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Stellated Dodecahedron

Stellated Dodecahedron
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2014-12-25 00:32
TrigonometrySolid GeometryMatrixGeometric Construction
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This example demonstrates the construction of a Stellated Dodecahedron using GeoGebra, a complex polyhedron formed by extending the vertices of a regular dodecahedron. Key mathematical concepts include solid geometry, trigonometric functions, matrix operations, and variance calculations. The construction involves commands and objects such as Point, Dodecahedron, Line, Intersect, and Pyramid, as summarized. Suitable for high school and advanced learners to explore polyhedron construction and spatial geometry relationships.

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, faceABCDE, faceAEIOJ, E, faceABFKJ, faceBCGLF, faceCDHMG, faceDEINH, faceINSTO, faceJKPTO, faceFKPQL, faceGLQRM, faceHMRSN, facePQRST, F, edgeAE, edgeAB, edgeBC, edgeCD, edgeDE, edgeAJ, edgeEI, edgeIO, edgeJO, edgeJK, G, edgeFK, edgeBF, edgeFL, edgeGL, edgeCG, edgeGM, edgeHM, edgeDH, edgeHN, edgeIN, H, edgeNS, edgeST, edgeOT, edgePT, edgeKP, edgePQ, edgeLQ, edgeQR, edgeMR, edgeRS, I, J, K, L
  3. Line(K, P) → b
  4. Line(P, Q) → c
  5. Line(Q, L) → d
  6. Line(L, F) → e
  7. Line(F, K) → f
  8. Line(B, F) → g
  9. Line(B, C) → h
  10. Line(C, G) → i
  11. Line(G, L) → j
  12. Line(Q, R) → k
  13. Line(R, M) → l
  14. Line(M, G) → m
  15. Line(R, S) → n
  16. Line(S, T) → p
  17. Line(T, P) → q
  18. Line(T, O) → r
  19. Line(O, J) → s
  20. Line(J, K) → t
  21. Line(B, A) → a_1
  22. Line(A, J) → b_1
  23. Line(M, H) → c_1
  24. Line(H, N) → d_1
  25. Line(N, S) → e_1
  26. Line(N, I) → f_1
  27. Line(I, O) → g_1
  28. Line(I, E) → h_1
  29. Line(E, D) → i_1
  30. Line(D, C) → j_1
  31. Line(D, H) → k_1
  32. Intersect(j, h) → U
  33. Intersect(h_1, k_1) → V
  34. Line(A, E) → l_1
  35. Intersect(j_1, l_1) → W
  36. Intersect(a_1, t) → Z
  37. Intersect(l_1, s) → A_1
  38. Intersect(f, c) → B_1
  39. Intersect(q, t) → C_1
  40. Intersect(n, c_1) → D_1
  41. Intersect(k, m) → E_1
  42. Intersect(d, a_1) → F_1
  43. Intersect(d, b) → G_1
  44. Intersect(s, h_1) → H_1
  45. Pyramid(facePQRST, G_1) → o, face5, edge1, face4, face3, face1, face2, edge5, edge4, edge3, edge2
  46. Pyramid(faceINSTO, H_1) → u, face7, edge10, face6, face8, face10, face9, edge8, edge6, edge7, edge9
  47. Pyramid(faceHMRSN, E_1) → v, face11, edge13, face13, face15, face14, face12, edge11, edge12, edge14, edge15
  48. Pyramid(faceDEINH, W) → w, faceDEW, edgeIW, faceDHW, faceHNW, faceINW, faceEIW, edgeEW, edgeDW, edgeHW, edgeNW
  49. Pyramid(faceGLQRM, D_1) → m_1, face16, edge18, face18, face20, face19, face17, edge16, edge17, edge19, edge20
  50. Pyramid(faceCDHMG, U) → n_1, faceCDU, edgeHU, faceCGU, faceGMU, faceHMU, faceDHU, edgeDU, edgeCU, edgeGU, edgeMU
  51. Pyramid(faceBCGLF, F_1) → o_1, face21, edge24, face22, face24, face25, face23, edge22, edge21, edge23, edge25
  52. Pyramid(faceFKPQL, C_1) → p_1, face26, edge28, face28, face30, face29, face27, edge26, edge27, edge29, edge30
  53. Pyramid(faceJKPTO, B_1) → q_1, face32, edge32, face34, face35, face33, face31, edge31, edge33, edge35, edge34
  54. Pyramid(faceAEIOJ, Z) → r_1, faceAJZ, edgeOZ, faceAEZ, faceEIZ, faceIOZ, faceJOZ, edgeJZ, edgeAZ, edgeEZ, edgeIZ
  55. Pyramid(faceABCDE, V) → s_1, faceAEV, edgeDV, faceABV, faceBCV, faceCDV, faceDEV, edgeEV, edgeAV, edgeBV, edgeCV
  56. Pyramid(faceABFKJ, A_1) → t_1, face36, edge38, face37, face40, face39, face38, edge37, edge36, edge39, edge40

AI Drawing Prompt Reference

Draw a Stellated Dodecahedron by extending the vertices of a regular dodecahedron to form a complex polyhedron. Start by constructing the basic shape of the dodecahedron, then extend lines from each vertex outward to create star-like protrusions. Pay attention to the intersections of lines and the faces formed, resulting in a polyhedron composed of multiple triangular pyramids and pentagonal faces.
The prompt above was compiled from the original drawing commands and is for reference only.