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Transformation from a Cube to a Rhombic Triacontahedron

Transformation from a Cube to a Rhombic Triacontahedron
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2019-02-18 18:11
Solid GeometryTransformationGeometric Construction
This example demonstrates the transformation of a cube into a rhombic triacontahedron through rotation and mirroring. Key mathematical concepts include geometric transformations and polyhedra. Using GeoGebra commands such as Rotate, Mirror, and Polygon, the process of transforming a 3D shape is visually illustrated. Suitable for high school or university geometry courses.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Circle(A, B) → c
  4. Point(c, 1 - b) → C
  5. Rotate(C, 90°, A) → C'
  6. Angle(C, A, C') → α
  7. Rotate(C', 90°, A) → C''
  8. Angle(C', A, C'') → β
  9. Point(Circle[C', Distance[C, C'], Segment[C, C']]) → D
  10. Cube(C, C', C'') → a, E, face5, face6, edge1, edge2, edge3, edge4, edge5, edge6, edge7, edge8, F, edge9, edge10, edge11, edge12, G, H, I, face1, face2, face3, face4
  11. Midpoint(M, J) → B_1
  12. Mirror(C_1, B_1) → H_1
  13. Midpoint(N, P) → R
  14. Midpoint(K, J) → T
  15. Segment(T, B_1) → f
  16. Intersect(f, Circle[C_1, f, Polygon[M, K, J, L]]) → U, V
  17. Mirror(V, B_1) → V'
  18. Polygon(V, C_1, V', H_1) → q1, v, c_1, v', h_1
  19. Mirror(S, R) → S'
  20. Segment(S', V) → g
  21. Mirror(V, W) → V'_1
  22. Mirror(C_1, W) → Z
  23. Mirror(V', W) → V''
  24. Mirror(H_1, W) → A_1
  25. Polygon(V'', Z, V'_1, A_1) → q2, v'', z_1, v'_1, a_1
  26. Midpoint(N, O) → D_1
  27. Segment(D_1, R) → h
  28. Polygon(N, O, P, Q) → q3, n_1, o, p, q
  29. Intersect(h, Circle[C_1, h, q3]) → E_1,
  30. Intersect(h, Circle[S, h, Polygon[N, O, P, Q]]) → G_1, G_2
  31. Mirror(G_2, R) → F_1
  32. Polygon(S, G_2, S', F_1) → q4, s, g_2, s', f_1
  33. Polygon(I_1, V, S', F_1) → q5, i_1, v_1, i, j
  34. Polygon(V, J_1, G_2, S') → q6, v_2, j_1, k, l
  35. Polygon(C_1, V, I_1, K_1) → q7, r, v_3, t, k_1
  36. Polygon(L_1, J_1, V, H_1) → q8, l_1, g_1, v_4, m_1
  37. Mirror(S, W) → S'_1
  38. Mirror(F_1, W) → M_1
  39. Mirror(S', W) → S''
  40. Mirror(G_2, W) → N_1
  41. Polygon(M_1, S'', N_1, S'_1) → q9, p_1, s'', q_1, s'_1
  42. Mirror(I_1, W) → O_1
  43. Mirror(K_1, W) → P_1
  44. Mirror(J_1, W) → Q_1
  45. Mirror(L_1, W) → R_1
  46. Polygon(V'_1, Q_1, R_1, A_1) → q10, r_1, s_1, t_1, b_1
  47. Polygon(Z, P_1, O_1, V'_1) → q11, z_2, d_1, o_1, e_1
  48. Polygon(V'_1, S'', N_1, Q_1) → q12, f_2, h_2, i_2, j_2
  49. Polygon(O_1, M_1, S'', V'_1) → q13, k_2, l_2, m_2, n_2
  50. Polygon(K_1, S_1, F_1, I_1) → q14, p_2, q_2, r_2, s_2
  51. Mirror(S_1, W) → T_1
  52. Polygon(P_1, T_1, M_1, O_1) → q15, t_2, a_2, b_2, c_2
  53. Polygon(V', U_1, K_1, C_1) → q16, d_2, u_1, e_2, f_3
  54. Mirror(U_1, W) → V_1
  55. Polygon(V'', V_1, P_1, Z) → q17, g_3, h_3, i_3, z_3
  56. Polygon(V', W_1, L_1, H_1) → q18, j_3, w_1, k_3, l_3
  57. Mirror(W_1, W) → Z_1
  58. Polygon(R_1, Z_1, V'', A_1) → q19, m_3, n_3, p_3, q_3
  59. Polygon(K_1, S_1, R_1, A_2) → q20, r_3, s_3, t_3, a_3
  60. Mirror(A_2, W) → B_2
  61. Polygon(P_1, B_2, L_1, T_1) → q21, b_3, c_3, d_3, e_3
  62. Polygon(V_1, G_2, B_2, P_1) → q22, f_4, g_4, h_4, i_4
  63. Polygon(B_2, G_2, J_1, L_1) → q23, j_4, k_4, l_4, m_4
  64. Polygon(T_1, L_1, W_1, M_1) → q24, n_4, p_4, q_4, r_4
  65. Polygon(V'', S, G_2, V_1) → q25, s_4, s_5, t_4, a_4
  66. Polygon(Z_1, F_1, S, V'') → q26, b_4, c_4, s_6, d_4
  67. Polygon(S_1, F_1, Z_1, R_1) → q27, e_4, f_5, g_5, h_5
  68. Polygon(N_1, A_2, R_1, Q_1) → q28, i_5, j_5, k_5, l_5
  69. Polygon(N_1, U_1, K_1, A_2) → q29, m_5, n_5, p_5, q_5
  70. Polygon(N_1, S'_1, V', U_1) → q30, r_5, t_5, a_5, b_5
  71. Polygon(W_1, V', S'_1, M_1) → q31, c_5, d_5, e_5, f_6