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Exploring Regular Polygon Phenomena

Exploring Regular Polygon Phenomena
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2019-03-21 10:32
PolygonAngleVectorGeometric Construction
This example demonstrates the combination of three regular polygons and the very interesting phenomena they generate. It involves core mathematical concepts such as geometric construction, transformation, vector, angle, and coordinate geometry. The construction utilizes commands and objects like Polygon, Segment, Angle, Midpoint, Dilate, Rotate, Vector, Line, Point, and Translate in GeoGebra. It is suitable for high school and above students to explore geometric transformations and polygon properties, ideal for teaching demonstrations or research purposes.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 5) → poly1, f, g, h, i, j, C, D, E
  2. Polygon(D, C, 10) → poly2, k, a, F, G, H, I, J, K, L, M, l, m, n, p, q, r, s, t
  3. Segment(E, H) → b
  4. Segment(E, M) → c
  5. Segment(M, H) → j_1
  6. Angle(E, M, H) → α
  7. Angle(H, E, M) → δ
  8. Segment(E, D) → e
  9. Segment(D, C) → f_1
  10. Segment(C, B) → g_1
  11. Segment(B, A) → h_1
  12. Segment(A, E) → i_1
  13. Segment(C, F) → k_1
  14. Segment(F, G) → l_1
  15. Segment(G, H) → m_1
  16. Segment(H, I) → n_1
  17. Segment(I, J) → p_1
  18. Segment(J, K) → q_1
  19. Segment(K, L) → r_1
  20. Segment(L, M) → s_1
  21. Midpoint(j_1) → N
  22. Polygon(G, F, 6) → poly3, t_1, R, a_1, b_1, c_1, d_1, e_1, O, P, Q
  23. Segment(M, D) → f_2
  24. Segment(B, R) → r_2
  25. Segment(F, O) → g_2
  26. Segment(O, P) → h_2
  27. Segment(P, Q) → i_2
  28. Segment(Q, R) → j_2
  29. Segment(R, G) → k_2
  30. Dilate(E, If[0 < u ≤ 1, 1 - u, 0], M) → E'
  31. Segment(E, E') → l_2
  32. Dilate(M, If[1 < u ≤ 2, 2 - u, 0], N) → M'
  33. Segment(M, M') → m_2
  34. Dilate(N, If[2 < u ≤ 3, 3 - u, 0], H) → N'
  35. Segment(N, N') → n_2
  36. Rotate(E, If[4 < u ≤ 5, (((u - 4) * 3) * π / 5), (3 * π / 5)], M) → E'_1
  37. Rotate(E', If[4 < u ≤ 5, (((u - 4) * 3) * π / 5), (3 * π / 5)], M) → E''
  38. Segment(E'_1, E'') → p_2
  39. Rotate(E'_1, If[5 < u ≤ 6, ((u - 5) * π), π], E'_1) → S
  40. Rotate(E'', If[5 < u ≤ 6, ((u - 5) * π), π], E'_1) → E'''
  41. Segment(S, E''') → q_2
  42. Dilate(B, If[6 < u ≤ 7, 7 - u, 0], R) → B'
  43. Segment(B, B') → s_2
  44. Dilate(H, If[8 < u ≤ 9, 9 - u, 0], R) → H'
  45. Vector(H, H') → v
  46. Line(R, q_2) → t_2
  47. Point(t_2) → T
  48. Angle(T, R, B) → ε
  49. Translate(S, v) → S'
  50. Translate(E''', v) → U
  51. Segment(S', U) → a_2
  52. Translate(E'_1, v) → V
  53. Translate(E'', v) → E'''_1
  54. Segment(V, E'''_1) → b_2
  55. Rotate(S', If[8 < u ≤ 9, ((u - 8) * ε), ε], U) → S''
  56. Rotate(U, If[8 < u ≤ 9, ((u - 8) * ε), ε], U) → U'
  57. Segment(S'', U') → c_2
  58. Rotate(V, If[8 < u ≤ 9, ((u - 8) * ε), ε], U) → V'
  59. Rotate(E'''_1, If[8 < u ≤ 9, ((u - 8) * ε), ε], U) → W
  60. Segment(V', W) → d_2

AI Drawing Prompt Reference

Draw three regular polygons: a pentagon and a decagon sharing a vertex, and a hexagon connected to part of the decagon. Connect the last vertex of the pentagon to the first vertex of the decagon, and the midpoint of the decagon to the last vertex of the pentagon. Calculate and mark the angles between these vertices, and identify key midpoints and vectors. Perform scaling and rotation transformations on the pentagon, and draw the transformed shapes.
The prompt above was compiled from the original drawing commands and is for reference only.