All Patterns of the 3D Cube

All Patterns of the 3D Cube
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2013-10-22 15:06
This example demonstrates various patterns and structures of a 3D cube, involving core mathematical concepts like solid geometry, coordinate geometry, trigonometry, and vector operations. It uses GeoGebra commands such as Vector, OrthogonalLine, Mirror, and Polygon for construction. Suitable for high school students and math enthusiasts exploring properties and transformations of three-dimensional shapes.

GeoGebra Commands & Parameters

  1. Vector(B_1, X) → u
  2. Vector(B_1, Z) → w
  3. OrthogonalLine(View3D, yAxis) → a
  4. OrthogonalLine(View3D, xAxis) → b_2
  5. Vector(B_1, Y) → v
  6. Mirror(J_1, V) → K_1
  7. Mirror(I_1, V) → L_1
  8. Polygon(K_1, L_1, J_1, I_1) → poly1, k_2, l_2, j_2, i_2
  9. Mirror(N_1, W) → S_1
  10. Mirror(M_1, W) → T_1
  11. Mirror(O_1, C) → U_1
  12. Mirror(C_1, C) → V_1
  13. Mirror(P_1, F_5) → W_1
  14. Mirror(F_1, F_5) → Z_1
  15. Mirror(R_1, Q) → A_2
  16. Mirror(Q_1, Q) → D_2
  17. Polygon(N_1, T_1, S_1, M_1) → poly2, n_2, t_1, s_1, m_2
  18. Polygon(F_1, P_1, Z_1, W_1) → poly3, f_2, p_2, z_1, w_1
  19. Polygon(C_1, O_1, V_1, U_1) → poly4, c_2, o_2, v_1, u_1
  20. Polygon(Q_1, R_1, D_2, A_2) → poly5, q_2, r_2, d_2, a_2
  21. Polygon(E_2, I_1, K_1, G_2) → poly6, e_2, i_3, k_3, g_2
  22. Polygon(H_2, I_2, I_1, E_2) → poly7, h_2, i_4, i_5, e_3
  23. Polygon(K_1, L_1, K_2, J_2) → poly8, k_4, l_3, k_5, j_3
  24. Polygon(J_1, L_2, M_2, L_1) → poly9, j_4, l_4, m_3, l_5
  25. Polygon(N_2, E_2, G_2, O_2) → poly10, n_3, e_4, g_3, o_3
  26. Polygon(R_2, S_1, M_1, P_2) → poly11, r_3, s_2, m_4, p_3
  27. Polygon(T_2, S_2, M_1, P_2) → poly12, t_2, s_3, m_5, p_4
  28. Polygon(R_2, S_1, U_2, V_2) → poly13, r_4, s_4, u_2, v_2
  29. Polygon(N_1, W_2, Z_2, T_1) → poly14, n_4, w_2, z_2, t_3
  30. Polygon(W_2, A_3, B_3, Z_2) → poly15, w_3, a_3, b_3, z_3
  31. Polygon(C_3, F_1, W_1, D_3) → poly16, c_4, f_3, w_4, d_4
  32. Polygon(C_3, E_3, F_3, F_1) → poly17, c_5, e_5, f_4, f_5
  33. Polygon(P_1, G_3, H_3, Z_1) → poly18, p_5, g_4, h_3, z_4
  34. Polygon(G_3, I_3, J_3, H_3) → poly19, g_5, i_6, j_5, h_4
  35. Polygon(Z_1, K_3, L_3, H_3) → poly20, z_5, k_6, l_6, h_5
  36. Polygon(M_3, Q_1, A_2, N_3) → poly21, m_6, q_1, a_4, n_5
  37. Polygon(M_3, P_3, O_3, Q_1) → poly22, m_7, p_6, o_1, q_3
  38. Polygon(R_1, Q_3, R_3, D_2) → poly23, r_5, q_4, r_6, d_5
  39. Polygon(A_2, D_2, S_3, T_3) → poly24, a_5, d_6, s_5, t_4
  40. Polygon(Q_3, U_3, V_3, R_3) → poly25, q_5, u_3, v_3, r_7
  41. Polygon(W_3, C_1, U_1, Z_3) → poly26, w_5, c_6, u_4, z_6
  42. Polygon(B_4, A_4, C_1, W_3) → poly27, b_4, a_6, c_7, w_6
  43. Polygon(O_1, C_4, D_4, V_1) → poly28, o_4, c_8, d_7, v_4
  44. Polygon(C_4, E_4, F_4, D_4) → poly29, c_9, e_7, f_7, d_8
  45. Polygon(D_4, F_4, G_4, H_4) → poly30, d_9, f_8, g_6, h_6
  46. Mirror(H, A) → F
  47. Mirror(E, A) → G
  48. Polygon(E, H, G, F) → poly31, e_8, h_7, g_7, f_9
  49. Polygon(M_4, E, H, N_4) → poly32, m_8, e_9, h_8, n_6
  50. Polygon(H, J_4, L_4, G) → poly33, h_9, j_6, l_7, g_8
  51. Polygon(G, O_4, P_4, F) → poly34, g_9, o_5, p_7, f_{10}
  52. Polygon(F, S, R, E) → poly35, f_{11}, s_9, r_8, e_{10}
  53. Polygon(S, K_4, I_4, R) → poly36, s_6, k_7, i_7, r_9
  54. Mirror(D, B) → D'
  55. Mirror(I, B) → I'
  56. Polygon(I', D', N, O) → poly37, i', d', n, o_6
  57. Polygon(I', D', I, D) → poly38, i'_1, d'_1, i, d_1
  58. Polygon(D, I, P, T) → poly39, d_3, i_1, p, t
  59. Polygon(A_1, D, T, U) → poly40, a_1, d_{10}, t_5, u_5
  60. Polygon(D', J, K, I) → poly41, d'_2, j, k_1, i_8
  61. Polygon(J, L, M, K) → poly42, j_1, l, m, k_8
  62. Mirror(E_1, D_1) → H_1
  63. Mirror(G_1, D_1) → B_2
  64. Polygon(E_1, G_1, H_1, B_2) → poly43, e_1, g_1, h_1, b_1
  65. Polygon(H_1, C_2, F_2, G_1) → poly44, h_{10}, c_1, f_1, g_{10}
  66. Polygon(C_2, Q_2, Q_4, F_2) → poly45, c_{10}, q_6, q_7, f_6
  67. Polygon(G_1, R_4, S_4, E_1) → poly46, g_{11}, r_1, s_7, e_6
  68. Polygon(T_4, E_1, S_4, U_4) → poly47, t_6, e_{11}, s_8, u_6
  69. Polygon(W_4, T_4, U_4, V_4) → poly48, w_7, t_7, u_7, v_5
  70. Mirror(B_5, Z_4) → C_5
  71. Mirror(A_5, Z_4) → D_5
  72. Polygon(D_5, E_5, J_5, K_5) → poly49, d_{11}, e_{12}, j_7, k_9
  73. Polygon(C_5, D_5, B_5, A_5) → poly50, c_{11}, d_{12}, b_5, a_7
  74. Polygon(D_5, E_5, G_5, B_5) → poly51, d_{13}, e_{13}, g_{12}, b_6
  75. Polygon(E_5, H_5, I_5, G_5) → poly52, e_{14}, h_{11}, i_9, g_{13}
  76. Polygon(A_5, B_5, L_5, M_5) → poly53, a_8, b_7, l_1, m_1
  77. Polygon(M_5, N_5, O_5, A_5) → poly54, m_9, n_1, o_7, a_9
  78. Mirror(R_5, P_5) → S_5
  79. Mirror(Q_5, P_5) → T_5
  80. Polygon(Q_5, S_5, T_5, R_5) → poly55, q_8, s_{10}, t_8, r_{10}
  81. Polygon(T_5, U_5, V_5, S_5) → poly56, t_9, u_8, v_6, s_{11}
  82. Polygon(U_5, W_5, Z_5, V_5) → poly57, u_9, w_8, z_7, v_7
  83. Polygon(B_6, A_6, W_5, U_5) → poly58, b_8, a_{10}, w_9, u_{10}
  84. Polygon(Q_5, S_5, C_6, D_6) → poly59, q_9, s_{12}, c_{12}, d_{14}
  85. Polygon(F_6, Q_5, D_6, E_6) → poly60, f_{12}, q_{10}, d_{15}, e_{15}
  86. Mirror(H_6, G_6) → J_6
  87. Mirror(I_6, G_6) → K_6
  88. Polygon(U_6, T_6, Q_6, J_6) → poly61, u_{11}, t_{10}, q_{11}, j_8
  89. Polygon(H_6, K_6, J_6, I_6) → poly62, h_{12}, k_{10}, j_9, i_{10}
  90. Polygon(I_6, J_6, Q_6, P_6) → poly63, i_{11}, j_{10}, q_{12}, p_1
  91. Polygon(I_6, L_6, M_6, H_6) → poly64, i_{12}, l_8, m_{10}, h_{13}
  92. Polygon(M_6, N_6, O_6, H_6) → poly65, m_{11}, n_7, o_8, h_{14}
  93. Polygon(Q_6, R_6, S_6, T_6) → poly66, q_{13}, r_{11}, s_{13}, t_{11}

AI Drawing Prompt Reference

Draw various patterns and structures of a 3D cube, showcasing core mathematical concepts like solid geometry, coordinate geometry, trigonometry, and vector operations. Start by constructing basic vectors of the cube, then use orthogonal lines and mirror operations to create symmetric patterns, and finally connect multiple points into polygons to form complex geometric structures, highlighting the cube's symmetry and spatial relationships.
The prompt above was compiled from the original drawing commands and is for reference only.