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Constructing Platonic Solids: Which Polygons Work?

Constructing Platonic Solids: Which Polygons Work?
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2016-10-21 05:33
PolygonTrigonometrySolid GeometryGeometric Construction
This example demonstrates how different polygons can be used to construct Platonic solids, revealing their geometric properties. It involves core mathematical concepts such as polygons, solid geometry, symmetry, and Euclidean geometry. Key GeoGebra commands and objects used include Polygon, OrthogonalLine, Intersect, Vector, Point, Circle, Segment, Rotate, and Angle. Suitable for high school students and math enthusiasts exploring spatial geometry and polyhedral constructions.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. Polygon(C, B, 3) → poly2, i, j, k, D
  3. Polygon(D, B, 3) → poly3, l, m, n, E
  4. OrthogonalLine(B, k) → p
  5. OrthogonalLine(C, j) → q
  6. Intersect(p, q) → F
  7. Vector(G, A) → u
  8. Vector(G, B) → v
  9. Vector(G, E) → w
  10. Vector(G, C) → b
  11. Vector(G, D) → c
  12. Vector(G, F) → d
  13. Point(point1) → H
  14. Point(point2) → I
  15. Point(point3) → J
  16. Polygon(C, H, D) → poly4, d_1, c_1, h_1
  17. Polygon(C, I, H) → poly5, h_2, c_2, i_1
  18. Polygon(D, J, H) → poly6, h_3, d_2, j_1
  19. Polygon(K, L, 3) → poly7, r, s, t, M
  20. Polygon(M, L, 3) → poly8, e, f_1, g_1, N
  21. Polygon(N, L, 3) → poly9, k_1, l_1, m_1, O
  22. Polygon(O, L, 3) → poly10, n_1, p_1, q_1, P
  23. Polygon(N, M, 4) → poly11, r_1, s_1, t_1, a_1, Q, R
  24. OrthogonalLine(L, g_1) → b_1
  25. OrthogonalLine(M, f_1) → e_1
  26. Intersect(b_1, e_1) → S
  27. Polygon(M, N, T) → poly12, t_2, m_2, n_2
  28. Polygon(M, U, T) → poly13, t_3, m_3, u_1
  29. Polygon(T, N, V) → poly14, v_1, t_4, n_3
  30. Polygon(T, W, V) → poly15, v_2, t_5, w_1
  31. Polygon(Z, A_1, 3) → poly16, f_2, g_2, i_2, B_1
  32. Polygon(B_1, A_1, 3) → poly17, j_2, k_2, l_2, C_1
  33. Polygon(C_1, A_1, 3) → poly18, p_2, q_2, r_2, D_1
  34. Polygon(D_1, A_1, 3) → poly19, s_2, a_2, b_2, E_1
  35. Polygon(E_1, A_1, 3) → poly20, e_2, f_3, g_3, F_1
  36. Polygon(C_1, B_1, 5) → poly21, i_3, j_3, k_3, l_3, p_3, G_1, H_1, I_1
  37. OrthogonalLine(B_1, k_2) → q_3
  38. OrthogonalLine(A_1, l_2) → r_3
  39. Intersect(r_3, q_3) → J_1
  40. Polygon(B_1, C_1, K_1) → poly22, k_4, b_3, c_3
  41. Polygon(C_1, L_1, K_1) → poly23, k_5, c_4, l_4
  42. Polygon(K_1, L_1, M_1) → poly24, m_4, k_6, l_5
  43. Polygon(B_1, O_1, K_1) → poly25, k_7, b_4, o_1
  44. Polygon(K_1, N_1, M_1) → poly26, m_5, k_8, n_4
  45. Polygon(Z_1, A_2, 5) → poly27, s_3, a_3, d_3, e_3, f_4, B_2, C_2, D_2
  46. Polygon(Z_1, D_2, 5) → poly28, g_4, h_4, i_4, j_4, p_4, E_2, F_2, G_2
  47. Polygon(E_2, D_2, 5) → poly29, q_4, r_4, s_4, a_4, d_4, H_2, I_2, J_2
  48. Circle(U_1, W_1, Q_1) → e_4
  49. Circle(U_1, 2) → c_5
  50. Intersect(e_4, c_5, 2) → K_2
  51. Circle(K_2, 2) → d_5
  52. Polygon(K_2, U_1, 6) → poly30, f_5, O_2, g_5, h_5, i_5, j_5, n_5, L_2, M_2, N_2
  53. Segment(L_2, R_1) → p_5
  54. Segment(R_1, K_2) → q_5
  55. Segment(R_1, Q_1) → r_5
  56. Polygon(U_2, P_2, R_1, T_2) → poly31, u_2, p_6, r_6, t_6
  57. Polygon(P_2, Q_2, R_2, R_1) → poly32, p_7, q_6, r_7, r_8
  58. Polygon(Q_1, R_1, T_2, S_2) → poly33, q_7, r_9, t_7, s_5
  59. Polygon(E_2, Z_1, 3) → poly34, a_5, b_5, e_5, W_2
  60. Rotate(W_2, (-(20°)), E_2) → Z_2
  61. Angle(Z_2, E_2, W_2) → α
  62. Segment(E_2, Z_2) → f_6
  63. Intersect(f_6, b_5) → A_3
  64. Rotate(Z_1, 20°, E_2) → B_3
  65. Angle(Z_1, E_2, B_3) → β
  66. Segment(B_3, E_2) → g_6
  67. Intersect(g_6, b_5) → C_3
  68. Circle(Z_1, C_3) → c_6
  69. Circle(W_2, A_3) → d_6
  70. Intersect(c_6, a_5) → D_3,
  71. Intersect(d_6, e_5) → E_3,
  72. Intersect(c_6, a_5, 1) → F_3
  73. Circle(D_3, Z_1) → e_6
  74. Circle(E_3, W_2) → h_6
  75. Intersect(e_6, a_5) → G_3, H_3
  76. Intersect(h_6, e_5) → I_3, J_3
  77. OrthogonalLine(Z_1, e_5) → i_6
  78. OrthogonalLine(E_2, b_5) → j_6
  79. Intersect(i_6, j_6) → K_3
  80. Polygon(Z_1, L_3, M_3, E_2, N_3) → poly35, z_1, l_6, m_6, e_7, n_6
  81. Polygon(Z_1, T_3, S_3, R_3, N_3) → poly36, z_2, t_8, s_6, r_{10}, n_7
  82. Polygon(N_3, E_2, O_3, P_3, Q_3) → poly37, n_8, e_8, o_3, p_8, q_8