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Schlegel Diagram Icosahedron

Schlegel Diagram Icosahedron
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2017-12-12 23:42
PolygonSolid GeometryGeometric Construction
This example demonstrates the Schlegel diagram of an icosahedron, showing how to project the 3D shape onto a 2D plane while preserving its structure. Key mathematical concepts include polygons, solid geometry, and symmetry. The construction utilizes GeoGebra commands like Circle, Intersect, Rotate, Point, Icosahedron, Midpoint, Segment, Line, OrthogonalPlane, Polygon, and Ray. It is suitable for high school students and educators exploring geometric properties of polyhedra.

GeoGebra Commands & Parameters

  1. Circle(OO, 1, xOyPlane) → c
  2. Intersect(xAxis, c, 1) → A
  3. Rotate(A, 72°) → B
  4. Point(Circle[Midpoint[A, B], (Distance[A, B] * sqrt(3) / 2), Segment[A, B]]) → C
  5. Icosahedron(A, B, C) → a, D, zijvlakABC, zijvlakACF, zijvlakAFG, zijvlakADG, zijvlakABD, zijvlakBDH, zijvlakBEH, zijvlakBCE, zijvlakCEI, zijvlakCFI, E, zijvlakEHK, zijvlakEIK, zijvlakIKL, zijvlakFIL, zijvlakHJK, zijvlakJKL, zijvlakGJL, zijvlakFGL, zijvlakDHJ, zijvlakDGJ, F, RibbeAC, RibbeAB, RibbeBC, RibbeAF, RibbeCF, RibbeAG, RibbeFG, RibbeAD, RibbeDG, RibbeBD, G, RibbeBH, RibbeDH, RibbeBE, RibbeEH, RibbeCE, RibbeCI, RibbeEI, RibbeFI, RibbeEK, RibbeHK, H, RibbeIK, RibbeIL, RibbeKL, RibbeFL, RibbeJK, RibbeHJ, RibbeJL, RibbeGL, RibbeGJ, RibbeDJ, I, J, K, L
  6. Midpoint(RibbeAC) → M
  7. Segment(F, M) → f
  8. Midpoint(RibbeCF) → N
  9. Segment(A, N) → g
  10. Intersect(f, g) → O
  11. Midpoint(RibbeHK) → P
  12. Midpoint(RibbeJK) → Q
  13. Segment(P, J) → h
  14. Segment(Q, H) → i
  15. Intersect(h, i) → R
  16. Line(O, R) → j
  17. Point(j) → S
  18. Point(j) → T
  19. OrthogonalPlane(T, j) → b
  20. Line(S, C) → k
  21. Line(S, A) → l
  22. Line(S, G) → m
  23. Line(S, F) → n
  24. Intersect(k, j) → U
  25. Line(S, L) → p
  26. Intersect(n, b) → V
  27. Intersect(l, b) → W
  28. Intersect(k, b) → Z
  29. Polygon(Z, W, V) → driehoek1, v, z_1, w
  30. Segment(S, W) → q
  31. Segment(S, V) → r
  32. Segment(Z, S) → s
  33. Line(S, D) → t
  34. Intersect(t, b) → A_1
  35. Segment(S, A_1) → d
  36. Ray(S, B) → e
  37. Intersect(e, b) → B_1
  38. Segment(S, B_1) → f_1
  39. Segment(Z, B_1) → g_1
  40. Segment(B_1, W) → h_1
  41. Line(S, L) → i_1
  42. Line(S, G) → j_1
  43. Intersect(i_1, b) → C_1
  44. Intersect(j_1, b) → D_1
  45. Segment(S, C_1) → k_1
  46. Segment(S, D_1) → l_1
  47. Segment(V, C_1) → m_1
  48. Segment(C_1, D_1) → n_1
  49. Segment(V, D_1) → p_1
  50. Segment(D_1, W) → q_1
  51. Segment(D_1, A_1) → r_1
  52. Segment(A_1, W) → s_1
  53. Segment(A_1, B_1) → t_1
  54. Line(S, K) → a_1
  55. Line(C, E) → b_1
  56. Line(S, I) → c_1
  57. Intersect(c_1, b) → E_1
  58. Intersect(a_1, b) → F_1
  59. Segment(S, F_1) → e_1
  60. Segment(S, E_1) → f_2
  61. Segment(Z, E_1) → g_2
  62. Segment(E_1, V) → j_2
  63. Segment(E_1, C_1) → k_2
  64. Segment(F_1, C_1) → n_2
  65. Segment(F_1, E_1) → p_2
  66. Line(S, J) → q_2
  67. Intersect(q_2, b) → H_1
  68. Segment(S, H_1) → r_2
  69. Segment(H_1, C_1) → s_2
  70. Segment(H_1, D_1) → t_2
  71. Segment(H_1, A_1) → a_2
  72. Segment(F_1, H_1) → b_2
  73. Line(S, H) → c_2
  74. Intersect(c_2, b) → I_1
  75. Segment(S, I_1) → d_2
  76. Segment(B_1, I_1) → e_2
  77. Segment(I_1, F_1) → h_3
  78. Segment(I_1, H_1) → i_3
  79. Line(S, E) → j_3
  80. Intersect(b, j_3) → G_1
  81. Segment(Z, G_1) → h_2
  82. Segment(E_1, G_1) → i_2
  83. Segment(G_1, B_1) → l_2
  84. Segment(F_1, G_1) → m_2
  85. Segment(G_1, I_1) → d_1
  86. Segment(I_1, A_1) → f_3

AI Drawing Prompt Reference

Draw a Schlegel diagram of an icosahedron, showing how to project a 3D shape onto a 2D plane while preserving its structure. Start by drawing a circle and rotating its intersections with an axis to generate multiple points. Use these points to construct the icosahedron. Then, find midpoints of key edges, and use them along with vertices to draw auxiliary lines and planes. Finally, create polygons using intersection points and auxiliary lines to form the 2D projection of the icosahedron.
The prompt above was compiled from the original drawing commands and is for reference only.