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Quadrature of a Cube

Quadrature of a Cube
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2013-10-13 22:18
VectorSolid GeometryGeometric Construction
This example demonstrates how to cover a cube with a square. It involves the geometric properties of a cube and the problem of quadrature. Using GeoGebra commands such as Polygon, Line, Circle, etc., a series of geometric objects are constructed. Suitable for high school mathematics education and students learning solid geometry.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Polygon(A, D, 4) → poly2, e, f, g, h, E, F
  3. Polygon(D, C, 4) → poly3, i, j, k, l, G, H
  4. Polygon(E, D, 4) → poly4, m, n, p, q, I, J
  5. Line(G, B) → r
  6. Circle(B, D) → s
  7. Intersect(s, r, 2) → K
  8. Polygon(B, K, 4) → poly5, t, a_1, b_1, c_1, L, M
  9. Midpoint(b_1) → N
  10. Semicircle(M, L) → d_1
  11. Midpoint(t) → O
  12. Ray(O, N) → e_1
  13. Intersect(d_1, e_1, 1) → P
  14. Polygon(M, P, 4) → poly6, f_1, g_1, h_1, i_1, Q, R
  15. Polygon(P, L, 4) → poly7, j_1, k_1, l_1, m_1, S, T
  16. Polygon(K, F, 4) → poly8, n_1, p_1, q_1, r_1, U, V
  17. Midpoint(M, Q) → W
  18. Polygon(P, W, Q) → poly9, q_2, p_2, w
  19. Polygon(P, M, W) → poly10, w_1, p_3, m_2
  20. Polygon(W, R, M) → poly11, m_3, w_2, r_2
  21. Polygon(W, R, Q) → poly12, q_3, w_3, r_3
  22. Midpoint(c_1) → Z
  23. Polygon(N, Z, M) → poly13, m_4, n_2, z
  24. Midpoint(a_1) → A_1
  25. Polygon(A_1, N, L) → poly14, l_2, a_2, n_3
  26. Polygon(A_1, O, K) → poly15, k_2, a_3, o
  27. Polygon(O, Z, B) → poly16, b_2, o_1, z_1
  28. Line(D, q_1) → s_1
  29. OrthogonalLine(D, s_1) → t_1
  30. Polygon(A_1, O, Z, N) → poly17, a_4, o_2, z_2, n_4
  31. Intersect(s_1, b) → B_1
  32. Intersect(s_1, q) → C_1
  33. Intersect(t_1, h) → D_1
  34. Intersect(t_1, k) → E_1
  35. Polygon(D_1, D, A) → poly18, a_5, d_2, d_3
  36. Polygon(A, D, B_1, B) → poly19, a_6, d_4, b_3, b_4
  37. Polygon(D, E_1, G, C) → poly20, d_5, e_2, g_2, c_2
  38. Polygon(B_1, D, C) → poly21, c_3, b_5, d_6
  39. Polygon(D_1, F, E, D) → poly22, d_7, f_2, e_3, d_8
  40. Polygon(E, C_1, D) → poly23, d_9, e_4, c_4
  41. Polygon(C_1, J, H, D) → poly24, c_5, j_2, h_2, d_{10}
  42. Polygon(D, H, E_1) → poly25, e_5, d_{11}, h_3
  43. Vector(D, F) → u
  44. Translate(D_1, u) → F_1
  45. Translate(D, u) → D'
  46. Translate(A, u) → A'
  47. Polygon(F_1, D', A') → poly18', a', f_3, d'
  48. Translate(A, u) → A'_1
  49. Translate(D, u) → D'_1
  50. Translate(B_1, u) → G_1
  51. Translate(B, u) → B'
  52. Polygon(A'_1, D'_1, G_1, B') → poly19', a'_1, d'_1, g_3, b'
  53. Vector(D, K) → v
  54. Translate(E, v) → E'
  55. Translate(C_1, v) → H_1
  56. Translate(D, v) → D'_2
  57. Polygon(E', H_1, D'_2) → poly23', d'_2, e', h_4
  58. Translate(D_1, v) → I_1
  59. Translate(F, v) → F'
  60. Translate(E, v) → E'_1
  61. Translate(D, v) → D'_3
  62. Polygon(I_1, F', E'_1, D'_3) → poly22', i_2, f', e'_1, d'_3
  63. Vector(D, U) → u_1
  64. Translate(B_1, u_1) → J_1
  65. Translate(D, u_1) → D'_4
  66. Translate(C, u_1) → C'
  67. Polygon(J_1, D'_4, C') → poly21', c', j_3, d'_4
  68. Translate(D, u_1) → D'_5
  69. Translate(E_1, u_1) → K_1
  70. Translate(G, u_1) → G'
  71. Translate(C, u_1) → C'_1
  72. Polygon(D'_5, K_1, G', C'_1) → poly20', d'_5, k_3, g', c'_1
  73. Vector(D, V) → v_1
  74. Translate(C_1, v_1) → L_1
  75. Translate(J, v_1) → J'
  76. Translate(H, v_1) → H'
  77. Translate(D, v_1) → D'_6
  78. Polygon(L_1, J', H', D'_6) → poly24', l_3, j', h', d'_6
  79. Translate(D, v_1) → D'_7
  80. Translate(H, v_1) → H'_1
  81. Translate(E_1, v_1) → M_1
  82. Polygon(D'_7, H'_1, M_1) → poly25', m_5, d'_7, h'_1
  83. Vector(W, J') → u_2
  84. Translate(W, u_2) → W'
  85. Translate(R, u_2) → R'
  86. Translate(Q, u_2) → Q'
  87. Polygon(W', R', Q') → poly12', q', w', r'
  88. Vector(W, G') → v_2
  89. Vector(W, F') → s_2
  90. Vector(W, B') → t_2
  91. Translate(W, s_2) → W'_1
  92. Translate(R, s_2) → R'_1
  93. Translate(M, s_2) → M'
  94. Polygon(W'_1, R'_1, M') → poly11', m', w'_1, r'_1
  95. Translate(P, t_2) → P'
  96. Translate(M, t_2) → M'_1
  97. Translate(W, t_2) → W'_2
  98. Polygon(P', M'_1, W'_2) → poly10', w'_2, p', m'_1
  99. Translate(P, v_2) → P'_1
  100. Translate(W, v_2) → W'_3
  101. Translate(Q, v_2) → Q'_1
  102. Polygon(P'_1, W'_3, Q'_1) → poly9', q'_1, p'_1, w'_3
  103. Vector(T, R'_1) → u_3
  104. Translate(poly7, u_3) → poly7'
  105. Midpoint(T, L) → N_1
  106. Midpoint(B, D) → O_1
  107. Midpoint(A, E) → P_1
  108. Midpoint(E, H) → Q_1
  109. Midpoint(C, H) → R_1

AI Drawing Prompt Reference

Draw a cube and cover it with a square. Start by constructing four squares using the vertices A, D, C, and E, then use these squares to create several lines and circles. Next, use these lines and circles to find intersection points, and finally, construct multiple squares using these points and vertices to cover the entire cube.
The prompt above was compiled from the original drawing commands and is for reference only.