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Duality of Polyhedra

Duality of Polyhedra
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2020-10-17 03:50
Solid GeometryTransformationGeometric Construction
This example demonstrates the duality of polyhedra. Key mathematical concepts include polyhedra and duality. Using GeoGebra commands such as Point, Dodecahedron, Midpoint, Line, etc., it constructs a polyhedron and its dual. Suitable for high school geometry learners and anyone interested in the duality of polyhedra.

GeoGebra Commands & Parameters

  1. Point(Circle[(((1 - sqrt(5)) * A) + ((3 + sqrt(5)) * B)) / 4, (Distance[A, B] * sqrt(10 + (2 * sqrt(5))) / 4), Segment[A, B]]) → C
  2. Dodecahedron(A, B, C) → a, D, M, N, O, P, Q, R, S, T, ABCDE, AEIOJ, E, ABFKJ, BCGLF, CDHMG, DEINH, INSTO, JKPTO, FKPQL, GLQRM, HMRSN, PQRST, F, AE, AB, BC, CD, DE, AJ, EI, IO, JO, JK, G, FK, BF, FL, GL, CG, GM, HM, DH, HN, IN, H, NS, ST, OT, PT, KP, PQ, LQ, QR, MR, RS, I, J, K, L
  3. Midpoint(IN) → U
  4. Midpoint(I, L) → V
  5. Line(V, U) → c
  6. Midpoint(NS) → W
  7. Midpoint(IO) → Z
  8. Line(V, W) → d
  9. Line(V, Z) → e
  10. OrthogonalPlane(W, d) → f
  11. OrthogonalPlane(U, c) → g
  12. OrthogonalPlane(Z, e) → h
  13. IntersectPath(h, f) → i
  14. IntersectPath(f, g) → j
  15. Intersect(j, i) → A_1
  16. Line(V, I) → k
  17. Rotate(A_1, 120°, k) → B_1
  18. Rotate(B_1, 120°, k) → C_1
  19. Icosahedron(A_1, C_1, B_1) → l, D_1, t1, t2, t5, t4, t3, t9, t10, t6, t7, t8, E_1, t13, t14, t19, t16, t18, t20, t17, t15, t12, t11, F_1, arete1, arete2, arete6, arete4, arete8, arete5, arete19, arete3, arete13, arete10, G_1, arete12, arete14, arete11, arete16, arete7, arete9, arete17, arete20, arete18, arete25, H_1, arete26, arete27, arete30, arete21, arete28, arete24, arete29, arete23, arete22, arete15, I_1, J_1, K_1, L_1
  20. Polygon(T, A_1, O, D_1) → poly1, t, a_1, o, d_1
  21. Polygon(A_1, T, G_1, S) → poly2, a_2, t_1, g_1, s
  22. Polygon(A_1, S, F_1, N) → poly3, a_3, s_1, f_1, n
  23. Polygon(A_1, O, C_1, I) → poly4, a_4, o_1, c_1, i_1
  24. Polygon(A_1, N, B_1, I) → poly5, a_5, n_1, b_1, i_2
  25. Polygon(T, D_1, P, G_1) → poly6, t_2, d_2, p, g_2
  26. Polygon(R, G_1, S, F_1) → poly7, r, g_3, s_2, f_2
  27. Polygon(F_1, H, B_1, N) → poly8, f_3, h_1, b_2, n_2
  28. Polygon(B_1, I, C_1, E) → poly9, b_3, i_3, c_2, e_1
  29. Polygon(C_1, J, D_1, O) → poly10, c_3, j_1, d_3, o_2
  30. Polygon(D_1, J, H_1, K) → poly11, d_4, j_2, h_2, k_1
  31. Polygon(D_1, K, J_1, P) → poly12, d_5, k_2, j_3, p_1
  32. Polygon(J_1, Q, G_1, P) → poly13, j_4, q, g_4, p_2
  33. Polygon(G_1, Q, L_1, R) → poly14, g_5, q_1, l_1, r_1
  34. Polygon(L_1, M, F_1, R) → poly15, l_2, m, f_4, r_2
  35. Polygon(F_1, M, I_1, H) → poly16, f_5, m_1, i_4, h_3
  36. Polygon(D, B_1, H, I_1) → poly17, d_6, b_4, h_4, i_5
  37. Polygon(D, E_1, E, B_1) → poly18, d_7, e_2, e_3, b_5
  38. Polygon(E, C_1, A, E_1) → poly19, e_4, c_4, a_6, e_5
  39. Polygon(C_1, J, H_1, A) → poly20, c_5, j_5, h_5, a_7
  40. Polygon(H_1, F, J_1, K) → poly21, h_6, f_6, j_6, k_3
  41. Polygon(F, K_1, L, J_1) → poly22, f_7, k_4, l_3, j_7
  42. Polygon(J_1, L, L_1, Q) → poly23, j_8, l_4, l_5, q_2
  43. Polygon(L_1, G, K_1, L) → poly24, l_6, g_6, k_5, l_7
  44. Polygon(L_1, G, I_1, M) → poly25, l_8, g_7, i_6, m_2
  45. Polygon(E_1, C, I_1, D) → poly26, e_6, c_6, i_7, d_8
  46. Polygon(C, I_1, G, K_1) → poly27, c_7, i_8, g_8, k_6
  47. Polygon(C, K_1, B, E_1) → poly28, c_8, k_7, b_6, e_7
  48. Polygon(B, H_1, A, E_1) → poly29, b_7, h_7, a_8, e_8
  49. Polygon(B, K_1, F, H_1) → poly30, b_8, k_8, f_8, h_8
  50. Distance(V, U) → u
  51. Sphere(V, (v / 5 * u)) → b

AI Drawing Prompt Reference

Construct in 3D a regular dodecahedron and its dual. Start by defining a base segment from which the initial geometry determines the dodecahedron vertices and its reference points. Use midpoints of specific edge-related point sets to create three key lines, then build three orthogonal planes through these lines. Intersect the corresponding plane pairs to obtain three primary intersection points, and from them form the dual polyhedron’s vertex set via a rotationally defined triangle. Finally, draw all polygonal faces as quadrilaterals by connecting the appropriate vertices in the specified cyclic orders, assembling the full dual polyhedron so that faces match the dodecahedron’s duality structure.
The prompt above was compiled from the original drawing commands and is for reference only.