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Fibonacci Golden Ratio

Fibonacci Golden Ratio
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2018-10-12 01:56
SequenceProbability & StatisticsGeometric Construction
In collections Symmetry Explorations
This example demonstrates the dynamic change of the ratio between consecutive terms in the Fibonacci sequence and its convergence to the golden ratio, involving concepts like sequences, ratios, golden ratio, and limits. It uses commands like Vector, Line, Mirror, and Translate in GeoGebra for construction, suitable for high school mathematics teaching to understand the practical application of sequence limits and the golden ratio.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. Line(C, u) → a
  3. Mirror(D, a) → D'
  4. Translate(D', u) → D''
  5. Mirror(D'', a) → D'''
  6. Translate(D''', u) → E
  7. Mirror(A, B) → A'
  8. Vector(A, A') → w
  9. Translate(D'', w) → D'''_1
  10. Translate(E, w) → E'
  11. Translate(D'''_1, w) → F
  12. Translate(E', w) → E''
  13. Translate(F, w) → F'
  14. Translate(E'', w) → E'''
  15. Translate(F', w) → F''
  16. Translate(E''', w) → G
  17. Translate(F'', w) → F'''
  18. Translate(G, w) → G'
  19. Translate(F''', w) → H
  20. Translate(G', w) → G''
  21. Translate(H, w) → H'
  22. Translate(G'', w) → G'''
  23. Translate(H', w) → H''
  24. Translate(G''', w) → I
  25. Translate(H'', w) → H'''
  26. Translate(I, w) → I'
  27. Mirror(B, A) → B'
  28. Vector(B, B') → b
  29. Translate(D, b) → D'_1
  30. Translate(D'', b) → D'''_2
  31. Translate(D'_1, b) → J
  32. Translate(D'''_2, b) → K
  33. Translate(J, b) → J'
  34. Translate(K, b) → K'
  35. Translate(J', b) → J''
  36. Translate(K', b) → K''
  37. Translate(J'', b) → J'''
  38. Translate(K'', b) → K'''
  39. Translate(J''', b) → L
  40. Translate(K''', b) → M

AI Drawing Prompt Reference

Draw a graph showing the dynamic change of the ratio between consecutive terms in the Fibonacci sequence and its convergence to the golden ratio. Start with two points, create a vector, draw a line using it, then generate a series of points through mirror and translation operations, connecting them to form a curve that represents the Fibonacci sequence geometrically and demonstrates the convergence to the golden ratio.
The prompt above was compiled from the original drawing commands and is for reference only.