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十二面体展开图 by Anthony OR 柯志明

十二面体展开图 by Anthony OR 柯志明
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2014-11-12 23:30
多边形向量立体几何变换几何作图
所属专题 3D Geometry & Cross Sections Surface Area and Volume Nets of 3D Figures 立体图形的折纸图样
①本范例演示十二面体的展开图及其构造过程；②涉及的核心数学概念包括多边形、立体几何、向量与变换；③作图要点：使用Point、Dodecahedron、Segment、Rotate、Circle、Center、OrthogonalLine、Polygon、TableText等GeoGebra命令绘制点、线、面，并通过旋转、正交线等操作构建展开图；④适合高中及以上学生及数学爱好者学习立体几何与几何作图。

GeoGebra 命令及参数

  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. Segment(U, U + (0, 0, 2)) → b
  4. Point(b) → V
  5. Rotate(Polygon[A, B, F, K, J], (α * t), Segment[A, B]) → poly1
  6. Rotate(Polygon[F, K, P, Q, L], (α * t), Segment[K, F]) → poly2
  7. Rotate(Polygon[P, Q, R, S, T], (α * t), Segment[P, Q]) → poly3
  8. Rotate(poly2, (α * t), Segment[A, B]) → poly2'
  9. Rotate(poly3, (α * t), Segment[K, F]) → poly3'
  10. Rotate(poly3', (α * t), Segment[A, B]) → poly3''
  11. Circle(A, B, D) → c
  12. Center(c) → W
  13. OrthogonalLine(W, Plane[A, B, D]) → d
  14. Rotate(poly1, 72°, d) → poly1'
  15. Rotate(poly1', 72°, d) → poly1''
  16. Rotate(poly1'', 72°, d) → poly1'''
  17. Rotate(poly1''', 72°, d) → poly4
  18. Rotate(poly2', 72°, d) → poly2''
  19. Rotate(poly2'', 72°, d) → poly2'''
  20. Rotate(poly2''', 72°, d) → poly5
  21. Rotate(poly5, 72°, d) → poly5'
  22. Polygon(A, B, C, D, E) → poly6, a_1, b_1, c_1, d_1, e
  23. TableText({{LaTeX[A, true, true]}, {LaTeX[B, true, true]}, {LaTeX[C, true, true]}, {LaTeX[D, true, true]}, {LaTeX[E, true, true]}, {LaTeX[F, true, true]}, {LaTeX[G, true, true]}, {LaTeX[H, true, true]}, {LaTeX[I, true, true]}, {LaTeX[J, true, true]}, {LaTeX[K, true, true]}, {LaTeX[L, true, true]}, {LaTeX[M, true, true]}, {LaTeX[N, true, true]}, {LaTeX[O, true, true]}, {LaTeX[P, true, true]}, {LaTeX[Q, true, true]}, {LaTeX[R, true, true]}, {LaTeX[S, true, true]}, {LaTeX[T, true, true]}, {LaTeX[U, true, true]}, {LaTeX[V, true, true]}, {LaTeX[W, true, true]}}) → text1
  24. Rotate(poly2''', 72°, d) → poly7
  25. Rotate(poly1'', 72°, d) → poly1'''_1
  26. Polygon(A, B, C, D, E) → poly8, a_2, b_2, c_2, d_2, e_1
  27. Rotate(poly1''', 72°, d) → poly9
  28. Rotate(poly5, 72°, d) → poly5'_1
  29. Rotate(Polygon[A, B, F, K, J], (α * t), Segment[A, B]) → poly10
  30. Rotate(poly2, (α * t), Segment[A, B]) → poly2'_1
  31. Rotate(poly3', (α * t), Segment[A, B]) → poly3''_1
  32. Rotate(poly1, 72°, d) → poly1'_1
  33. Rotate(poly2', 72°, d) → poly2''_1
  34. Rotate(poly1', 72°, d) → poly1''_1
  35. Rotate(poly2'', 72°, d) → poly2'''_1

AI 绘图提示语参考

绘制一个十二面体的展开图，展示其构造过程。首先画出十二个正五边形，然后通过旋转和连接这些五边形，形成一个立体的十二面体。在绘制过程中，注意使用旋转和正交线来确保形状的准确性，并标注关键点和边，最终呈现出一个清晰、可理解的十二面体展开图。
以上提示语为根据原始绘图命令整理而来，仅供参考。