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大十二面体：开普勒-庞索多面体的三维构造

大十二面体：开普勒-庞索多面体的三维构造
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2014-12-31 02:02
多边形立体几何几何作图
所属专题 My 3D Collection
本范例演示了如何利用GeoGebra的3D功能构造大十二面体（Great dodecahedron），这是一种非凸的正则星形多面体，属于开普勒-庞索多面体家族。核心数学概念包括正多面体、星形多面体、欧拉公式以及三维几何中的顶点、棱和面关系。作图要点是从正二十面体（Icosahedron）出发，通过顶点（Vertex）、直线（Line）、交点（Intersect）和多边形（Polygon）命令，逐步提取并连接特定顶点，形成大十二面体的星形五边形面。构造中涉及大量顶点对象（如A1至A20）和棱边对象（如krawędźBC等），体现了复杂的空间几何关系。适合高中及以上学生或数学爱好者用于探索非凸多面体的结构，也可作为3D几何作图的进阶案例。

GeoGebra 命令及参数

  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 绘图提示语参考

请绘制一个大十二面体，这是一个非凸正星多面体，属于开普勒-施泰因多面体家族。构造从正二十面体开始，提取其20个顶点，然后连接这些顶点形成星五边形面。具体来说，需要识别哪些顶点构成大十二面体的面，并用线段连接它们。确保图形呈现三维立体结构，展示其复杂的空间关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。