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二十面体与十二面体结合体

二十面体与十二面体结合体
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2022-04-25 18:52
多边形立体几何几何作图
所属专题 Archimedean Solids
①本范例演示了二十面体与十二面体结合体的几何构造与性质；②涉及的核心数学概念包括多边形、立体几何、相交线、面积计算等；③作图要点——从下方『构造摘要』提炼用到了Polygon（多边形）、Icosahedron（二十面体）、Sphere（球）、Intersect（相交）、Segment（线段）等GeoGebra命令/对象；④适合高中及以上学生学习和理解复杂多面体的构造与关系。

GeoGebra 命令及参数

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. Icosahedron(A, B, C) → a, D, faceABC, faceACF, faceAFG, faceADG, faceABD, faceBDH, faceBEH, faceBCE, faceCEI, faceCFI, E, faceEHK, faceEIK, faceIKL, faceFIL, faceHJK, faceJKL, faceGJL, faceFGL, faceDHJ, faceDGJ, F, edgeAC, edgeAB, edgeBC, edgeAF, edgeCF, edgeAG, edgeFG, edgeAD, edgeDG, edgeBD, G, edgeBH, edgeDH, edgeBE, edgeEH, edgeCE, edgeCI, edgeEI, edgeFI, edgeEK, edgeHK, H, edgeIK, edgeIL, edgeKL, edgeFL, edgeJK, edgeHJ, edgeJL, edgeGL, edgeGJ, edgeDJ, I, J, K, L
  3. Sphere(J, m) → b
  4. Intersect(edgeJK, b, 1) → M
  5. Intersect(edgeJL, b, 1) → N
  6. Intersect(edgeHJ, b, 1) → O
  7. Intersect(edgeGJ, b, 1) → P
  8. Intersect(edgeDJ, b, 1) → Q
  9. Sphere(K, m) → c
  10. Intersect(edgeKL, c) → R, I_2
  11. Intersect(edgeIK, c) → S, J_2
  12. Intersect(edgeJK, c) → T, K_2
  13. Intersect(edgeHK, c) → U, L_2
  14. Intersect(edgeEK, c) → V, M_2
  15. Sphere(I, m) → d
  16. Intersect(edgeIL, d) → W, N_2
  17. Intersect(edgeIK, d) → Z, O_2
  18. Intersect(edgeFI, d) → A_1, P_2
  19. Intersect(edgeCI, d) → B_1, Q_2
  20. Intersect(edgeEI, d) → C_1, R_2
  21. Sphere(F, m) → e
  22. Intersect(edgeFG, e) → D_1, S_2
  23. Intersect(edgeAF, e) → E_1, T_2
  24. Intersect(edgeFI, e, 1) → F_1
  25. Intersect(edgeFL, e, 1) → G_1
  26. Intersect(edgeCF, e, 1) → H_1
  27. Sphere(G, m) → i
  28. Intersect(edgeGJ, i, 1) → I_1
  29. Intersect(edgeAG, i, 1) → J_1
  30. Intersect(edgeDG, i, 1) → K_1
  31. Intersect(edgeFG, i, 1) → L_1
  32. Intersect(edgeGL, i, 1) → M_1
  33. Sphere(L, m) → j
  34. Intersect(edgeIL, j, 1) → N_1
  35. Intersect(edgeKL, j, 1) → O_1
  36. Intersect(edgeJL, j, 1) → P_1
  37. Intersect(edgeGL, j, 1) → Q_1
  38. Intersect(edgeFL, j, 1) → R_1
  39. Sphere(E, m) → k
  40. Intersect(edgeEK, k, 1) → S_1
  41. Intersect(edgeEH, k, 1) → T_1
  42. Intersect(edgeBE, k, 1) → U_1
  43. Intersect(edgeEI, k, 1) → V_1
  44. Intersect(edgeCE, k, 1) → W_1
  45. Sphere(D, m) → l
  46. Intersect(edgeDG, l) → Z_1, U_2
  47. Intersect(edgeAD, l) → A_2, V_2
  48. Intersect(edgeBD, l) → B_2, W_2
  49. Intersect(edgeDH, l) → C_2, Z_2
  50. Intersect(edgeDJ, l) → D_2, A_3
  51. Sphere(A, m) → n
  52. Intersect(edgeAD, n, 1) → E_2
  53. Intersect(edgeAG, n, 1) → F_2
  54. Intersect(edgeAF, n, 1) → G_2
  55. Sphere(B, m) → o
  56. Intersect(edgeBH, o, 1) → H_2
  57. Sphere(H, m) → p
  58. Intersect(edgeHJ, p, 1) → B_3
  59. Intersect(edgeDH, p, 1) → C_3
  60. Intersect(edgeBH, p, 1) → D_3
  61. Intersect(edgeHK, p, 1) → E_3
  62. Intersect(edgeEH, p, 1) → F_3
  63. Polygon(C_3, D_3, F_3, E_3, B_3) → poly2, c_3, d_3, f_3, e_3, b_3
  64. Polygon(U_1, W_1, V_1, S_1, T_1) → poly3, u_1, w_1, v_1, s_1, t_1
  65. Polygon(C_1, B_1, A_1, W, Z) → poly4, c_1, b_1, a_1, w, z_1
  66. Polygon(P, Q, O, M, N) → poly5, p_1, q, o_1, m_1, n_1
  67. Segment(L, G) → s
  68. Polygon(T, U, V, S, R) → poly6, t, u, v, s_2, r_1
  69. Intersect(edgeBD, o, 1) → G_3
  70. Intersect(edgeBE, o, 1) → H_3
  71. Intersect(s, j, 1) → I_3
  72. Polygon(N_1, R_1, I_3, P_1, O_1) → poly7, f_1, g_1, i_3, h_1, i_1
  73. Intersect(edgeFG, i, 1) → J_3
  74. Intersect(s, i) → K_3, P_3
  75. Polygon(J_3, J_1, K_1, I_1, K_3) → poly8, j_3, j_1, k_1, l_1, k_3
  76. Polygon(Z_1, A_2, B_2, C_2, D_2) → poly9, q_1, a_2, b_2, c_2, d_2
  77. Intersect(edgeBC, o) → L_3, Q_3
  78. Intersect(edgeAB, o) → M_3, R_3
  79. Polygon(M_3, L_3, H_3, H_2, G_3) → poly10, m_3, l_3, h_3, h_2, g_3
  80. Polygon(F_1, H_1, E_1, D_1, G_1) → poly11, d_1, e_1, f_2, g_2, i_2
  81. Intersect(edgeAC, n) → N_3, S_3
  82. Intersect(edgeAB, n) → O_3, T_3
  83. Polygon(N_3, O_3, E_2, F_2, G_2) → poly12, n_3, o_3, e_2, j_2, k_2
  84. Sphere(C, m) → l_2
  85. Intersect(edgeCF, l_2, 1) → U_3
  86. Intersect(edgeCE, l_2, 1) → V_3
  87. Intersect(edgeBC, l_2, 1) → W_3
  88. Intersect(edgeAC, l_2, 1) → Z_3
  89. Intersect(edgeCI, l_2, 1) → A_4
  90. Polygon(W_3, V_3, A_4, U_3, Z_3) → poly13, w_3, v_3, a_4, u_3, z_3
  91. Polygon(M, T, R, O_1, P_1, N) → poly14, m_2, t_2, r_2, n_2, p_2, n_4
  92. Polygon(O, B_3, E_3, U, T, M) → poly15, o_2, q_2, p_3, u_2, t_3, m_4
  93. Polygon(E_3, F_3, T_1, S_1, V, U) → poly16, q_3, r_3, s_3, a_3, v_2, u_4
  94. Polygon(V, S_1, V_1, C_1, Z, S) → poly17, v_4, f_4, g_4, h_4, z_2, s_4
  95. Polygon(R, O_1, N_1, W, Z, S) → poly18, r_4, i_4, j_4, w_2, z_4, s_5
  96. Polygon(V_1, W_1, V_3, A_4, B_1, C_1) → poly19, k_4, l_4, p_4, q_4, t_4, b_4
  97. Polygon(B_1, A_4, U_3, H_1, F_1, A_1) → poly20, c_4, d_4, e_4, f_5, g_5, h_5
  98. Polygon(W, A_1, F_1, G_1, R_1, N_1) → poly21, w_4, i_5, j_5, k_5, l_5, m_5
  99. Polygon(R_1, G_1, D_1, J_3, K_3, I_3) → poly22, n_5, p_5, q_5, r_5, t_5, a_5
  100. Polygon(P_1, I_3, K_3, I_1, P, N) → poly23, b_5, c_5, d_5, e_5, p_6, n_6
  101. Polygon(P, I_1, K_1, Z_1, D_2, Q) → poly24, p_7, f_6, g_6, h_6, i_6, q_6
  102. Polygon(Q, D_2, C_2, C_3, B_3, O) → poly25, q_7, j_6, k_6, l_6, m_6, o_4
  103. Polygon(C_2, B_2, G_3, H_2, D_3, C_3) → poly26, r_6, s_6, t_6, a_6, b_6, c_6
  104. Polygon(D_3, F_3, T_1, U_1, H_3, H_2) → poly27, d_6, e_6, f_7, g_7, h_7, i_7
  105. Polygon(V_3, W_1, U_1, H_3, L_3, W_3) → poly28, j_7, k_7, l_7, m_7, n_7, r_7
  106. Polygon(Z_3, N_3, O_3, M_3, L_3, W_3) → poly29, s_7, t_7, a_7, b_7, c_7, d_7
  107. Polygon(H_1, U_3, Z_3, N_3, G_2, E_1) → poly30, e_7, f_8, g_8, h_8, i_8, j_8
  108. Polygon(J_3, J_1, F_2, G_2, E_1, D_1) → poly31, k_8, l_8, m_8, n_8, p_8, q_8
  109. Polygon(J_1, F_2, E_2, A_2, Z_1, K_1) → poly32, r_8, s_8, t_8, a_8, b_8, c_8
  110. Polygon(E_2, O_3, M_3, G_3, B_2, A_2) → poly33, d_8, e_8, f_9, g_9, h_9, i_9

AI 绘图提示语参考

绘制一个二十面十二面体，首先画出三角形ABC，然后以A、B、C为顶点构造二十面体。接着以J为中心画球体，与边JK、JL、HJ、GJ、DJ相交得到M、N、O、P、Q点。再以K为中心画球体，与边KL、IK、JK、HK相交得到R、S、T、U点。以此类推，构造二十面体的所有顶点和边。
以上提示语为根据原始绘图命令整理而来，仅供参考。