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Morphing Polyhedron

Morphing Polyhedron
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2019-02-08 17:21
Solid GeometryTransformationGeometric Construction
This example demonstrates the morphing process of polyhedra. Key mathematical concepts include solid geometry and transformation. Using GeoGebra commands such as Intersect, Point, Rotate, and Dodecahedron, one can construct polyhedra and observe their transformations. Suitable for high school or university-level solid geometry studies.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Circle(A, B) → c
  4. Point(c, Mod[1 - o_2, 0.4]) → C
  5. Rotate(C, 72°, A) → C'
  6. Angle(C, A, C') → α
  7. Point(Circle[((C * (1 - sqrt(5))) + (C' * (3 + sqrt(5)))) / 4, (Distance[C, C'] * sqrt(10 + (2 * sqrt(5))) / 4), Segment[C, C']]) → D
  8. Dodecahedron(C, C', D) → a, E, N, O, P, Q, R, S, T, U, faceCC'DEF, faceCFJPK, F, faceCC'GLK, faceC'DHMG, faceDEINH, faceEFJOI, faceJOTUP, faceKLQUP, faceGLQRM, faceHMRSN, faceINSTO, faceQRSTU, G, edgeCF, edgeCC', edgeC'D, edgeDE, edgeEF, edgeCK, edgeFJ, edgeJP, edgeKP, edgeKL, H, edgeGL, edgeC'G, edgeGM, edgeHM, edgeDH, edgeHN, edgeIN, edgeEI, edgeIO, edgeJO, I, edgeOT, edgeTU, edgePU, edgeQU, edgeLQ, edgeQR, edgeMR, edgeRS, edgeNS, edgeST, J, K, L, M
  9. Line(Q, R) → f
  10. Line(G, M) → g
  11. Intersect(g, f) → W
  12. Polygon(Z, A_1, B_1, C_1, D_1) → poly1, z_1, a_1, b_1, c_1, d_1
  13. Mirror(poly1, V) → poly1'
  14. Line(M, R) → d
  15. Line(L, Q) → e
  16. Intersect(e, d) → E_1
  17. Polygon(J_1, F_1, G_1, H_1, I_1) → poly2, j_1, f_1, g_1, i_1, k_1
  18. Mirror(poly2, V) → poly2'
  19. Line(U, Q) → k
  20. Line(S, R) → s
  21. Intersect(s, k) → O_1
  22. Polygon(R_1, Q_1, P_1, T_1, S_1) → poly3, a_2, b_2, c_2, d_2, e_2
  23. Midpoint(E_1, W) → K_1
  24. Line(K_1, edgeRS) → r_1
  25. Plane(E_1, L_1, W) → v
  26. Line(V, R) → f_2
  27. Intersect(f_2, v) → M_1
  28. Polygon(V_1, I_1, H_1, W_1) → q3, v_1, g_2, m_2, w_1
  29. Polygon(W_1, A_1, Z, V_1) → q8, n_2, p_2, z_3, s_2
  30. Mirror(q3, V) → q3'
  31. Mirror(q8, V) → q8'
  32. Mirror(V_1, V) → Z_1
  33. Mirror(W_1, V) → A_2
  34. Mirror(H_1, V) → B_2
  35. Mirror(I_1, V) → C_2
  36. Mirror(Z, V) → Z'
  37. Mirror(A_1, V) → D_2
  38. Polygon(Z_1, Z', D_2, A_2) → q16, b_3, z', c_3, d_3
  39. Polygon(Z_1, A_2, B_2, C_2) → q17, e_3, f_4, h_4, i_4
  40. Midpoint(E_1, O_1) → E_2
  41. Plane(E_1, F_2, O_1) → w
  42. Line(V, Q) → j_4
  43. Intersect(j_4, w) → G_2
  44. Polygon(H_2, J_1, I_1, V_1) → q18, k_4, n_4, p_4, s_4
  45. Polygon(H_2, V_1, P_1, Q_1) → q19, t_4, a_4, b_4, c_4
  46. Mirror(P_1, V) → U_1
  47. Mirror(Q_1, V) → I_2
  48. Mirror(H_2, V) → J_2
  49. Polygon(Z_1, U_1, I_2, J_2) → q1, h, u_1, i_2, j_2
  50. Mirror(J_1, V) → K_2
  51. Polygon(Z_1, J_2, K_2, C_2) → q2, i, j, k_2, l
  52. Mirror(poly3, V) → poly3'
  53. Rotate(V_1, 108°, H_2) → L_2
  54. Polygon(L_2, F_1, J_1, H_2) → q4, l_2, m, n, h_2
  55. Mirror(L_2, V) → M_2
  56. Mirror(F_1, V) → N_2
  57. Polygon(K_2, J_2, M_2, N_2) → q5, p, q, r, t
  58. Line(T, U) → h_1
  59. Line(J, P) → l_1
  60. Intersect(l_1, h_1) → P_2
  61. Polygon(H_2, Q_2, R_2, L_2) → q6, m_1, q_2, r_2, n_1
  62. Mirror(R_2, V) → S_2
  63. Mirror(Q_2, V) → T_2
  64. Polygon(J_2, T_2, S_2, M_2) → q7, p_1, t_2, q_1, s_1
  65. Rotate(H_2, 108°, L_2) → U_2
  66. Mirror(U_2, V) → V_2
  67. Polygon(U_2, G_1, F_1, L_2) → q9, u_2, t_1, e_1, f_3
  68. Polygon(W_1, H_1, G_1, U_2) → q10, g_3, h_3, i_3, j_3
  69. Mirror(G_1, V) → W_2
  70. Polygon(V_2, W_2, N_2, M_2) → q11, v_2, w_2, k_3, l_3
  71. Polygon(A_2, B_2, W_2, V_2) → q12, m_3, n_3, p_3, q_3
  72. Line(R, S) → r_3
  73. Line(U, T) → s_3
  74. Intersect(r_3, s_3) → O_2
  75. Mirror(Z_2, V) → B_3
  76. Mirror(A_3, V) → C_3
  77. Polygon(W_1, U_2, A_3, Z_2) → q13, t_3, a_3, g_4, z_2
  78. Polygon(V_2, C_3, B_3, A_2) → q14, l_4, m_4, q_4, r_4
  79. Line(S, T) → d_4
  80. Line(Q, U) → e_4
  81. Intersect(e_4, d_4) → D_3
  82. Mirror(E_3, V) → G_3
  83. Mirror(F_3, V) → H_3
  84. Polygon(U_2, L_2, F_3, E_3) → q15, f_5, g_5, h_5, i_5
  85. Polygon(M_2, H_3, G_3, V_2) → q20, j_5, k_5, l_5, m_5
  86. Mirror(S_1, V) → I_3
  87. Mirror(R_1, V) → J_3
  88. Mirror(T_1, V) → K_3
  89. Mirror(C_1, V) → L_3
  90. Mirror(D_1, V) → M_3
  91. Mirror(B_1, V) → N_3
  92. Midpoint(Q_2, Q_1) → P_3
  93. Mirror(H_2, P_3) → Q_3
  94. Rotate(V_1, (-(108°)), H_2, faceGLQRM) → R_3
  95. Polygon(H_2, R_3, R_1, Q_1) → q21, n_5, p_5, q_5, r_5
  96. Mirror(R_3, V) → S_3
  97. Rotate(H_2, (-(108°)), R_3, faceGLQRM) → T_3
  98. Polygon(R_3, R_1, S_1, T_3) → q22, s_5, t_5, a_5, b_5
  99. Mirror(T_3, V) → U_3
  100. Polygon(S_3, U_3, I_3, J_3) → q23, c_5, u_3, d_5, e_5
  101. Polygon(J_3, I_3, K_3, U_1, I_2) → poly4, f_6, g_6, h_6, i_6, j_6
  102. Polygon(J_2, S_3, J_3, I_2) → q24, k_6, l_6, m_6, n_6
  103. Rotate(R_3, (-(108°)), T_3, faceGLQRM) → V_3
  104. Mirror(V_3, V) → W_3
  105. Polygon(S_1, T_1, V_3, T_3) → q25, p_6, q_6, v_3, r_6
  106. Polygon(V_1, V_3, T_1, P_1) → q26, s_6, t_6, a_6, b_6
  107. Polygon(U_3, W_3, K_3, I_3) → q27, c_6, w_3, d_6, e_6
  108. Polygon(W_3, Z_1, U_1, K_3) → q28, f_7, g_7, h_7, i_7
  109. Mirror(Z_3, V) → C_4
  110. Mirror(A_4, V) → D_4
  111. Mirror(B_4, V) → E_4
  112. Polygon(R_2, Q_2, A_4, B_4, Z_3) → poly5, j_7, k_7, l_7, m_7, n_7
  113. Polygon(R_3, A_4, Q_2, H_2) → q29, p_7, q_7, r_7, s_7
  114. Polygon(S_2, C_4, E_4, D_4, T_2) → poly6, t_7, a_7, b_7, c_7, d_7
  115. Polygon(T_2, D_4, S_3, J_2) → q30, e_7, f_8, g_8, h_8
  116. Polygon(V_3, V_1, Z, D_1) → q31, i_8, j_8, z_4, k_8
  117. Polygon(Z_1, W_3, M_3, Z') → q32, l_8, m_8, n_8, p_8
  118. Rotate(E_3, 108°, F_3, faceJOTUP) → F_4
  119. Rotate(F_3, 108°, F_4, faceJOTUP) → G_4
  120. Rotate(F_4, 108°, G_4, faceJOTUP) → H_4
  121. Mirror(H_4, V) → I_4
  122. Mirror(G_4, V) → J_4
  123. Mirror(F_4, V) → K_4
  124. Polygon(F_3, F_4, G_4, H_4, E_3) → poly7, q_8, r_8, s_8, t_8, a_8
  125. Polygon(H_3, K_4, J_4, I_4, G_3) → poly8, b_8, c_8, d_8, e_8, f_9
  126. Rotate(V_3, 108°, R_3) → L_4
  127. Polygon(L_4, L_2, R_2, Z_3) → q33, g_9, h_9, i_9, j_9
  128. Mirror(L_4, V) → M_4
  129. Polygon(M_2, M_4, K_4, H_3) → q34, k_9, l_9, m_9, n_9
  130. Rotate(H_2, 108°, R_3, faceKLQUP) → N_4
  131. Mirror(N_4, V) → O_4
  132. Polygon(L_4, Z_3, B_4, N_4) → q35, p_9, q_9, r_9, s_9
  133. Polygon(N_4, B_4, A_4, R_3) → q36, t_9, a_9, b_9, c_9
  134. Polygon(L_4, F_4, F_3, L_2) → q37, d_9, e_9, f_{10}, g_{10}
  135. Polygon(M_3, L_3, N_3, D_2, Z') → poly9, h_{10}, i_{10}, j_{10}, k_{10}, l_{10}
  136. Polygon(L_4, N_4, N_3, L_3) → q38, m_{10}, n_{10}, p_{10}, q_{10}
  137. Polygon(T_3, V_3, J_4, I_4) → q39, r_{10}, s_{10}, t_{10}, a_{10}
  138. Polygon(V_3, M_4, K_4, J_4) → q40, b_{10}, c_{10}, d_{10}, e_{10}
  139. Polygon(V_3, D_1, C_1, M_4) → q41, f_{11}, g_{11}, h_{11}, i_{11}
  140. Polygon(M_4, C_1, B_1, O_4) → q42, j_{11}, k_{11}, l_{11}, o_4
  141. Polygon(O_4, B_1, A_1, W_1) → q43, m_{11}, n_{11}, p_{11}, q_{11}
  142. Polygon(M_2, M_4, C_4, S_2) → q44, r_{11}, s_{11}, t_{11}, a_{11}
  143. Polygon(M_4, O_4, E_4, C_4) → q45, b_{11}, c_{11}, d_{11}, e_{11}
  144. Polygon(O_4, S_3, D_4, E_4) → q46, f_{12}, g_{12}, h_{12}, i_{12}
  145. Polygon(W_3, L_4, L_3, M_3) → q47, j_{12}, k_{12}, l_{12}, m_{12}
  146. Polygon(W_3, G_4, F_4, L_4) → q48, n_{12}, p_{12}, q_{12}, r_{12}
  147. Polygon(U_3, H_4, G_4, W_3) → q49, s_{12}, t_{12}, a_{12}, b_{12}
  148. Polygon(U_2, E_3, H_4, U_3) → q50, c_{12}, d_{12}, e_{12}, f_{13}
  149. Polygon(D_2, N_3, N_4, A_2) → q51, g_{13}, h_{13}, i_{13}, j_{13}
  150. Polygon(V_2, T_3, I_4, G_3) → q52, k_{13}, l_{13}, m_{13}, n_{13}
  151. Rotate(Z_2, 108°, A_3, faceINSTO) → P_4
  152. Mirror(P_4, V) → Q_4
  153. Polygon(A_3, U_2, U_3, P_4) → q53, p_{13}, q_{13}, r_{13}, s_{13}
  154. Polygon(C_3, Q_4, T_3, V_2) → q54, t_{13}, a_{13}, b_{13}, c_{13}
  155. Rotate(A_3, 108°, P_4, faceINSTO) → R_4
  156. Mirror(R_4, V) → S_4
  157. Polygon(S_3, R_4, P_4, U_3) → q55, d_{13}, e_{13}, f_{14}, g_{14}
  158. Polygon(R_3, T_3, Q_4, S_4) → q56, h_{14}, i_{14}, j_{14}, k_{14}
  159. Rotate(P_4, 108°, R_4, faceINSTO) → T_4
  160. Mirror(T_4, V) → U_4
  161. Polygon(O_4, T_4, R_4, S_3) → q57, l_{14}, m_{14}, n_{14}, p_{14}
  162. Polygon(W_1, Z_2, T_4, O_4) → q58, q_{14}, r_{14}, s_{14}, t_{14}
  163. Polygon(Z_2, A_3, P_4, R_4, T_4) → poly10, a_{14}, b_{14}, c_{14}, d_{14}, e_{14}
  164. Polygon(N_4, R_3, S_4, U_4) → q59, f_{15}, g_{15}, h_{15}, u_4
  165. Polygon(N_4, U_4, B_3, A_2) → q60, i_{15}, j_{15}, k_{15}, l_{15}
  166. Polygon(S_4, Q_4, C_3, B_3, U_4) → poly11, m_{15}, n_{15}, p_{15}, q_{15}, r_{15}