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Unfolding Golden Rectangle Fibonacci

Unfolding Golden Rectangle Fibonacci
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2016-05-24 05:09
FunctionSequenceTransformationGeometric Construction
This example demonstrates the geometric unfolding of a Golden Rectangle through semicircles, angle bisectors, intersections, reflections, and polygon constructions, visually linking the Fibonacci sequence and the Golden Ratio. Core mathematical concepts include the Golden Rectangle, Fibonacci sequence, logarithmic functions, and geometric transformations (rotation, translation, dilation). GeoGebra commands and objects used are Semicircle, LineBisector, Intersect, Segment, Mirror, Polygon, Vector, Translate, Dilate, Rotate, CircleArc, and objects like B, A, g, f, F, e, F', C, poly1, a, b, c, d, k, u, A', F'', B'_1, F'_1, poly1', a', f', b', f'_1, A'', F'''. Suitable for high school and above students to understand geometric construction techniques, the connection between the Golden Ratio and Fibonacci sequence, and applications of geometric transformations.

GeoGebra Commands & Parameters

  1. Semicircle(A, B) → g
  2. LineBisector(A, B) → f
  3. Intersect(g, f) → F,
  4. Segment(A, B) → e
  5. Mirror(F, e) → F'
  6. Polygon(A, F', B, F) → poly1, a, b, c, d
  7. Vector(A, B) → u
  8. Translate(A, u) → A'
  9. Translate(F', u) → F''
  10. Translate(B, u) → B'_1
  11. Translate(F, u) → F'_1
  12. Polygon(A', F'', B'_1, F'_1) → poly1', a', f'', b', f'_1
  13. Dilate(A', 0.6180339887498948, B) → A''
  14. Dilate(F'', 0.6180339887498948, B) → F'''
  15. Dilate(B'_1, 0.6180339887498948, B) → B''_1
  16. Dilate(F'_1, 0.6180339887498948, B) → D
  17. Polygon(A'', F''', B''_1, D) → poly1'', a'', f''', b'', d_1
  18. Rotate(A'', ((k * 180))°, B) → A'''
  19. Rotate(F''', ((k * 180))°, B) → E
  20. Rotate(B''_1, ((k * 180))°, B) → B'''_1
  21. Rotate(D, ((k * 180))°, B) → D'
  22. Polygon(A''', E, B'''_1, D') → poly1''', a''', e_1, b''', d'
  23. Translate(B, u) → B'
  24. Dilate(B', 0.61803, B) → B''
  25. CircleArc(C, B, A) → c_1
  26. Intersect(e_1, a''') → O
  27. similphi(A, B, k') → G, poly2
  28. similphi(B, G, k') → H, poly3
  29. similphi(G, H, k') → I, poly4
  30. similphi(H, I, k') → J, poly5
  31. similphi(I, J, k') → K, poly6
  32. similphi(J, K, k') → L, poly7
  33. similphi(K, L, k') → M, poly8
  34. similphi(L, M, k') → N, poly9
  35. similphi(M, N, k') → P, poly10

AI Drawing Prompt Reference

Draw a golden rectangle and unfold it using semicircles and angle bisectors to illustrate the connection between the Fibonacci sequence and the golden ratio. Start by constructing semicircles across the rectangle's sides and bisecting the angles, finding intersections and reflections to form polygons. Use vector translation and dilation by the golden ratio to create new shapes. Finally, rotate these figures and add arcs to connect them, showcasing geometric transformations and the sequence's relationship.
The prompt above was compiled from the original drawing commands and is for reference only.