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Dividing a Cube into Equal Parts with Equal Volume and Surface Area

Dividing a Cube into Equal Parts with Equal Volume and Surface Area
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2021-07-09 05:37
PolygonSolid GeometryTransformationOptimizationGeometric Construction
In collections Dissecting a cube into equal parts
① This example demonstrates how to divide a cube into several parts such that each part has equal volume and equal surface area, a classic problem in solid geometry. ② Core mathematical concepts include volume and surface area calculations of a cube, symmetric spatial partitioning, and constructing cutting planes using polygons, involving geometric transformations and equality relations of area/volume. ③ Construction highlights: from the construction summary, many point objects (e.g., A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, Z) and subscripted points (e.g., 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) are used, and the Polygon command creates polygonal faces, with auxiliary objects (e.g., q1, j) possibly aiding the division. ④ Suitable for high school or early college solid geometry teaching, also useful for mathematical modeling or geometric intuition training, helping students understand spatial partitioning and symmetry.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a cube and divide it into several equal parts by using multiple cutting planes, so that each small polyhedron has equal volume and surface area. First establish the 8 vertices of the cube, then create polygonal cutting surfaces from the interior of the cube in various directions. These intersecting cutting planes divide the cube into multiple irregular polyhedron blocks, each defined by vertices and polygonal faces. Maintain symmetry so that each divided piece is as evenly distributed as possible in geometry.
The prompt above was compiled from the original drawing commands and is for reference only.