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Regular Decagon Magic Demo

Regular Decagon Magic Demo
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2018-07-05 08:06
PolygonAngleTrigonometryGeometric Construction
This dynamic GeoGebra example illustrates several fascinating phenomena within a regular decagon. It covers core math concepts like polygon properties, angle calculations, and geometric transformations (rotation, dilation). Constructed using commands such as Line, Polygon, Segment, LineBisector, Intersect, Rotate, Dilate, and objects like Angle, Circle, Vector, the demo is suitable for high school students and math enthusiasts exploring geometric wonders.

GeoGebra Commands & Parameters

  1. Line((0, 0, 0), (1, 0, 0)) → m_4
  2. Line((0, 0, 0), (0, 1, 0)) → n_4
  3. Line((0, 0, 0), (0, 0, 1)) → p_4
  4. Polygon(A, B, 10) → poly3, r, s_1, C, D, I, J, K, S, T, U, t, i_1, j_1, k_1, l_1, n_1, p_1, r_1
  5. Segment(S, K) → t_1
  6. Segment(K, J) → a_1
  7. Segment(J, I) → b_1
  8. Segment(I, D) → c_1
  9. Segment(D, C) → d_1
  10. Segment(C, B) → e_1
  11. Segment(B, A) → f_2
  12. Segment(A, U) → g_2
  13. Segment(U, T) → h_2
  14. Segment(T, S) → i_2
  15. LineBisector(a_1) → j_2
  16. LineBisector(i_2) → k_2
  17. Intersect(j_2, k_2) → V
  18. Line(T, I) → l_2
  19. Line(A, J) → m_2
  20. Line(C, K) → n_2
  21. Intersect(l_2, m_2) → W
  22. Rotate(poly3, If[10 < b ≤ 11, ((b - 11) * π / 5), π / 5], V) → poly3'
  23. Rotate(T, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → T'
  24. Rotate(S, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → S'
  25. Segment(T', S') → p_2
  26. Rotate(S, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → S'_1
  27. Rotate(K, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → K'
  28. Segment(S'_1, K') → q_2
  29. Rotate(K, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → K'_1
  30. Rotate(J, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → J'
  31. Segment(K'_1, J') → r_2
  32. Rotate(J, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → J'_1
  33. Rotate(I, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → I'
  34. Segment(J'_1, I') → s_2
  35. Rotate(I, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → I'_1
  36. Rotate(D, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → D'
  37. Segment(I'_1, D') → t_2
  38. Rotate(D, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → D'_1
  39. Rotate(C, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → C'
  40. Segment(D'_1, C') → a_2
  41. Rotate(C, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → C'_1
  42. Rotate(B, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → B'
  43. Segment(C'_1, B') → b_2
  44. Rotate(B, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → B'_1
  45. Rotate(A, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → A'
  46. Segment(B'_1, A') → c_2
  47. Rotate(A, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → A'_1
  48. Rotate(U, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → U'
  49. Segment(A'_1, U') → d_2
  50. Rotate(U, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → U'_1
  51. Rotate(T, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → T'_1
  52. Rotate(C'_1, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → Z
  53. Rotate(B', If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → B''
  54. Rotate(U, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → U'_2
  55. Rotate(T, If[10 < b ≤ 11, ((b - 10) * π / 5), π / 5], V) → T'_2
  56. Segment(U'_2, T'_2) → e_2
  57. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], I) → T'_3
  58. Segment(T, T'_3) → g_3
  59. Dilate(A, If[13 < b ≤ 14, 14 - b, 0], W) → A'_2
  60. Segment(A, A'_2) → h_3
  61. Dilate(W, If[14 < b ≤ 15, 15 - b, 0], J) → W'
  62. Segment(W, W') → i_3
  63. Dilate(C, If[15 < b ≤ 16, 16 - b, 0], W) → C'_2
  64. Segment(C, C'_2) → j_3
  65. Dilate(W, If[16 < b ≤ 17, 17 - b, 0], K) → W'_1
  66. Segment(W, W'_1) → k_3
  67. Angle(I, C, W) → η
  68. Circle(C, (0.4 * r_2)) → c
  69. Rotate(C, 36°, W) → C'_3
  70. Angle(C, W, C'_3) → α
  71. Rotate(C, (-(36°)), I) → C'_4
  72. Angle(C'_4, I, C) → β
  73. Ray(W, C'_3) → f
  74. Ray(I, C'_4) → h
  75. Intersect(j_3, h) → H
  76. Line(I, C) → i
  77. Intersect(i, f) → M
  78. Rotate(C, If[17 < b ≤ 18, ((-(b - 17)) * π / 5), ((-π)) / 5], C) → C'_5
  79. Rotate(C'_2, If[17 < b ≤ 18, ((-(b - 17)) * π / 5), ((-π)) / 5], C) → N
  80. Segment(C'_5, N) → j
  81. Intersect(c, j_3, 1) → O
  82. Rotate(O, If[18 < b ≤ 19, ((-(b - 18)) * π / 5), ((-π)) / 5], C) → O'
  83. CircleSector(C, O', O) → d
  84. Dilate(C, If[20 < b ≤ 21, 21 - b, 0], W) → C'_6
  85. Vector(C, C'_6) → u
  86. Translate(d, u) → d'
  87. Dilate(W, If[18 < b ≤ 19, 19 - b, 0], M) → W'_2
  88. Segment(W, W'_2) → k
  89. Line(C, k) → l
  90. Rotate(d', If[20 < b ≤ 21, (((-(b - 20)) * 3) * π / 5), (-3 * π / 5)], C'_6) → d''
  91. Rotate(d'', If[22 < b ≤ 23, ((-(b - 22)) * π / 5), ((-π)) / 5], W) → d'''
  92. Polygon(C, W, I, J, K, S, T, U, A, B) → poly1, c_3, b_3, w, i_4, j_4, k_4, s, t_3, u_1, a_3
  93. Polygon(C, D, I) → t1, i_5, c_4, d_3
  94. Polygon(C, I, W) → t2, w_1, c_5, i_6
  95. Segment(T, W) → m
  96. Rotate(W, If[23 < b ≤ 24, (((-(b - 23)) * 3) * π / 5), (-3 * π / 5)], W) → W'_3
  97. Rotate(W'_1, If[23 < b ≤ 24, (((-(b - 23)) * 3) * π / 5), (-3 * π / 5)], W) → F
  98. Segment(W'_3, F) → n
  99. Rotate(W, If[24 < b ≤ 25, (((b - 24) * 2) * π / 5), (2 * π / 5)], J) → W'_4
  100. Rotate(W', If[24 < b ≤ 25, (((b - 24) * 2) * π / 5), (2 * π / 5)], J) → W''
  101. Segment(W'_4, W'') → p
  102. OrthogonalLine(C, b_2) → q
  103. Circle(C, d_2) → e
  104. Intersect(e, q) → P, Q
  105. Circle(Q, d_2) → f_1
  106. Intersect(f_1, q, 2) → R
  107. Segment(C, D) → g_1
  108. Rotate(C, If[25 < b ≤ 26, ((-(b - 25)) * π / 2), ((-π)) / 2], C) → C'_7
  109. Rotate(D, If[25 < b ≤ 26, ((-(b - 25)) * π / 2), ((-π)) / 2], C) → D'_2
  110. Segment(C'_7, D'_2) → h_1
  111. Angle(Q, C, D) → γ
  112. Ray(Q, R) → m_1
  113. Rotate(C'_7, If[26 < b ≤ 27, ((-(b - 26)) * π), (-π)], Q) → A_1
  114. Rotate(D'_2, If[26 < b ≤ 27, ((-(b - 26)) * π), (-π)], Q) → B_1
  115. Segment(A_1, B_1) → q_1
  116. Line(D, R) → f_3
  117. Line(W, f_3) → l_3
  118. Point(l_3) → C_1
  119. Angle(A, W, C_1) → δ
  120. Dilate(W, If[27 < b ≤ 28, 28 - b, 0], D) → W'_5
  121. Vector(W, W'_5) → v
  122. Translate(A, v) → A'_3
  123. Translate(A'_2, v) → D_1
  124. Segment(A'_3, D_1) → m_3
  125. Rotate(A'_3, If[27 < b ≤ 28, ((b - 27) * δ), δ], D_1) → E_1
  126. Rotate(D_1, If[27 < b ≤ 28, ((b - 27) * δ), δ], D_1) → F_1
  127. Segment(E_1, F_1) → n_3
  128. Polygon(R, D, C) → t3, c_6, r_3, d_4