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将立方体分割成等体积等表面积的部分

将立方体分割成等体积等表面积的部分
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2021-07-09 05:37
多边形立体几何变换最值优化几何作图
所属专题 Dissecting a cube into equal parts
① 本范例演示如何将一个立方体分割成若干部分，使得每个部分的体积和表面积都相等，属于立体几何中的等分构造问题。② 核心数学概念包括立方体的体积与表面积计算、空间对称分割、以及利用多边形（Polygon）构造切割面，涉及几何变换与面积/体积的等量关系。③ 作图要点：从构造摘要可见使用了大量点对象（如A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, Z等）以及带下标的点（如A_1, B_1, C_1, D_1, F_1, I_1, J_1, O_1, P_1, Q_1, R_1, S_1, T_1等），并通过Polygon命令创建多边形面，可能还涉及辅助对象（如q1, j）来辅助分割。④ 适合高中或大学低年级的立体几何教学，也可用于数学建模或几何直观训练，帮助学生理解空间等分与对称性。

GeoGebra 命令及参数

  1. Polygon(J, I, O, M) → q1, j, i, o, m
  2. Polygon(A, B, G, C) → q2, a_1, b, g, c
  3. Polygon(L, J, M, P) → q3, l, j_1, m_1, p
  4. Polygon(G, B, E, H) → q4, g_1, b_1, e, h
  5. Polygon(A, B, E, D) → q5, a_2, b_2, e_1, d
  6. Polygon(J, I, K, L) → q6, j_2, i_1, k, l_1
  7. Polygon(M, P, N, O) → q7, m_2, p_1, n, o_1
  8. Polygon(C, G, H, F) → q8, c_1, g_2, h_1, f
  9. Polygon(B_1, T, R, D) → q9, q, t, r, d_1
  10. Polygon(A_1, K, Q, S) → q10, s, k_1, q_1, s_1
  11. Polygon(A_1, P, L) → t1, l_2, f_1, p_2
  12. Polygon(P, W, U, E_1) → poly1, n_1, r_1, t_1, f_2
  13. Polygon(P, E_1, N) → t2, n_2, p_3, h_2
  14. Polygon(A_1, S, W, P) → q11, i_2, s_2, w, p_4
  15. Polygon(W, Q, R, Z) → q12, w_1, q_2, r_2, z_1
  16. Polygon(C_1, D_1, Z, W) → q13, k_2, t_2, z_2, w_2
  17. Polygon(T, B_1, H, Z) → t3, c_2, d_2, e_2, f_3
  18. Polygon(Z, V, F_1, H) → q14, z_3, v, g_3, h_3
  19. Polygon(H, F, F_1) → t4, i_3, h_4, f_4
  20. Polygon(W, S, C_1) → poly2, j_3, k_3, l_3
  21. Polygon(Z, T, D_1) → t5, m_3, n_3, r_3
  22. Polygon(B_1, E, H) → t6, h_5, q_3, e_3
  23. Polygon(D_1, C_1, Q, R) → q15, s_3, t_3, q_4, r_4
  24. Polygon(W, Q, U) → t7, u, w_3, q_5
  25. Polygon(Z, V, R) → t8, r_5, z_4, v_1
  26. Polygon(K, I, O, N) → q16, k_4, i_4, o_2, n_4
  27. Polygon(K, E_1, U, Q) → t9, a_3, b_3, c_3, d_3
  28. Polygon(F, C, A, D) → q17, f_5, c_4, a_4, d_4
  29. Polygon(V, F_1, D, R) → q18, v_2, g_4, d_5, r_6
  30. Polygon(H_1, G_1, J_1, I_1) → q19, j_4, l_4, m_4, s_4
  31. Polygon(L_1, K_1, H_1, G_1) → q20, t_4, b_4, e_4, g_5
  32. Polygon(L_1, K_1, I_1, J_1) → q21, i_5, j_5, k_5, l_5
  33. Polygon(I_1, P_1, O_1, S_1) → q22, m_5, n_5, p_5, s_5
  34. Polygon(R_1, Q_1, J_1, T_1) → poly3, t_5, a_5, b_5, c_5
  35. Polygon(O_1, P_1, H_1, M_1) → q23, e_5, f_6, g_6, h_6
  36. Polygon(G_1, N_1, R_1, Q_1) → q24, i_6, j_6, k_6, l_6
  37. Polygon(A_2, Z_1, W_1, S_1) → q25, m_6, n_6, p_6, q_6
  38. Polygon(T_1, B_2, V_1, U_1) → t10, s_6, t_6, a_6, b_6
  39. Polygon(Z_1, D_2, M_1, W_1) → q26, c_6, d_6, e_6, f_7
  40. Polygon(N_1, C_2, V_1, U_1) → q27, g_7, h_7, i_7, u_1
  41. Polygon(D_2, Z_1, A_2, E_2) → q28, j_7, k_7, l_7, m_7
  42. Polygon(C_2, V_1, B_2, F_2) → q29, n_7, p_7, q_7, r_7
  43. Polygon(I_1, P_1, H_1) → t11, s_7, t_7, a_7
  44. Polygon(W_1, S_1, O_1) → t12, b_7, c_7, d_7
  45. Polygon(G_1, Q_1, J_1) → t13, e_7, f_8, g_8
  46. Polygon(T_1, U_1, R_1) → t14, h_8, i_8, j_8
  47. Polygon(E_2, A_2, S_1, G_2) → q30, k_8, l_8, m_8, n_8
  48. Polygon(F_2, H_2, T_1, B_2) → q31, p_8, q_8, r_8, s_8
  49. Polygon(S_1, I_2, J_2, I_1) → q32, t_8, a_8, b_8, c_8
  50. Polygon(K_2, L_2, T_1, J_1) → q33, d_8, e_8, f_9, g_9
  51. Polygon(I_2, G_2, S_1) → t15, h_9, i_9, j_9
  52. Polygon(K_1, J_2, I_1) → t16, k_9, l_9, m_9
  53. Polygon(L_1, K_2, J_1) → t18, r_9, s_9, t_9
  54. Polygon(F_2, C_2, N_1, H_2) → q34, a_9, b_9, c_9, d_9
  55. Polygon(L_2, K_2, G_1, N_1) → q35, e_9, f_{10}, g_{10}, h_{10}
  56. Polygon(M_1, I_2, J_2, H_1) → q37, m_{10}, n_{10}, p_{10}, q_{10}
  57. Polygon(D_2, E_2, G_2, M_1) → q36, i_{10}, j_{10}, k_{10}, l_{10}
  58. Polygon(H_3, I_3, S_2, R_2) → q38, n_9, p_9, q_9, r_{10}
  59. Polygon(Q_2, R_2, C_3, B_3) → q39, s_{10}, t_{10}, a_{10}, b_{10}
  60. Polygon(O_2, D_3, E_3, N_2) → q40, c_{10}, d_{10}, e_{10}, f_{11}
  61. Polygon(F_3, G_3, N_2, M_2) → q41, g_{11}, h_{11}, i_{11}, j_{11}
  62. Polygon(F_3, T_2, J_3, M_2) → q43, p_{11}, q_{11}, r_{11}, s_{11}
  63. Polygon(U_2, K_3, J_3, T_2) → q44, u_2, t_{11}, a_{11}, b_{11}
  64. Polygon(L_3, M_3, Z_2, A_3) → q45, c_{11}, d_{11}, e_{11}, f_{12}
  65. Polygon(Z_2, W_2, B_3, C_3) → q46, g_{12}, h_{12}, i_{12}, j_{12}
  66. Polygon(E_3, U_2, V_2, D_3) → q47, k_{12}, l_{12}, m_{12}, n_{12}
  67. Polygon(Q_2, O_2, V_2, W_2) → q48, p_{12}, q_{12}, r_{12}, s_{12}
  68. Polygon(Z_2, Q_3, P_3, W_2) → q49, t_{12}, a_{12}, b_{12}, c_{12}
  69. Polygon(V_2, O_3, N_3, U_2) → q50, d_{12}, o_3, e_{12}, f_{13}
  70. Polygon(W_2, S_3, R_3, V_2) → q51, g_{13}, h_{13}, i_{13}, j_{13}
  71. Polygon(I_3, H_3, Z_2, A_3) → q52, k_{13}, l_{13}, m_{13}, n_{13}
  72. Polygon(G_3, F_3, T_2, U_2) → q53, p_{13}, q_{13}, r_{13}, s_{13}
  73. Polygon(H_3, Z_2, C_3) → t17, t_{13}, a_{13}, b_{13}
  74. Polygon(B_3, W_2, Q_2) → t19, c_{13}, d_{13}, e_{13}
  75. Polygon(O_2, D_3, V_2) → t20, f_{14}, g_{14}, h_{14}
  76. Polygon(G_3, E_3, U_2) → t21, i_{14}, j_{14}, k_{14}
  77. Polygon(S_2, R_2, M_3, L_3) → q54, l_{14}, m_{14}, n_{14}, p_{14}
  78. Polygon(Q_3, P_3, Q_2, R_2) → q55, q_{14}, r_{14}, s_{14}, t_{14}
  79. Polygon(S_3, R_3, O_2, Q_2) → q56, a_{14}, b_{14}, c_{14}, d_{14}
  80. Polygon(N_3, N_2, O_2, O_3) → q57, e_{14}, f_{15}, g_{15}, h_{15}
  81. Polygon(N_2, K_3, J_3, M_2) → q58, i_{15}, j_{15}, k_{15}, l_{15}
  82. Polygon(U_2, N_3, K_3) → t22, m_{15}, n_{15}, p_{15}
  83. Polygon(V_2, O_3, R_3) → t23, q_{15}, r_{15}, s_{15}
  84. Polygon(W_2, S_3, P_3) → t24, t_{15}, a_{15}, b_{15}
  85. Polygon(Z_2, M_3, Q_3) → t25, c_{15}, d_{15}, e_{15}
  86. Polygon(M_4, L_4, Z_3, A_4) → q42, k_{11}, l_{11}, m_{11}, n_{11}
  87. Polygon(A_4, B_4, C_4, Z_3) → q59, f_{16}, g_{16}, h_{16}, i_{16}
  88. Polygon(N_4, O_4, T_3, U_3) → q60, j_{16}, o_4, k_{16}, u_3
  89. Polygon(U_3, D_4, E_4, T_3) → q61, l_{16}, m_{16}, n_{16}, p_{16}
  90. Polygon(G_4, F_4, W_3, V_3) → q62, q_{16}, r_{16}, s_{16}, v_3
  91. Polygon(V_3, T_4, U_4, W_3) → q63, t_{16}, a_{16}, u_4, b_{16}
  92. Polygon(T_4, V_3, R_4) → t26, c_{16}, d_{16}, e_{16}
  93. Polygon(U_4, W_3, Q_4) → t27, f_{17}, g_{17}, h_{17}
  94. Polygon(U_3, S_4, D_4) → t28, i_{17}, j_{17}, k_{17}
  95. Polygon(S_4, R_4, V_3, U_3) → q64, l_{17}, m_{17}, n_{17}, p_{17}
  96. Polygon(Q_4, P_4, Z_3, W_3) → q65, q_{17}, r_{17}, s_{17}, t_{17}
  97. Polygon(P_4, C_4, Z_3) → t29, a_{17}, b_{17}, c_{17}
  98. Polygon(H_4, K_4, Z_3, W_3) → q66, d_{17}, e_{17}, f_{18}, g_{18}
  99. Polygon(I_4, J_4, U_3, V_3) → q67, h_{18}, i_{18}, j_{18}, k_{18}
  100. Polygon(N_4, J_4, U_3) → t30, l_{18}, m_{18}, n_{18}
  101. Polygon(V_3, G_4, I_4) → t31, p_{18}, q_{18}, r_{18}
  102. Polygon(W_3, F_4, H_4) → t32, s_{18}, t_{18}, a_{18}
  103. Polygon(Z_3, K_4, L_4) → t33, b_{18}, c_{18}, d_{18}
  104. Polygon(U_4, T_4, G_4, F_4) → q68, e_{18}, f_{19}, g_{19}, h_{19}
  105. Polygon(A_5, Z_4, C_4, B_4) → q69, i_{19}, j_{19}, k_{19}, l_{19}
  106. Polygon(A_5, Z_4, L_4, M_4) → q70, m_{19}, n_{19}, p_{19}, q_{19}
  107. Polygon(O_4, N_4, W_4, V_4) → q71, r_{19}, s_{19}, w_4, v_4
  108. Polygon(T_3, O_4, V_4, E_4) → q72, t_{19}, a_{19}, b_{19}, c_{19}
  109. Polygon(E_4, D_4, W_4, V_4) → q73, d_{19}, e_{19}, f_{20}, g_{20}
  110. Polygon(B_4, A_4, M_4, A_5) → q74, h_{20}, i_{20}, j_{20}, k_{20}
  111. Polygon(S_4, R_4, G_4, W_4) → q75, l_{20}, m_{20}, n_{20}, p_{20}
  112. Polygon(G_4, W_4, J_4, I_4) → q76, q_{20}, r_{20}, s_{20}, t_{20}
  113. Polygon(Z_4, F_4, Q_4, P_4) → q78, e_{20}, f_{21}, g_{21}, h_{21}
  114. Polygon(K_4, H_4, F_4, Z_4) → q77, a_{20}, b_{20}, c_{20}, d_{20}

AI 绘图提示语参考

画一个立方体，然后通过多个切割平面将其分割成若干个相等的部分，使得每个小多面体具有相等的体积和表面积。构造时先建立立方体的8个顶点，然后从立方体内部向各个方向创建多边形切割面，这些切割面相互交错，将立方体分割成多个不规则的多面体块，每个块都通过顶点和多边形面来界定。注意保持对称性，使得分割后的每个小块在几何上尽可能均匀分布。
以上提示语为根据原始绘图命令整理而来，仅供参考。