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Schlegel Diagram of Dodecahedron

Schlegel Diagram of Dodecahedron
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2017-12-12 21:33
Solid GeometryTransformationGeometric Construction
This example demonstrates the Schlegel diagram of a dodecahedron. A Schlegel diagram is a projection of a polyhedron into a lower dimension, showing its internal structure. Key mathematical concepts include polyhedra, geometric transformations, and projections. Using GeoGebra commands such as Circle, Intersect, and Rotate, the Schlegel diagram of a dodecahedron is constructed. Suitable for high school geometry education and learning.

GeoGebra Commands & Parameters

  1. Circle(OO, 1, xOyPlane) → c
  2. Intersect(xAxis, c, 1) → A
  3. Rotate(A, 60°) → B
  4. Point(Circle[((A * (1 - sqrt(5))) + (B * (3 + sqrt(5)))) / 4, (Distance[A, B] * sqrt(10 + (2 * sqrt(5))) / 4), Segment[A, B]]) → C
  5. Dodecahedron(A, B, C) → a, D, M, N, O, P, Q, R, S, T, zijvlakABCDE, zijvlakAEIOJ, E, zijvlakABFKJ, zijvlakBCGLF, zijvlakCDHMG, zijvlakDEINH, zijvlakINSTO, zijvlakJKPTO, zijvlakFKPQL, zijvlakGLQRM, zijvlakHMRSN, zijvlakPQRST, F, RibbeAE, RibbeAB, RibbeBC, RibbeCD, RibbeDE, RibbeAJ, RibbeEI, RibbeIO, RibbeJO, RibbeJK, G, RibbeFK, RibbeBF, RibbeFL, RibbeGL, RibbeCG, RibbeGM, RibbeHM, RibbeDH, RibbeHN, RibbeIN, H, RibbeNS, RibbeST, RibbeOT, RibbePT, RibbeKP, RibbePQ, RibbeLQ, RibbeQR, RibbeMR, RibbeRS, I, J, K, L
  6. Midpoint(RibbeAB) → U
  7. Segment(K, U) → f
  8. Midpoint(RibbeFK) → V
  9. Segment(A, V) → g
  10. Intersect(f, g) → W
  11. Midpoint(RibbeRS) → Z
  12. Segment(H, Z) → h
  13. Midpoint(RibbeNS) → A_1
  14. Segment(M, A_1) → i
  15. Midpoint(i) → WW
  16. Line(W, WW) → j
  17. Point(j) → B_1
  18. Point(j) → C_1
  19. OrthogonalPlane(C_1, j) → b
  20. Line(B_1, A) → k
  21. Line(B_1, B) → l
  22. Intersect(l, k) → D_1
  23. Line(B_1, F) → m
  24. Line(B_1, K) → n
  25. Line(B_1, J) → p
  26. Intersect(p, b) → E_1
  27. Intersect(n, b) → F_1
  28. Intersect(m, b) → G_1
  29. Intersect(l, b) → H_1
  30. Intersect(k, b) → I_1
  31. Polygon(I_1, H_1, G_1, F_1, E_1) → veelhoek1, i_1, h_1, g_1, f_1, e_1
  32. Segment(B_1, E_1) → q
  33. Segment(B_1, F_1) → r
  34. Segment(B_1, G_1) → s
  35. Segment(B_1, H_1) → t
  36. Segment(B_1, I_1) → d
  37. Line(B_1, P) → e
  38. Line(B_1, Q) → j_1
  39. Intersect(e, b) → J_1
  40. Intersect(j_1, b) → K_1
  41. Segment(B_1, J_1) → l_1
  42. Segment(B_1, K_1) → m_1
  43. Intersect(RibbeJO, RibbeIO) → L_1
  44. Line(B_1, L_1) → n_1
  45. Line(B_1, T) → p_1
  46. Intersect(n_1, b) → M_1
  47. Intersect(p_1, b) → N_1
  48. Polygon(E_1, M_1, N_1, J_1, F_1) → veelhoek2, e_2, m_2, n_2, j_3, f_3
  49. Line(B_1, L) → k_1
  50. Intersect(k_1, b) → O_1
  51. Polygon(F_1, J_1, K_1, O_1, G_1) → veelhoek3, f_2, j_2, k_2, o_1, g_2
  52. Segment(B_1, M_1) → q_1
  53. Segment(B_1, N_1) → r_1
  54. Segment(B_1, O_1) → s_1
  55. Line(B_1, G) → t_1
  56. Line(B_1, C) → a_1
  57. Intersect(t_1, b) → P_1
  58. Intersect(a_1, b) → Q_1
  59. Polygon(G_1, O_1, P_1, Q_1, H_1) → veelhoek4, g_3, o_2, p_2, q_2, h_2
  60. Intersect(a_1, t_1) → R_1
  61. Line(B_1, E) → b_1
  62. Intersect(t_1, a_1) → S_1
  63. Line(B_1, D) → c_1
  64. Line(B_1, I) → d_1
  65. Intersect(d_1, b) → T_1
  66. Intersect(b_1, b) → U_1
  67. Segment(B_1, P_1) → i_2
  68. Intersect(c_1, b_1) → V_1
  69. Segment(B_1, Q_1) → l_2
  70. Segment(B_1, T_1) → r_2
  71. Segment(B_1, U_1) → s_2
  72. Polygon(I_1, U_1, T_1, M_1, E_1) → veelhoek5, i_3, u_1, t_2, m_3, e_3
  73. Line(B_1, H) → a_2
  74. Line(B_1, M) → b_2
  75. Intersect(b_2, b) → W_1
  76. Intersect(c_1, b) → Z_1
  77. Intersect(a_2, b) → A_2
  78. Polygon(I_1, H_1, Q_1, Z_1, U_1) → veelhoek6, i_4, h_3, q_3, z_1, u_2
  79. Segment(B_1, A_2) → c_2
  80. Line(B_1, D) → d_2
  81. Segment(B_1, Z_1) → k_3
  82. Line(B_1, N) → l_3
  83. Intersect(l_3, b) → B_2
  84. Segment(B_1, B_2) → n_3
  85. Segment(B_1, A_2) → p_3
  86. Segment(B_1, W_1) → r_3
  87. Polygon(T_1, U_1, Z_1, A_2, B_2) → veelhoek7, t_3, u_3, z_2, a_3, b_3
  88. Polygon(Z_1, Q_1, P_1, W_1, A_2) → veelhoek8, z_3, q_4, p_4, w_1, a_4
  89. Line(B_1, S) → s_3
  90. Intersect(s_3, b) → C_2
  91. Segment(B_1, C_2) → c_3
  92. Polygon(M_1, T_1, B_2, C_2, N_1) → veelhoek9, m_4, t_4, b_4, c_4, n_4
  93. Line(B_1, R) → d_3
  94. Intersect(d_3, b) → D_2
  95. Segment(B_1, D_2) → f_4
  96. Segment(C_2, D_2) → g_4
  97. Segment(K_1, D_2) → h_4
  98. Segment(D_2, W_1) → j_4