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Dissecting a Cube into 4 Equal Parts

Dissecting a Cube into 4 Equal Parts
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2020-09-08 03:27
VectorSolid GeometryTransformationGeometric Construction
① Demonstrates how a cube can be dissected into four congruent parts using geometric transformations. ② Core concepts include solid geometry, geometric transformations (reflection, rotation, translation), similarity transformations, and vectors. ③ Construction highlights: commands such as Intersect, Point, Polygon, Midpoint, Segment, Circle, LineBisector, Line, Mirror, Rotate, Prism, Vector, and Translate are employed to create vertices, edges, faces, midpoints, mirrored and rotated objects. ④ Suitable for high‑school solid geometry lessons, geometric construction practice, and university‑level geometry demonstrations.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → E
  3. Polygon(E, F, 4) → poly2, j, k, l, m, G, H
  4. Midpoint(j) → L
  5. Midpoint(l) → K
  6. Segment(K, L) → p
  7. Point(p) → U
  8. Point(p) → T
  9. Midpoint(U, T) → H_1
  10. Circle(H_1, 0.95) → d_2
  11. LineBisector(T, U) → e
  12. Intersect(d_2, e, 2) → I_1
  13. Segment(H_1, I_1) → l_1
  14. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  15. Midpoint(h) → I
  16. Midpoint(f) → J
  17. Segment(I, J) → n
  18. Point(n) → M
  19. Midpoint(I, J) → N
  20. Circle(M, 0.2) → c
  21. Intersect(c, n, 2) → O
  22. Line(M, h) → q
  23. Line(O, q) → r
  24. Point(q) → P
  25. Line(P, n) → s
  26. Intersect(r, s) → Q
  27. Mirror(P, N) → P'
  28. Mirror(Q, N) → Q'
  29. Mirror(O, N) → O'
  30. Mirror(M, N) → M'
  31. Polygon(D, I, J, A, M, P, Q, O, O', Q', P', M') → poly3, d, m', j_1, a, i_1, m_1, p_1, q_1, o, o', q', p'
  32. Polygon(C, I, J, B, M, P, Q, O, O', Q', P', M') → poly4, c_1, m'_1, j_2, b, i_2, m_2, p_2, q_2, o_1, o'_1, q'_1, p'_1
  33. Midpoint(m) → R
  34. Midpoint(k) → S
  35. Segment(R, S) → t
  36. Intersect(t, p) → W
  37. Rotate(T, 90°, W) → T'
  38. Point(l_1) → V
  39. Rotate(V, 90°, W) → V'
  40. Rotate(U, 90°, W) → U'
  41. Polygon(K, H, R, T', V', U', W, U, V, T) → poly5, k_1, t_1, h_1, r_1, t', v', u', w, u_3, v
  42. Rotate(K, 90°, W) → K'
  43. Rotate(H, 90°, W) → H'
  44. Rotate(R, 90°, W) → R'
  45. Rotate(T', 90°, W) → T''
  46. Rotate(V', 90°, W) → V''
  47. Rotate(U', 90°, W) → U''
  48. Rotate(W, 90°, W) → W'
  49. Rotate(U, 90°, W) → U'_1
  50. Rotate(V, 90°, W) → V'_1
  51. Rotate(T, 90°, W) → T'_1
  52. Polygon(K', H', R', T'', V'', U'', W', U'_1, V'_1, T'_1) → poly5', k', t'_1, h', r', t'', v'', u'', w', u'_1, v'_1
  53. Rotate(K', 90°, W) → K''
  54. Rotate(H', 90°, W) → H''
  55. Rotate(R', 90°, W) → R''
  56. Rotate(T'', 90°, W) → T'''
  57. Rotate(V'', 90°, W) → V'''
  58. Rotate(U'', 90°, W) → U'''
  59. Rotate(W', 90°, W) → W''
  60. Rotate(U'_1, 90°, W) → Z
  61. Rotate(V'_1, 90°, W) → A_1
  62. Rotate(T'_1, 90°, W) → B_1
  63. Polygon(K'', H'', R'', T''', V''', U''', W'', Z, A_1, B_1) → poly5'', k'', b_1, h'', r'', t''', v''', u''', w'', z_1, a_1
  64. Rotate(K'', 90°, W) → K'''
  65. Rotate(H'', 90°, W) → H'''
  66. Rotate(R'', 90°, W) → R'''
  67. Rotate(T''', 90°, W) → C_1
  68. Rotate(V''', 90°, W) → D_1
  69. Rotate(U''', 90°, W) → E_1
  70. Rotate(W'', 90°, W) → W'''
  71. Rotate(Z, 90°, W) → Z'
  72. Rotate(A_1, 90°, W) → F_1
  73. Rotate(B_1, 90°, W) → G_1
  74. Polygon(K''', H''', R''', C_1, D_1, E_1, W''', Z', F_1, G_1) → poly5''', k''', g_1, h''', r''', c_2, d_1, e_1, w''', z', f_1
  75. Prism(poly5, n_1) → prism1, R_1, C_2, face12, face13, face15, face20, face21, face19, face18, face17, face16, S_1, face14, face22, edge22, edge21, edge23, edge28, edge29, edge30, edge27, edge25, T_1, edge26, edge24, edge31, edge33, edge34, edge35, edge36, edge37, edge38, edge39, U_1, edge40, edge32, V_1, W_1, Z_1, A_2, B_2
  76. Segment(W, K_1) → s_1
  77. Segment(W, L_1) → f_2
  78. Circle(W, (a_2 * s_1)) → e_2
  79. Intersect(s_1, e_2, 1) → M_1
  80. Intersect(f_2, e_2, 1) → N_1
  81. Vector(W, M_1) → u
  82. Vector(W, N_1) → u_1
  83. Translate(prism1, Vector[(-u_1)]) → prism1'
  84. Prism(poly5''', b_2) → prism2, K_2, T_2, face23, face25, face26, face27, face28, face29, face30, face31, face32, L_2, face24, face33, edge41, edge42, edge43, edge44, edge45, edge46, edge47, edge48, M_2, edge49, edge50, edge51, edge53, edge54, edge55, edge56, edge57, edge58, edge59, N_2, edge60, edge52, O_2, P_2, Q_2, R_2, S_2
  85. Translate(prism2, u) → prism2'
  86. Prism(poly5', g_2) → prism4, U_2, F_3, face34, face36, face37, face38, face39, face40, face41, face42, face43, V_2, face35, face44, edge61, edge62, edge63, edge64, edge65, edge66, edge67, edge68, W_2, edge69, edge70, edge71, edge73, edge74, edge75, edge76, edge77, edge78, edge79, Z_2, edge80, edge72, A_3, B_3, C_3, D_3, E_3
  87. Prism(poly5'', k_2) → prism3, G_3, P_3, face45, face47, face48, face49, face50, face51, face52, face53, face54, H_3, face46, face55, edge81, edge82, edge83, edge84, edge85, edge86, edge87, edge88, I_3, edge89, edge90, edge91, edge93, edge94, edge95, edge96, edge97, edge98, edge99, J_3, edge100, edge92, K_3, L_3, M_3, N_3, O_3
  88. Translate(prism3, u_1) → prism3'
  89. Translate(prism4, Vector[(-u)]) → prism4'

AI Drawing Prompt Reference

Draw a cube and dissect it into four congruent parts. Start by creating the cube's vertices and edges using transformations like reflection, rotation, and translation. Then, use midpoints, segments, and circles to construct the cube's faces and division lines. Finally, apply mirroring and rotation to divide the cube into four congruent parts.
The prompt above was compiled from the original drawing commands and is for reference only.