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Visualizing the Fibonacci Sequence

Visualizing the Fibonacci Sequence
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2018-11-27 11:39
TriangleSequenceCoordinate GeometryGeometric ConstructionGraphing
In collections Graphs of Sequences
This example dynamically demonstrates the Fibonacci sequence's geometric distribution, involving concepts like equilateral triangles, centroids, integral calculus, and right triangles. It utilizes GeoGebra commands such as Mod, If, Length, CurveCartesian, and Element. Ideal for high school and above, exploring the connection between sequences and geometry through dynamic visualization.

GeoGebra Commands & Parameters

  1. Mod(n, 2) → test
  2. If(test ≟ 1, (2 * π), π) → umax
  3. If(test ≟ 1, π, 0) → umin
  4. Length(Seq) → lng
  5. CurveCartesian((radArc * cos(u)) + mid, (radArc * sin(u)), u, umin, umax) → curveSeq
  6. If(n < 5, Element[Seq, 4], Element[Seq, n]) → lastElement
  7. Element(Seq, n) → n1
  8. Element(Seq, n + 1) → n2
  9. CurveCartesian((abs((Element[Seq, 2] - Element[Seq, 1]) / 2) * cos(u)) + (Element[Seq, 2] + Element[Seq, 1]) / 2, (abs((Element[Seq, 2] - Element[Seq, 1]) / 2) * sin(u)), u, If[Mod[Element[Seq, 2], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 2], 2] ≟ 1, (2 * π), π]) → B1
  10. CurveCartesian((abs((Element[Seq, 3] - Element[Seq, 2]) / 2) * cos(u)) + (Element[Seq, 3] + Element[Seq, 2]) / 2, (abs((Element[Seq, 3] - Element[Seq, 2]) / 2) * sin(u)), u, If[Mod[Element[Seq, 3], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 3], 2] ≟ 1, (2 * π), π]) → B2
  11. CurveCartesian((abs((Element[Seq, 4] - Element[Seq, 3]) / 2) * cos(u)) + (Element[Seq, 4] + Element[Seq, 3]) / 2, (abs((Element[Seq, 4] - Element[Seq, 3]) / 2) * sin(u)), u, If[Mod[Element[Seq, 4], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 4], 2] ≟ 1, (2 * π), π]) → B3
  12. CurveCartesian((abs((Element[Seq, 5] - Element[Seq, 4]) / 2) * cos(u)) + (Element[Seq, 5] + Element[Seq, 4]) / 2, (abs((Element[Seq, 5] - Element[Seq, 4]) / 2) * sin(u)), u, If[Mod[Element[Seq, 5], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 5], 2] ≟ 1, (2 * π), π]) → B4
  13. CurveCartesian((abs((Element[Seq, 6] - Element[Seq, 5]) / 2) * cos(u)) + (Element[Seq, 6] + Element[Seq, 5]) / 2, (abs((Element[Seq, 6] - Element[Seq, 5]) / 2) * sin(u)), u, If[Mod[Element[Seq, 6], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 6], 2] ≟ 1, (2 * π), π]) → B5
  14. CurveCartesian((abs((Element[Seq, 7] - Element[Seq, 6]) / 2) * cos(u)) + (Element[Seq, 7] + Element[Seq, 6]) / 2, (abs((Element[Seq, 7] - Element[Seq, 6]) / 2) * sin(u)), u, If[Mod[Element[Seq, 7], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 7], 2] ≟ 1, (2 * π), π]) → B6
  15. CurveCartesian((abs((Element[Seq, 8] - Element[Seq, 7]) / 2) * cos(u)) + (Element[Seq, 8] + Element[Seq, 7]) / 2, (abs((Element[Seq, 8] - Element[Seq, 7]) / 2) * sin(u)), u, If[Mod[Element[Seq, 8], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 8], 2] ≟ 1, (2 * π), π]) → B7
  16. CurveCartesian((abs((Element[Seq, 9] - Element[Seq, 8]) / 2) * cos(u)) + (Element[Seq, 9] + Element[Seq, 8]) / 2, (abs((Element[Seq, 9] - Element[Seq, 8]) / 2) * sin(u)), u, If[Mod[Element[Seq, 9], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 9], 2] ≟ 1, (2 * π), π]) → B8
  17. CurveCartesian((abs((Element[Seq, 10] - Element[Seq, 9]) / 2) * cos(u)) + (Element[Seq, 10] + Element[Seq, 9]) / 2, (abs((Element[Seq, 10] - Element[Seq, 9]) / 2) * sin(u)), u, If[Mod[Element[Seq, 10], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 10], 2] ≟ 1, (2 * π), π]) → B9
  18. CurveCartesian((abs((Element[Seq, 11] - Element[Seq, 10]) / 2) * cos(u)) + (Element[Seq, 11] + Element[Seq, 10]) / 2, (abs((Element[Seq, 11] - Element[Seq, 10]) / 2) * sin(u)), u, If[Mod[Element[Seq, 11], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 11], 2] ≟ 1, (2 * π), π]) → B10
  19. CurveCartesian((abs((Element[Seq, 12] - Element[Seq, 11]) / 2) * cos(u)) + (Element[Seq, 12] + Element[Seq, 11]) / 2, (abs((Element[Seq, 12] - Element[Seq, 11]) / 2) * sin(u)), u, If[Mod[Element[Seq, 12], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 12], 2] ≟ 1, (2 * π), π]) → B11
  20. CurveCartesian((abs((Element[Seq, 13] - Element[Seq, 12]) / 2) * cos(u)) + (Element[Seq, 13] + Element[Seq, 12]) / 2, (abs((Element[Seq, 13] - Element[Seq, 12]) / 2) * sin(u)), u, If[Mod[Element[Seq, 13], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 13], 2] ≟ 1, (2 * π), π]) → B12
  21. CurveCartesian((abs((Element[Seq, 14] - Element[Seq, 13]) / 2) * cos(u)) + (Element[Seq, 14] + Element[Seq, 13]) / 2, (abs((Element[Seq, 14] - Element[Seq, 13]) / 2) * sin(u)), u, If[Mod[Element[Seq, 14], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 14], 2] ≟ 1, (2 * π), π]) → B13
  22. CurveCartesian((abs((Element[Seq, 15] - Element[Seq, 14]) / 2) * cos(u)) + (Element[Seq, 15] + Element[Seq, 14]) / 2, (abs((Element[Seq, 15] - Element[Seq, 14]) / 2) * sin(u)), u, If[Mod[Element[Seq, 15], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 15], 2] ≟ 1, (2 * π), π]) → B14
  23. CurveCartesian((abs((Element[Seq, 16] - Element[Seq, 15]) / 2) * cos(u)) + (Element[Seq, 16] + Element[Seq, 15]) / 2, (abs((Element[Seq, 16] - Element[Seq, 15]) / 2) * sin(u)), u, If[Mod[Element[Seq, 16], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 16], 2] ≟ 1, (2 * π), π]) → B15
  24. CurveCartesian((abs((Element[Seq, 17] - Element[Seq, 16]) / 2) * cos(u)) + (Element[Seq, 17] + Element[Seq, 16]) / 2, (abs((Element[Seq, 17] - Element[Seq, 16]) / 2) * sin(u)), u, If[Mod[Element[Seq, 17], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 17], 2] ≟ 1, (2 * π), π]) → B16
  25. CurveCartesian((abs((Element[Seq, 18] - Element[Seq, 17]) / 2) * cos(u)) + (Element[Seq, 18] + Element[Seq, 17]) / 2, (abs((Element[Seq, 18] - Element[Seq, 17]) / 2) * sin(u)), u, If[Mod[Element[Seq, 18], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 18], 2] ≟ 1, (2 * π), π]) → B17
  26. CurveCartesian((abs((Element[Seq, 19] - Element[Seq, 18]) / 2) * cos(u)) + (Element[Seq, 19] + Element[Seq, 18]) / 2, (abs((Element[Seq, 19] - Element[Seq, 18]) / 2) * sin(u)), u, If[Mod[Element[Seq, 19], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 19], 2] ≟ 1, (2 * π), π]) → B18
  27. CurveCartesian((abs((Element[Seq, 20] - Element[Seq, 19]) / 2) * cos(u)) + (Element[Seq, 20] + Element[Seq, 19]) / 2, (abs((Element[Seq, 20] - Element[Seq, 19]) / 2) * sin(u)), u, If[Mod[Element[Seq, 20], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 20], 2] ≟ 1, (2 * π), π]) → B19
  28. CurveCartesian((abs((Element[Seq, 21] - Element[Seq, 20]) / 2) * cos(u)) + (Element[Seq, 21] + Element[Seq, 20]) / 2, (abs((Element[Seq, 21] - Element[Seq, 20]) / 2) * sin(u)), u, If[Mod[Element[Seq, 21], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 21], 2] ≟ 1, (2 * π), π]) → B20
  29. CurveCartesian((abs((Element[Seq, 22] - Element[Seq, 21]) / 2) * cos(u)) + (Element[Seq, 22] + Element[Seq, 21]) / 2, (abs((Element[Seq, 22] - Element[Seq, 21]) / 2) * sin(u)), u, If[Mod[Element[Seq, 22], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 22], 2] ≟ 1, (2 * π), π]) → B21
  30. CurveCartesian((abs((Element[Seq, 23] - Element[Seq, 22]) / 2) * cos(u)) + (Element[Seq, 23] + Element[Seq, 22]) / 2, (abs((Element[Seq, 23] - Element[Seq, 22]) / 2) * sin(u)), u, If[Mod[Element[Seq, 23], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 23], 2] ≟ 1, (2 * π), π]) → B22
  31. CurveCartesian((abs((Element[Seq, 24] - Element[Seq, 23]) / 2) * cos(u)) + (Element[Seq, 24] + Element[Seq, 23]) / 2, (abs((Element[Seq, 24] - Element[Seq, 23]) / 2) * sin(u)), u, If[Mod[Element[Seq, 24], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 24], 2] ≟ 1, (2 * π), π]) → B23
  32. CurveCartesian((abs((Element[Seq, 25] - Element[Seq, 24]) / 2) * cos(u)) + (Element[Seq, 25] + Element[Seq, 24]) / 2, (abs((Element[Seq, 25] - Element[Seq, 24]) / 2) * sin(u)), u, If[Mod[Element[Seq, 25], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 25], 2] ≟ 1, (2 * π), π]) → B24
  33. CurveCartesian((abs((Element[Seq, 26] - Element[Seq, 25]) / 2) * cos(u)) + (Element[Seq, 26] + Element[Seq, 25]) / 2, (abs((Element[Seq, 26] - Element[Seq, 25]) / 2) * sin(u)), u, If[Mod[Element[Seq, 26], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 26], 2] ≟ 1, (2 * π), π]) → B25
  34. CurveCartesian((abs((Element[Seq, 27] - Element[Seq, 26]) / 2) * cos(u)) + (Element[Seq, 27] + Element[Seq, 26]) / 2, (abs((Element[Seq, 27] - Element[Seq, 26]) / 2) * sin(u)), u, If[Mod[Element[Seq, 27], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 27], 2] ≟ 1, (2 * π), π]) → B26
  35. CurveCartesian((abs((Element[Seq, 28] - Element[Seq, 27]) / 2) * cos(u)) + (Element[Seq, 28] + Element[Seq, 27]) / 2, (abs((Element[Seq, 28] - Element[Seq, 27]) / 2) * sin(u)), u, If[Mod[Element[Seq, 28], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 28], 2] ≟ 1, (2 * π), π]) → B27
  36. CurveCartesian((abs((Element[Seq, 29] - Element[Seq, 28]) / 2) * cos(u)) + (Element[Seq, 29] + Element[Seq, 28]) / 2, (abs((Element[Seq, 29] - Element[Seq, 28]) / 2) * sin(u)), u, If[Mod[Element[Seq, 29], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 29], 2] ≟ 1, (2 * π), π]) → B28
  37. CurveCartesian((abs((Element[Seq, 30] - Element[Seq, 29]) / 2) * cos(u)) + (Element[Seq, 30] + Element[Seq, 29]) / 2, (abs((Element[Seq, 30] - Element[Seq, 29]) / 2) * sin(u)), u, If[Mod[Element[Seq, 30], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 30], 2] ≟ 1, (2 * π), π]) → B29
  38. CurveCartesian((abs((Element[Seq, 31] - Element[Seq, 30]) / 2) * cos(u)) + (Element[Seq, 31] + Element[Seq, 30]) / 2, (abs((Element[Seq, 31] - Element[Seq, 30]) / 2) * sin(u)), u, If[Mod[Element[Seq, 31], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 31], 2] ≟ 1, (2 * π), π]) → B30
  39. CurveCartesian((abs((Element[Seq, 32] - Element[Seq, 31]) / 2) * cos(u)) + (Element[Seq, 32] + Element[Seq, 31]) / 2, (abs((Element[Seq, 32] - Element[Seq, 31]) / 2) * sin(u)), u, If[Mod[Element[Seq, 32], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 32], 2] ≟ 1, (2 * π), π]) → B31
  40. CurveCartesian((abs((Element[Seq, 33] - Element[Seq, 32]) / 2) * cos(u)) + (Element[Seq, 33] + Element[Seq, 32]) / 2, (abs((Element[Seq, 33] - Element[Seq, 32]) / 2) * sin(u)), u, If[Mod[Element[Seq, 33], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 33], 2] ≟ 1, (2 * π), π]) → B32
  41. CurveCartesian((abs((Element[Seq, 34] - Element[Seq, 33]) / 2) * cos(u)) + (Element[Seq, 34] + Element[Seq, 33]) / 2, (abs((Element[Seq, 34] - Element[Seq, 33]) / 2) * sin(u)), u, If[Mod[Element[Seq, 34], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 34], 2] ≟ 1, (2 * π), π]) → B33
  42. CurveCartesian((abs((Element[Seq, 35] - Element[Seq, 34]) / 2) * cos(u)) + (Element[Seq, 35] + Element[Seq, 34]) / 2, (abs((Element[Seq, 35] - Element[Seq, 34]) / 2) * sin(u)), u, If[Mod[Element[Seq, 35], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 35], 2] ≟ 1, (2 * π), π]) → B34
  43. CurveCartesian((abs((Element[Seq, 36] - Element[Seq, 35]) / 2) * cos(u)) + (Element[Seq, 36] + Element[Seq, 35]) / 2, (abs((Element[Seq, 36] - Element[Seq, 35]) / 2) * sin(u)), u, If[Mod[Element[Seq, 36], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 36], 2] ≟ 1, (2 * π), π]) → B35
  44. CurveCartesian((abs((Element[Seq, 37] - Element[Seq, 36]) / 2) * cos(u)) + (Element[Seq, 37] + Element[Seq, 36]) / 2, (abs((Element[Seq, 37] - Element[Seq, 36]) / 2) * sin(u)), u, If[Mod[Element[Seq, 37], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 37], 2] ≟ 1, (2 * π), π]) → B36
  45. CurveCartesian((abs((Element[Seq, 38] - Element[Seq, 37]) / 2) * cos(u)) + (Element[Seq, 38] + Element[Seq, 37]) / 2, (abs((Element[Seq, 38] - Element[Seq, 37]) / 2) * sin(u)), u, If[Mod[Element[Seq, 38], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 38], 2] ≟ 1, (2 * π), π]) → B37
  46. CurveCartesian((abs((Element[Seq, 39] - Element[Seq, 38]) / 2) * cos(u)) + (Element[Seq, 39] + Element[Seq, 38]) / 2, (abs((Element[Seq, 39] - Element[Seq, 38]) / 2) * sin(u)), u, If[Mod[Element[Seq, 39], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 39], 2] ≟ 1, (2 * π), π]) → B38
  47. CurveCartesian((abs((Element[Seq, 40] - Element[Seq, 39]) / 2) * cos(u)) + (Element[Seq, 40] + Element[Seq, 39]) / 2, (abs((Element[Seq, 40] - Element[Seq, 39]) / 2) * sin(u)), u, If[Mod[Element[Seq, 40], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 40], 2] ≟ 1, (2 * π), π]) → B39
  48. CurveCartesian((abs((Element[Seq, 41] - Element[Seq, 40]) / 2) * cos(u)) + (Element[Seq, 41] + Element[Seq, 40]) / 2, (abs((Element[Seq, 41] - Element[Seq, 40]) / 2) * sin(u)), u, If[Mod[Element[Seq, 41], 2] ≟ 1, π, 0], If[Mod[Element[Seq, 41], 2] ≟ 1, (2 * π), π]) → B40

AI Drawing Prompt Reference

Visualize the geometric distribution of the Fibonacci sequence by drawing a series of circular arcs based on its elements, connecting the midpoints of these arcs to form curves. Then, draw multiple triangles using adjacent elements, with each triangle's vertices at the corresponding points and the base connecting the midpoints of the sides. Combine these curves and triangles to create a dynamic illustration showcasing the Fibonacci sequence's geometric beauty.
The prompt above was compiled from the original drawing commands and is for reference only.