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Fibonacci Spiral II

Fibonacci Spiral II
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2016-04-25 03:27
CirclePolygonFunctionSequenceGeometric Construction
In collections Regularity
This example demonstrates the construction of a Fibonacci spiral using the Fibonacci sequence, circles, and polygons. It involves key mathematical concepts like sequences, circles, tangents, circumcircles, and geometric constructions. The construction uses commands such as Point, Sequence, Flatten, Join, Intersect, CircleArc, Polygon, Last, and Take in GeoGebra. Suitable for high school mathematics education or exploration by math enthusiasts.

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
  56. Sequence(Midpoint[Element[lista33, i], Element[lista33, i + 1]], i, 1, n + 4) → lista40
  57. Sequence(Rotate[Translate[Dilate[imagen1, Element[lista34, i + 1], Element[lista40, 1]], Vector[Element[lista40, 1], Element[lista40, i + 1]]], (0*° * i), Element[lista40, i + 1]], i, 1, n + 4) → lista41

AI Drawing Prompt Reference

Draw a Fibonacci spiral starting from four points on the axes, C, F, H, and L. Use the Fibonacci sequence to generate a series of points for concentric arcs, forming the spiral shape. The spiral is composed of a series of polygons, with each polygon's vertices being the endpoints of the arcs. The spiral starts at (3,5) and the radius of each arc increases gradually, forming the spiral shape.
The prompt above was compiled from the original drawing commands and is for reference only.