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Dodecahedron Inverted in a Cube

Dodecahedron Inverted in a Cube
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2016-09-07 17:22
PolygonSolid Geometry
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This example demonstrates the inversion of a dodecahedron inside a cube, showcasing spatial relationships and symmetry. It involves core math concepts like solid geometry, polygons, transformations, and spatial symmetry. The construction uses commands like Point, Dodecahedron, Cube, Mirror, Segment, Polygon, and objects A, B, C, etc. It is suitable for high school students and math enthusiasts exploring spatial geometry and polyhedron properties.

GeoGebra Commands & Parameters

  1. Point(Circle[(((0, 0) * (1 - sqrt(5))) + ((0, 1) * (3 + sqrt(5)))) / 4, (Distance[(0, 0), (0, 1)] * sqrt(10 + (2 * sqrt(5))) / 4), Segment[(0, 0), (0, 1)]]) → A
  2. Dodecahedron((0, 0), (0, 1), A) → a, B, K, L, M, N, O, P, Q, R, face1, face3, C, face2, face4, faceABFKE, faceBCGLF, faceGLQRM, faceHINRM, faceDINOJ, faceEJOPK, faceFKPQL, faceNOPQR, D, edge2, edge1, edge4, edgeAB, edgeBC, edge3, edgeCG, edgeGM, edgeHM, edgeHI, E, edgeDI, edge5, edgeDJ, edgeEJ, edgeAE, edgeEK, edgeFK, edgeBF, edgeFL, edgeGL, F, edgeLQ, edgeQR, edgeMR, edgeNR, edgeIN, edgeNO, edgeJO, edgeOP, edgeKP, edgePQ, G, H, I, J
  3. Point(edge4) → S
  4. Cube(E, O, Q) → b, T, faceQTZW, faceUVWZ, edgeET, edgeEO, edgeOQ, edgeQT, edgeEU, edgeTZ, edgeUZ, edgeUV, U, edgeOV, edgeVW, edgeQW, edgeWZ, V, W, Z, faceEOQT, faceETZU, faceEOVU, faceOQWV
  5. Mirror(N, faceOQWV) → N'
  6. Mirror(R, faceOQWV) → R'
  7. Mirror(J, faceEOVU) → J'
  8. Mirror(D, faceEOVU) → D'
  9. Mirror(A, faceETZU) → A'
  10. Mirror(B, faceETZU) → B'
  11. Mirror(H, faceUVWZ) → H'
  12. Mirror(A_1, faceUVWZ) → B_1
  13. Mirror(L, faceQTZW) → L'
  14. Mirror(G, faceQTZW) → G'
  15. Mirror(K, faceEOQT) → K'
  16. Mirror(P, faceEOQT) → P'
  17. Mirror(K, faceEOQT) → K'_1
  18. Segment(J', O) → h
  19. Segment(J', E) → i
  20. Segment(A', E) → k
  21. Segment(A', S) → l
  22. Segment(B', T) → m
  23. Segment(B', Z) → n
  24. Segment(B', A') → p
  25. Segment(B_1, S) → q
  26. Segment(B_1, Z) → r
  27. Segment(D', V) → f
  28. Segment(D', S) → g
  29. Segment(D', J') → j
  30. Segment(H', V) → s
  31. Segment(H', W) → t
  32. Segment(H', B_1) → c
  33. Segment(W, G') → d
  34. Segment(G', Z) → e
  35. Segment(Q, L') → f_1
  36. Segment(L', T) → g_1
  37. Segment(Q, R') → i_1
  38. Segment(R', W) → j_1
  39. Segment(N', O) → k_1
  40. Segment(N', V) → l_1
  41. Segment(O, P') → m_1
  42. Segment(Q, P') → n_1
  43. Segment(R', N') → q_1
  44. Segment(L', G') → p_1
  45. Polygon(O, N', R', Q) → poly1, o, n', r', q_2
  46. Polygon(N', V, W, R') → poly2, n'_1, v, w, r'_1
  47. Polygon(O, J', D', V) → poly3, o_1, j', d', v_1
  48. Polygon(E, J', D', S) → poly5, e_1, j'_1, d'_1, s_1
  49. Polygon(G', L', Q, W) → poly4, g', l'_2, q_3, w_1
  50. Polygon(Z, G', L', T) → poly8, z_1, g'_1, l'_3, t_2
  51. Polygon(Q, P', K', T) → poly6, q_4, p', k', t_1
  52. Polygon(P', O, E, K') → poly7, p'_1, o_2, e_2, k'_1
  53. Polygon(T, E, A', B') → poly9, t_3, e_3, a', b'
  54. Polygon(Z, B', A', S) → poly10, z_2, b'_1, a'_1, s_2
  55. Polygon(H', W, Z, B_1) → poly11, h', w_2, z_3, b_1
  56. Polygon(V, H', B_1, S) → poly12, v_2, h'_1, b_2, s_3