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From Cube to Tetrahedron

From Cube to Tetrahedron
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2022-01-20 19:42
Coordinate GeometrySolid GeometryCalculusGeometric Construction
This example demonstrates the continuous transformation from a cube to a tetrahedron. It involves core mathematical concepts such as solid geometry, continuity, calculus, and coordinate geometry. The construction utilizes GeoGebra commands and objects like Point, Cube, Sphere, Intersect, Polygon, Segment, and Rotate. Suitable for high school students and math enthusiasts exploring spatial geometric transformations and calculus applications.

GeoGebra Commands & Parameters

  1. Point(Circle[A, Distance[B, A], Segment[B, A]]) → I
  2. Cube(B, A, I) → a, J, faceIJNM, faceKLMN, edgeBJ, edgeAB, edgeAI, edgeIJ, edgeBK, edgeJN, edgeKN, edgeKL, K, edgeAL, edgeLM, edgeIM, edgeMN, L, M, N, faceABJI, faceBJNK, faceABKL, faceAIML
  3. Sphere(G, s) → b
  4. Sphere(F, s) → c
  5. Sphere(H, s) → d
  6. Sphere(E, s) → e
  7. Sphere(I, s) → f
  8. Sphere(C, s) → g
  9. Sphere(B, s) → h
  10. Sphere(A, s) → i
  11. Intersect(edgeKL, e) → O, O_1
  12. Intersect(edgeKL, c) → P_1, P
  13. Intersect(edgeAB, h) → Q_1, Q
  14. Intersect(edgeAB, i) → R, R_1
  15. Intersect(edgeLM, d) → S, S_1
  16. Intersect(edgeMN, b) → T, T_1
  17. Intersect(edgeMN, d) → U_1, U
  18. Intersect(edgeIM, d) → V_1, V
  19. Intersect(edgeKN, b) → W_1, W
  20. Intersect(edgeJN, b) → Z, Z_1
  21. Intersect(edgeAL, e) → A_2, A_1
  22. Intersect(edgeLM, e) → B_2, B_1
  23. Intersect(edgeIM, f) → C_1, C_2
  24. Intersect(edgeIJ, f) → D_2, D_1
  25. Intersect(edgeIJ, g) → E_1, E_2
  26. Intersect(edgeJN, g) → F_2, F_1
  27. Intersect(edgeBJ, g) → G_2, G_1
  28. Intersect(edgeBJ, h) → H_1, H_2
  29. Intersect(edgeBK, h) → I_1, I_2
  30. Intersect(edgeBK, c) → J_2, J_1
  31. Intersect(edgeKN, c) → K_1, K_2
  32. Polygon(K_1, J_1, P) → t1, p, k_1, j_1
  33. Polygon(B_1, O, A_1) → t2, a_1, b_1, o
  34. Polygon(S, V, U) → t3, u, s_1, v
  35. Polygon(T, Z, W) → t4, w, t, z_1
  36. Intersect(edgeAL, i) → L_1, L_2
  37. Intersect(edgeAI, f) → M_1, M_2
  38. Intersect(edgeAI, i) → N_2, N_1
  39. Polygon(L_1, N_1, R) → t5, r, l_1, n_1
  40. Polygon(D_1, M_1, C_1) → t6, c_1, d_1, m_1
  41. Polygon(F_1, G_1, E_1) → t7, e_1, f_1, g_1
  42. Polygon(I_1, H_1, Q) → t8, q, i_1, h_1
  43. Segment(K_1, W) → j
  44. Segment(T, U) → k
  45. Segment(S, B_1) → l
  46. Segment(J_1, I_1) → m
  47. Segment(P, O) → n
  48. Segment(Z, F_1) → p_1
  49. Segment(A_1, L_1) → q_1
  50. Segment(R, Q) → r_1
  51. Segment(G_1, H_1) → t_1
  52. Segment(D_1, E_1) → f_2
  53. Segment(V, C_1) → g_2
  54. Segment(N_1, M_1) → h_2
  55. Segment(T, Z) → i_2
  56. Segment(F_1, E_1) → j_2
  57. Segment(C_1, D_1) → k_2
  58. Segment(V, U) → l_2
  59. Intersect(l_2, i_2) → O_2
  60. Intersect(l_2, k_2) → P_2
  61. Intersect(j_2, k_2) → Q_2
  62. Intersect(j_2, i_2) → R_2
  63. Polygon(O_2, P_2, Q_2, R_2) → q1, o_2, p_2, q_2, r_2
  64. Rotate(q1, 90°, zAxis) → q1'
  65. Rotate(q1, 180°, zAxis) → q1'_{1}
  66. Rotate(q1, 270°, zAxis) → q1'_{2}
  67. Rotate(O_2, 90°, zAxis) → S_2
  68. Rotate(P_2, 90°, zAxis) → T_2
  69. Rotate(Q_2, 90°, zAxis) → U_2
  70. Rotate(R_2, 90°, zAxis) → V_2
  71. Rotate(O_2, 180°, zAxis) → W_2
  72. Rotate(P_2, 180°, zAxis) → Z_2
  73. Rotate(Q_2, 180°, zAxis) → A_3
  74. Rotate(R_2, 180°, zAxis) → B_3
  75. Rotate(O_2, 270°, zAxis) → C_3
  76. Rotate(P_2, 270°, zAxis) → D_3
  77. Rotate(Q_2, 270°, zAxis) → E_3
  78. Rotate(R_2, 270°, zAxis) → F_3
  79. Rotate(q1, 90°, yAxis) → q1'_{3}
  80. Rotate(O_2, 90°, yAxis) → G_3
  81. Rotate(P_2, 90°, yAxis) → H_3
  82. Rotate(Q_2, 90°, yAxis) → I_3
  83. Rotate(R_2, 90°, yAxis) → J_3
  84. Rotate(q1, -90°, yAxis) → q1'_{4}
  85. Rotate(O_2, -90°, yAxis) → K_3
  86. Rotate(P_2, -90°, yAxis) → L_3
  87. Rotate(Q_2, -90°, yAxis) → M_3
  88. Rotate(R_2, -90°, yAxis) → N_3
  89. Polygon(R_2, T_2, S_2, N_3, M_3, O_2) → poly1, m_2, t_2, s_2, n_3, m_3, n_2
  90. Polygon(N_3, K_3, W_2, Z_2, V_2, S_2) → poly2, a_2, k_3, w_2, z_2, v_2, b_2
  91. Polygon(W_2, B_3, D_3, C_3, L_3, K_3) → poly3, c_2, b_3, d_3, c_3, l_3, d_2
  92. Polygon(C_3, F_3, P_2, O_2, M_3, L_3) → poly4, e_2, f_3, g_3, h_3, i_3, j_3
  93. Polygon(U_2, J_3, I_3, A_3, Z_2, V_2) → poly5, u_2, p_3, q_3, a_3, r_3, s_3
  94. Polygon(A_3, I_3, H_3, E_3, D_3, B_3) → poly6, t_3, e_3, f_4, g_4, h_4, i_4
  95. Polygon(E_3, H_3, G_3, Q_2, P_2, F_3) → poly7, j_4, k_4, l_4, m_4, n_4, p_4
  96. Polygon(R_2, Q_2, G_3, J_3, U_2, T_2) → poly8, q_4, r_4, s_4, t_4, a_4, b_4
  97. Polygon(S, U, T, W, K_1, P, O, B_1) → poly9, s_5, u_1, t_5, w_1, c_4, p_5, o_1, d_4
  98. Rotate(poly9, 90°, xAxis) → poly9'
  99. Rotate(poly9, 180°, xAxis) → poly9'_{1}
  100. Rotate(poly9, 270°, xAxis) → poly9'_{2}
  101. Rotate(K_1, -90°, yAxis) → O_3
  102. Rotate(J_1, -90°, yAxis) → P_3
  103. Segment(O_3, P_3) → p'
  104. Rotate(poly9'_{2}, 90°, zAxis) → poly10
  105. Rotate(poly9'_{2}, -90°, zAxis) → poly11

AI Drawing Prompt Reference

Start with a cube and gradually transform it into a tetrahedron. First, create a cube, then use rotation and intersection to gradually change the cube's shape until it forms a tetrahedron. Pay attention to the changes in key points and faces at each step.
The prompt above was compiled from the original drawing commands and is for reference only.