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Icosahedron as a Planar Graph Visualization

Icosahedron as a Planar Graph Visualization
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2017-11-17 14:22
PolygonSolid GeometryTransformationGeometric ConstructionGraphing
This example demonstrates how to represent an icosahedron, a regular polyhedron with 20 triangular faces, as a planar graph by projecting its vertices and edges onto a two-dimensional plane while preserving the topological structure. The core mathematical concepts include planar graph representation of polyhedra, Euler's formula (V - E + F = 2), the symmetry of the regular icosahedron, and the topological properties of polygonal meshes. Construction highlights: using GeoGebra's Polygon command to create multiple triangular faces (e.g., veelhoek1, veelhoek2, etc.) and Segment command to connect vertices (A, B, E, F, G, H, I, J, K, L, M, N, O, P, etc.) as edges, along with transformations (afbeelding1) and auxiliary objects (f, g, h, etc.) to precisely position vertices, resulting in a planar embedding of twenty triangular faces. This example is suitable for high school or university courses in solid geometry, graph theory, or mathematical visualization, helping students intuitively understand the correspondence between polyhedra and planar graphs and how to represent 3D structures in 2D.

GeoGebra Commands & Parameters

  1. Polygon(E, F, 3) → veelhoek1, f, g, h, G
  2. Polygon(H, I, 3) → veelhoek2, i, j, k, J
  3. Polygon(K, L, 3) → veelhoek3, l, m, n, M
  4. Polygon(N, O, 3) → veelhoek4, p, q, r, P
  5. Segment(G, J) → s
  6. Segment(P, L) → d
  7. Segment(P, J) → e
  8. Segment(J, O) → f_1
  9. Segment(P, O) → g_1
  10. Segment(O, I) → h_1
  11. Segment(I, F) → i_1
  12. Segment(H, N) → j_1
  13. Segment(N, I) → k_1
  14. Segment(P, N) → l_1
  15. Segment(N, O) → m_1
  16. Segment(H, E) → n_1
  17. Segment(P, H) → p_1
  18. Segment(O, K) → q_1
  19. Segment(N, M) → t_1
  20. Segment(G, E) → a_1
  21. Segment(E, F) → b_1
  22. Segment(F, G) → c_1
  23. Segment(R, Q) → d_1
  24. Segment(Q, S) → e_1
  25. Segment(Q, V) → f_2
  26. Segment(Q, T) → g_2
  27. Segment(T, U) → h_2
  28. Segment(U, V) → i_2
  29. Segment(R, A_1) → j_2
  30. Segment(T, B_1) → k_2
  31. Segment(R, Z) → l_2
  32. Segment(U, B_1) → m_2
  33. Segment(U, C_1) → n_2
  34. Segment(V, C_1) → p_2
  35. Segment(V, W) → q_2
  36. Segment(A_1, Z) → r_2
  37. Segment(Z, W) → s_2
  38. Segment(A_1, D_1) → t_2
  39. Segment(B_1, D_1) → a_2
  40. Segment(A_1, B_1) → b_2
  41. Segment(B_1, C_1) → c_2
  42. Segment(C_1, W) → d_2
  43. Segment(D_1, W) → e_2
  44. Segment(C_1, D_1) → f_3
  45. Segment(Z, D_1) → g_3
  46. Segment(S, V) → h_3
  47. Segment(Q, U) → i_3
  48. Segment(S, Z) → j_3
  49. Segment(R, S) → k_3
  50. Segment(T, A_1) → l_3
  51. Segment(R, T) → m_3
  52. Segment(S, W) → n_3
  53. Segment(G, L) → t
  54. Segment(L, H) → p_3
  55. Segment(G, K) → r_1
  56. Segment(K, I) → q_3
  57. Segment(E, M) → a
  58. Segment(M, I) → r_3
  59. Segment(H, M) → b
  60. Segment(M, F) → s_3
  61. Segment(E, L) → s_1
  62. Segment(L, J) → t_3
  63. Segment(J, K) → c
  64. Segment(K, F) → a_3