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Painted Cube 4x4x4

Painted Cube 4x4x4
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2016-03-06 20:54
AngleSolid GeometryTransformationGeometric Construction
① This example demonstrates a 4×4×4 painted cube, allowing users to explore its three‑dimensional structure and symmetry through interactive geometry. ② The core mathematical concepts include solid geometry (vertices, faces, edges of a cube), translation transformations, planar symmetry, and the geometry of right triangles formed by the cube's diagonals. ③ The construction uses GeoGebra commands such as Point, Intersect, Cube, Segment, and PointIn to create vertices (A, B, C, …), face objects (faceABCD, …), and edge objects, enabling visualization of painted faces and measurement of small cube sections. ④ It is suitable for high‑school geometry classes or modeling projects, helping students understand spatial properties of solids and practice dynamic geometric exploration.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Intersect(xAxis, zAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. Point(edgeAB) → I
  6. Point(edgeEF) → J
  7. Segment(I, J) → b
  8. Point(edgeAB) → K
  9. Point(edgeEF) → L
  10. Segment(K, L) → c
  11. PointIn(faceABFE) → M
  12. Point(edgeEF) → N
  13. Segment(M, N) → d
  14. Point(edgeCD) → O
  15. Point(edgeGH) → P
  16. Segment(O, P) → e
  17. Point(edgeCD) → Q
  18. PointIn(faceCDHG) → R
  19. Segment(Q, R) → f
  20. Point(edgeCD) → S
  21. Point(edgeGH) → T
  22. Segment(S, T) → g
  23. Segment(J, T) → h
  24. Segment(L, R) → i
  25. Segment(P, N) → j
  26. Point(xAxis) → U
  27. Point(edgeFG) → V
  28. Segment(U, V) → k
  29. Point(xAxis) → W
  30. PointIn(faceBCGF) → Z
  31. Segment(W, Z) → l
  32. PointIn(faceBCGF) → B_1
  33. Segment(A_1, B_1) → m
  34. Segment(U, C_1) → n
  35. Segment(W, D_1) → p
  36. Point(edgeAD) → E_1
  37. Segment(A_1, E_1) → q
  38. Segment(O, M) → r
  39. Segment(Q, K) → s
  40. Segment(S, I) → t
  41. Point(edgeEH) → F_1
  42. Segment(C_1, F_1) → a_1
  43. Intersect(p, edgeAD) → G_1
  44. Point(edgeEH) → H_1
  45. Segment(D_1, H_1) → b_1
  46. Point(edgeEH) → I_1
  47. Segment(E_1, I_1) → c_1
  48. Point(zAxis) → J_1
  49. Point(edgeCG) → K_1
  50. Segment(J_1, K_1) → d_1
  51. Point(zAxis) → L_1
  52. Point(edgeCG) → M_1
  53. Segment(L_1, M_1) → e_1
  54. Point(edgeBF) → N_1
  55. Point(edgeCG) → O_1
  56. Segment(N_1, O_1) → f_1
  57. Point(edgeCG) → P_1
  58. Point(edgeDH) → Q_1
  59. Segment(P_1, Q_1) → g_1
  60. PointIn(faceCDHG) → R_1
  61. Point(edgeDH) → S_1
  62. Segment(R_1, S_1) → h_1
  63. Point(edgeCG) → T_1
  64. Point(edgeDH) → U_1
  65. Segment(T_1, U_1) → i_1
  66. Point(edgeAE) → V_1
  67. Segment(Q_1, V_1) → j_1
  68. PointIn(faceADHE) → W_1
  69. Segment(S_1, W_1) → k_1
  70. Point(edgeAE) → Z_1
  71. Segment(U_1, Z_1) → l_1
  72. Intersect(zAxis, f_1) → A_2
  73. Segment(Z_1, A_2) → m_1
  74. Point(zAxis) → B_2
  75. Segment(B_2, W_1) → n_1
  76. Point(zAxis) → C_2
  77. Segment(V_1, C_2) → p_1
  78. Intersect(edgeEH, a_1) → D_2
  79. Point(edgeFG) → E_2
  80. Segment(D_2, E_2) → q_1
  81. Intersect(edgeFG, l) → F_2
  82. Intersect(edgeEH, b_1) → G_2
  83. Segment(F_2, G_2) → r_1
  84. Intersect(edgeEH, c_1) → H_2
  85. Intersect(edgeFG, m) → I_2
  86. Segment(H_2, I_2) → s_1

AI Drawing Prompt Reference

Draw a 4×4×4 cube showing its vertices, edges, faces, and the internal grid of 64 smaller cubes. All six faces are painted and marked with equally spaced lines parallel to the edges, forming a 4×4 square grid on each face. Inside the cube, draw three sets of parallel cross‑sections along the three axes to create a 3D lattice. Also depict the space diagonals and the right triangles formed with face diagonals, illustrating translational and planar symmetry. The figure should display the full vertex‑edge‑face relationships of the solid, making the painted faces and small cube sections easy to locate.
The prompt above was compiled from the original drawing commands and is for reference only.