use of commands like Point, Sequence, and CircleArc. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Point(xAxis) → C
  2. Point(yAxis) → F
  3. Point(xAxis) → H
  4. Point(yAxis) → L
  5. Sequence(Element[lista1, 1 + (4 * i)], i, 0, n) → lista2
  6. Sequence(Element[lista1, 2 + (4 * i)], i, 0, n) → lista3
  7. Sequence(Element[lista1, 3 + (4 * i)], i, 0, n) → lista4
  8. Sequence(Element[lista1, 4 + (4 * i)], i, 0, n) → lista5
  9. Sequence((Element[lista2, i], 0), i, 1, n) → lista6
  10. Sequence((0, (-Element[lista2, i])), i, 1, n) → lista7
  11. Sequence((0, (-Element[lista3, i])), i, 1, n) → lista8
  12. Sequence(((-Element[lista3, i]), 0), i, 1, n) → lista9
  13. Sequence(((-Element[lista4, i]), 0), i, 1, n) → lista10
  14. Sequence((0, Element[lista4, i]), i, 1, n) → lista11
  15. Sequence((0, Element[lista5, i]), i, 1, n) → lista12
  16. Sequence((Element[lista5, i], 0), i, 1, n) → lista13
  17. Sequence({Element[lista6, i], Element[lista7, i], Element[lista8, i], Element[lista9, i], Element[lista10, i], Element[lista11, i], Element[lista12, i], Element[lista13, i]}, i, 1, n) → lista14
  18. Flatten(lista14) → lista15
  19. Sequence((3, 5) + Sum[Take[lista15, 1, i]], i, 1, (4 * n)) → lista16
  20. Join({{(3, 5)}, lista16}) → lista17
  21. Sequence((0, (-Element[lista18, 1 + (4 * i)])), i, 0, n) → lista19
  22. Sequence(((-Element[lista18, 2 + (4 * i)]), 0), i, 0, n) → lista20
  23. Sequence((0, Element[lista18, 3 + (4 * i)]), i, 0, n) → lista21
  24. Sequence((Element[lista18, 4 + (4 * i)], 0), i, 0, n) → lista22
  25. Flatten(Sequence[{Element[lista19, i], Element[lista20, i], Element[lista21, i], Element[lista22, i]}, i, 1, n]) → lista23
  26. Sequence((3, 2) + Sum[Take[lista23, 1, i]], i, 1, n) → lista24
  27. Sequence(Element[lista17, 1 + (2 * i)], i, 0, n) → lista26
  28. Sequence(Element[lista17, 3 + (2 * i)], i, 0, n) → lista27
  29. Sequence(CircleArc[Element[lista24, i], Element[lista27, i], Element[lista26, i]], i, 1, n) → lista25
  30. Sequence(Element[lista27, i] - Vector[Element[lista17, (2 * i) - 1], Element[lista17, (2 * i)]], i, 1, n) → lista28
  31. Sequence(Polygon[Append[Take[lista17, (2 * i) - 1, (2 * i) + 1], Element[lista28, i]]], i, 1, n) → lista29
  32. Point(yAxis) → O
  33. Point(yAxis) → P
  34. Point(xAxis) → S
  35. Intersect(xAxis, yAxis) → T
  36. CircleArc(R, P, S) → e
  37. CircleArc((3, 2), (3, 5), (0, 2)) → c
  38. Polygon((3, 2), (3, 5), (0, 5), (0, 2)) → polígono1, g, h, i, j
  39. Polygon(T, S, R, P) → polígono2, t, s, r, p
  40. CircleArc((2, 1), (2, 0), (3, 1)) → k
  41. Polygon((2, 0), (3, 0), (3, 1), (2, 1)) → polígono3, l, m, q, a_1
  42. CircleArc((2, 1), (3, 1), (2, 2)) → c_1
  43. Polygon((2, 1), (3, 1), (3, 2), (2, 2)) → polígono4, b_1, d_1, e_1, f_1
  44. Last(lista29) → lista30
  45. Polygon((3, 2), (3, 5), (0, 5), (0, 2)) → polígono6, g_1, h_1, i_1, j_1
  46. Polygon(T, S, R, P) → polígono7, t_1, s_1, r_1, p_1
  47. Polygon((2, 0), (3, 0), (3, 1), (2, 1)) → polígono8, k_1, l_1, m_1, n_1
  48. Polygon((2, 1), (3, 1), (3, 2), (2, 2)) → polígono9, q_1, a_2, b_2, c_2
  49. Join({lista32, lista26}) → lista33
  50. Join({{1, 1, 2, 3}, lista1}) → lista34
  51. Sequence(Text[Element[lista34, i], Midpoint[Element[lista33, i], Element[lista33, i + 1]]], i, 1, n + 4) → lista35
  52. Flatten(Append[{c_1, k, e, c}, lista25]) → lista36
  53. Flatten(Append[{polígono4, polígono3, polígono2, polígono1}, lista29]) → lista38
  54. Take(lista38, 1, n + 4 - 1 / 3) → lista39
  55. Take(lista36, 1, n + 4 - 2 / 3) → lista37