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Great Dodecahedron: Constructing a Kepler-Poinsot Polyhedron

Great Dodecahedron: Constructing a Kepler-Poinsot Polyhedron
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2014-12-31 02:02
PolygonSolid GeometryGeometric Construction
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This example demonstrates how to construct the Great Dodecahedron, a non-convex regular star polyhedron belonging to the Kepler-Poinsot family, using GeoGebra's 3D capabilities. The core mathematical concepts include regular polyhedra, star polyhedra, Euler's formula, and the relationships among vertices, edges, and faces in three-dimensional geometry. The construction starts from an icosahedron (Icosahedron) and uses commands such as Vertex, Line, Intersect, and Polygon to extract and connect specific vertices, forming the star pentagonal faces of the Great Dodecahedron. The construction involves numerous vertex objects (e.g., A1 to A20) and edge objects (e.g., krawędźBC), illustrating complex spatial relationships. This is suitable for high school and above students or math enthusiasts to explore non-convex polyhedra, and serves as an advanced example of 3D geometric construction.

GeoGebra Commands & Parameters

  1. Icosahedron(C, A, B) → a, D, A1, A6, A10, A9, A3, A4, A5, A2, A7, A8, E, A13, A14, A19, A16, A18, A20, A17, A15, A12, A11, F, krawędźBC, krawędźAC, krawędźAB, krawędźCF, krawędźBF, krawędźCG, krawędźFG, krawędźCD, krawędźDG, krawędźAD, G, krawędźAH, krawędźDH, krawędźAE, krawędźEH, krawędźBE, krawędźBI, krawędźEI, krawędźFI, krawędźEK, krawędźHK, H, krawędźIK, krawędźIL, krawędźKL, krawędźFL, krawędźJK, krawędźHJ, krawędźJL, krawędźGL, krawędźGJ, krawędźDJ, I, J, K, L
  2. Vertex(A1, 1) → B1
  3. Vertex(A1, 2) → C1
  4. Vertex(A1, 3) → D1
  5. Vertex(A2, 1) → B2
  6. Vertex(A2, 2) → C2
  7. Vertex(A2, 3) → D2
  8. Vertex(A3, 1) → B3
  9. Vertex(A3, 2) → C3
  10. Vertex(A3, 3) → D3
  11. Vertex(A4, 1) → B4
  12. Vertex(A4, 2) → C4
  13. Vertex(A4, 3) → D4
  14. Vertex(A5, 1) → B5
  15. Vertex(A5, 2) → C5
  16. Vertex(A5, 3) → D5
  17. Vertex(A6, 1) → B6
  18. Vertex(A6, 2) → C6
  19. Vertex(A6, 3) → D6
  20. Vertex(A7, 1) → B7
  21. Vertex(A7, 2) → C7
  22. Vertex(A7, 3) → D7
  23. Vertex(A8, 1) → B8
  24. Vertex(A8, 2) → C8
  25. Vertex(A8, 3) → D8
  26. Vertex(A9, 1) → B9
  27. Vertex(A9, 2) → C9
  28. Vertex(A9, 3) → D9
  29. Vertex(A10, 1) → B10
  30. Vertex(A10, 2) → C10
  31. Vertex(A10, 3) → D10
  32. Vertex(A11, 1) → B11
  33. Vertex(A11, 2) → C11
  34. Vertex(A11, 3) → D11
  35. Vertex(A12, 1) → B12
  36. Vertex(A12, 2) → C12
  37. Vertex(A12, 3) → D12
  38. Vertex(A13, 1) → B13
  39. Vertex(A13, 2) → C13
  40. Vertex(A13, 3) → D13
  41. Vertex(A14, 1) → B14
  42. Vertex(A14, 2) → C14
  43. Vertex(A14, 3) → D14
  44. Vertex(A15, 1) → B15
  45. Vertex(A15, 2) → C15
  46. Vertex(A15, 3) → D15
  47. Vertex(A16, 1) → B16
  48. Vertex(A16, 2) → C16
  49. Vertex(A16, 3) → D16
  50. Vertex(A17, 1) → B17
  51. Vertex(A17, 2) → C17
  52. Vertex(A17, 3) → D17
  53. Vertex(A18, 1) → B18
  54. Vertex(A18, 2) → C18
  55. Vertex(A18, 3) → D18
  56. Vertex(A19, 1) → B19
  57. Vertex(A19, 2) → C19
  58. Vertex(A19, 3) → D19
  59. Vertex(A20, 1) → B20
  60. Vertex(A20, 2) → C20
  61. Vertex(A20, 3) → D20
  62. Line(C, E) → b
  63. Line(B, D) → c
  64. Intersect(c, b) → M
  65. Polygon(B1, C1, F1) → G1, H1, I1, J1
  66. Polygon(C1, D1, F1) → K1, L1, M1, N1
  67. Polygon(D1, B1, F1) → O1, P1, Q1, R1
  68. Polygon(B2, C2, F2) → G2, H2, I2, J2
  69. Polygon(B3, C3, F3) → G3, H3, I3, J3
  70. Polygon(B4, C4, F4) → G4, H4, I4, J4
  71. Polygon(B5, C5, F5) → G5, H5, I5, J5
  72. Polygon(B6, C6, F6) → G6, H6, I6, J6
  73. Polygon(B7, C7, F7) → G7, H7, I7, J7
  74. Polygon(B8, C8, F8) → G8, H8, I8, J8
  75. Polygon(B9, C9, F9) → G9, H9, I9, J9
  76. Polygon(B10, C10, F10) → G10, H10, I10, J10
  77. Polygon(B11, C11, F11) → G11, H11, I11, J11
  78. Polygon(B12, C12, F12) → G12, H12, I12, J12
  79. Polygon(B13, C13, F13) → G13, H13, I13, J13
  80. Polygon(B14, C14, F14) → G14, H14, I14, J14
  81. Polygon(B15, C15, F15) → G15, H15, I15, J15
  82. Polygon(B16, C16, F16) → G16, H16, I16, J16
  83. Polygon(B17, C17, F17) → G17, H17, I17, J17
  84. Polygon(B18, C18, F18) → G18, H18, I18, J18
  85. Polygon(B19, C19, F19) → G19, H19, I19, J19
  86. Polygon(B20, C20, F20) → G20, H20, I20, J20
  87. Polygon(C2, D2, F2) → K2, L2, M2, N2
  88. Polygon(C3, D3, F3) → K3, L3, M3, N3
  89. Polygon(C4, D4, F4) → K4, L4, M4, N4
  90. Polygon(C5, D5, F5) → K5, L5, M5, N5
  91. Polygon(C6, D6, F6) → K6, L6, M6, N6
  92. Polygon(C7, D7, F7) → K7, L7, M7, N7
  93. Polygon(C8, D8, F8) → K8, L8, M8, N8
  94. Polygon(C9, D9, F9) → K9, L9, M9, N9
  95. Polygon(C10, D10, F10) → K10, L10, M10, N10
  96. Polygon(C11, D11, F11) → K11, L11, M11, N11
  97. Polygon(C12, D12, F12) → K12, L12, M12, N12
  98. Polygon(C13, D13, F13) → K13, L13, M13, N13
  99. Polygon(C14, D14, F14) → K14, L14, M14, N14
  100. Polygon(C15, D15, F15) → K15, L15, M15, N15
  101. Polygon(C16, D16, F16) → K16, L16, M16, N16
  102. Polygon(C17, D17, F17) → K17, L17, M17, N17
  103. Polygon(C18, D18, F18) → K18, L18, M18, N18
  104. Polygon(C19, D19, F19) → K19, L19, M19, N19
  105. Polygon(C20, D20, F20) → K20, L20, M20, N20
  106. Polygon(D2, B2, F2) → O2, P2, Q2, R2
  107. Polygon(D3, B3, F3) → O3, P3, Q3, R3
  108. Polygon(D4, B4, F4) → O4, P4, Q4, R4
  109. Polygon(D5, B5, F5) → O5, P5, Q5, R5
  110. Polygon(D6, B6, F6) → O6, P6, Q6, R6
  111. Polygon(D7, B7, F7) → O7, P7, Q7, R7
  112. Polygon(D8, B8, F8) → O8, P8, Q8, R8
  113. Polygon(D9, B9, F9) → O9, P9, Q9, R9
  114. Polygon(D10, B10, F10) → O10, P10, Q10, R10
  115. Polygon(D11, B11, F11) → O11, P11, Q11, R11
  116. Polygon(D12, B12, F12) → O12, P12, Q12, R12
  117. Polygon(D13, B13, F13) → O13, P13, Q13, R13
  118. Polygon(D14, B14, F14) → O14, P14, Q14, R14
  119. Polygon(D15, B15, F15) → O15, P15, Q15, R15
  120. Polygon(D16, B16, F16) → O16, P16, Q16, R16
  121. Polygon(D17, B17, F17) → O17, P17, Q17, R17
  122. Polygon(D18, B18, F18) → O18, P18, Q18, R18
  123. Polygon(D19, B19, F19) → O19, P19, Q19, R19
  124. Polygon(D20, B20, F20) → O20, P20, Q20, R20

AI Drawing Prompt Reference

Draw a great dodecahedron, a non-convex regular star polyhedron belonging to the Kepler-Poinsot family. Start from a regular icosahedron, extract its 20 vertices, and connect these vertices to form star pentagonal faces. Specifically, identify which vertices form the faces of the great dodecahedron and connect them with line segments. Ensure the 3D structure is clearly shown, highlighting its complex spatial relationships.
The prompt above was compiled from the original drawing commands and is for reference only.