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构建柏拉图立体：哪些多边形适用？

构建柏拉图立体：哪些多边形适用？
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2016-10-21 05:33
多边形三角函数立体几何几何作图
①本范例演示如何通过不同多边形构建柏拉图立体，揭示其几何特性；②涉及核心数学概念包括多边形、立体几何、对称性、欧几里得几何；③作图要点：运用了Polygon（多边形）、OrthogonalLine（垂线）、Intersect（交点）、Vector（向量）、Point（点）、Circle（圆）、Segment（线段）、Rotate（旋转）、Angle（角度）等GeoGebra命令与对象；④适合高中及以上学生及数学爱好者，用于探索空间几何与多面体构造。

GeoGebra 命令及参数

  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