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Cube Dissection Demo

Cube Dissection Demo
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2019-06-03 22:10
VectorSolid GeometryTransformationGeometric Construction
In collections Dissecting a cube into equal parts From POST-IT to 3D PRINTING: playing with squares and cubes
This example demonstrates how to dissect a cube into four equal parts. It involves core mathematical concepts such as solid geometry, geometric transformations (reflection, rotation, translation), and similarity transformations. Key GeoGebra commands and objects used in the construction include Intersect, Point, Polygon, Midpoint, Segment, Circle, LineBisector, Line, Mirror, Rotate, Prism, Vector, and Translate. Suitable for high school and above students studying solid geometry, geometric transformations, and spatial reasoning.

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 demonstrate how to dissect it into four equal parts. Start by plotting the intersection point of the coordinate axes and the vertices of the cube, then create the faces of the cube. Find midpoints of key points and draw segments and circles. Use intersections of midpoints and circles to create auxiliary segments and new points. Apply mirror and rotation transformations to parts of the cube, then combine all sections to show the dissection effect.
The prompt above was compiled from the original drawing commands and is for reference only.