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Regular Polygon Phenomena 3

Regular Polygon Phenomena 3
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2019-03-27 21:12
PolygonAngleTransformationGeometric Construction
This example demonstrates the construction of regular polygons and their rotational symmetry phenomena. Core math concepts include regular polygons, rotation, central symmetry, and angle calculations. It utilizes GeoGebra commands/objects like Polygon, Segment, Circle, Intersect, Center, Dilate, Angle, and Rotate. Suitable for high school students studying geometric transformations, symmetry, and polygon properties.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 12) → poly1, f, p, q, r, C, D, E, F, G, H, I, g, J, K, L, h, i, j, k, l, m, n
  2. Segment(G, D) → s_2
  3. Polygon(F, E, 6) → poly2, a_1, U, c_1, d_1, e_1, i_2, k_2, N, O, T
  4. Circle(G, Distance[U, E]) → c_2
  5. Intersect(c_2, s_2, 1) → W
  6. Polygon(I, H, 3) → poly5, n_1, p_1, q_1, V
  7. Segment(A, B) → s
  8. Segment(B, C) → t
  9. Segment(A, L) → a
  10. Segment(L, K) → b
  11. Segment(K, J) → c
  12. Segment(J, I) → j_1
  13. Segment(I, H) → k_1
  14. Segment(H, G) → l_1
  15. Segment(G, F) → m_1
  16. Segment(F, E) → t_1
  17. Segment(E, D) → b_1
  18. Center(Circle[J, I, H]) → M
  19. Segment(I, V) → r_1
  20. Segment(V, H) → s_1
  21. Segment(E, N) → e
  22. Segment(N, O) → f_1
  23. Segment(O, T) → g_1
  24. Segment(T, U) → h_1
  25. Segment(U, F) → i_1
  26. Segment(F, E) → f_2
  27. Dilate(M, If[0 < d ≤ 1, 1 - d, 0], G) → M'
  28. Segment(M, M') → l_2
  29. Dilate(G, If[1 < d ≤ 2, 2 - d, 0], V) → G'
  30. Dilate(V, If[2 < d ≤ 3, 3 - d, 0], J) → V'
  31. Segment(V, V') → m_2
  32. Dilate(J, If[3 < d ≤ 4, 4 - d, 0], M) → J'
  33. Angle(V, G, M) → α
  34. Angle(G, M, J) → β
  35. Angle(M, J, V) → γ
  36. Dilate(V, If[1 < d ≤ 2, 2 - d, 0], G) → V'_1
  37. Segment(V, V'_1) → g_2
  38. Dilate(M, If[2 < d < 3, 3 - d, 0], J) → M'_1
  39. Segment(M, M'_1) → n_2
  40. Rotate(M, If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 2), π / 2], M) → M'_2
  41. Rotate(M', If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 2), π / 2], M) → M''
  42. Segment(M'_2, M'') → p_2
  43. Rotate(M'_2, If[4.5 < d ≤ 5.5, ((d - 4.5) * π / 2), π / 2], J) → R
  44. Rotate(M'', If[4.5 < d ≤ 5.5, ((d - 4.5) * π / 2), π / 2], J) → M'''
  45. Segment(R, M''') → q_2
  46. Rotate(R, If[5.5 < d ≤ 6.5, ((d - 5.5) * π / 2), π / 2], V) → R'
  47. Rotate(M''', If[5.5 < d ≤ 6.5, ((d - 5.5) * π / 2), π / 2], V) → S
  48. Segment(R', S) → r_2
  49. Dilate(G, If[7 < d ≤ 8, 8 - d, 0], W) → G'_1
  50. Polygon(J, M, G, V) → q1, j_3, m_3, g_3, v
  51. Segment(G, G'_1) → h_2
  52. Dilate(W, If[8 < d ≤ 9, 9 - d, 0], D) → W'
  53. Segment(W, W') → t_2
  54. Dilate(U, If[7 < d ≤ 9, (0.5 * (9 - d)), 0], E) → U'
  55. Segment(U, U') → a_2
  56. Rotate(U, If[10 < d ≤ 11, ((d - 10) * π / 6), π / 6], D) → U'_1
  57. Rotate(U', If[10 < d ≤ 11, ((d - 10) * π / 6), π / 6], D) → U''
  58. Segment(U'_1, U'') → b_2
  59. Rotate(E, If[10 < d ≤ 11, ((d - 10) * π / 6), π / 6], D) → E'
  60. Rotate(D, If[10 < d ≤ 11, ((d - 10) * π / 6), π / 6], D) → D'
  61. Segment(E', D') → d_2

AI Drawing Prompt Reference

Draw a regular dodecagon and demonstrate its rotational symmetry. Start by constructing a regular dodecagon, then draw a circle centered at the midpoint of one of its sides. Find points on this circle corresponding to the vertices of the dodecagon and connect them to form a smaller regular hexagon. Next, draw a circle centered at one vertex of the hexagon, intersecting the larger circle at two points. Connect these points to a vertex of the dodecagon to form a smaller equilateral triangle. Finally, use rotation and scaling to illustrate the rotational symmetry of the dodecagon.
The prompt above was compiled from the original drawing commands and is for reference only.