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刘徽金字塔体积证明 by Anthony OR 柯志明 (2)

刘徽金字塔体积证明 by Anthony OR 柯志明 (2)
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2014-11-02 15:51
三角函数立体几何几何作图
所属专题 Area, Surface Area, Volume, Cross Section, 3D Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes and Surface Areas of Pyramids, Cones and Spheres More about Volumes and Surface Areas Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes and Surface Areas of Pyramids, Cones and Spheres
①本范例演示了刘徽（公元3世纪）关于金字塔体积的证明方法；②涉及的核心数学概念包括体积公式、几何体、截面相似性、极限思想；③作图要点——使用了Vector（向量）、Polygon（多边形）、Prism（棱柱）、Translate（平移）、Midpoint（中点）、Rotate（旋转）、Sequence（序列）等GeoGebra命令与对象；④适合高中及以上学生学习和理解立体几何体积定理。

GeoGebra 命令及参数

  1. Vector((0, 0, 0), (3, 0, 3)) → v
  2. Polygon(A, D, E) → poly1, e, a, d
  3. Prism(poly1, B) → b, F_1, edgeBF, edgeFG, edgeBG, F, faceABFD, faceDEGF, faceABGE, faceBFG, edgeAB, edgeDF, edgeEG
  4. Translate(B, Vector[(s * v)]) → B'
  5. Translate(C, Vector[(s * v)]) → C'
  6. Translate(F, Vector[(s * v)]) → F'
  7. Translate(E, Vector[(s * v)]) → E'
  8. Translate(Pyramid[Polygon[B, C, F], E], Vector[(s * v)]) → c
  9. Translate(Pyramid[Polygon[A, B, C, D], E], Vector[(0 * v)]) → f
  10. Midpoint(A, E) → G
  11. Midpoint(D, E) → H
  12. Midpoint(B, F) → I
  13. Midpoint(C, F) → J
  14. Midpoint(A, B) → K
  15. Midpoint(D, C) → L
  16. Midpoint(E, F) → M
  17. Midpoint(B, C) → N
  18. Midpoint(A, D) → O
  19. Midpoint(M, L) → P
  20. Midpoint(M, K) → Q
  21. Rotate(Prism[Polygon[D, L, P, H], O], 0°, zAxis) → g
  22. Rotate(Prism[Polygon[A, O, G], K], 0°, zAxis) → h
  23. Midpoint(K, L) → R
  24. Rotate(Prism[Polygon[Q, R, N], P], 0°, zAxis) → i
  25. Translate(M, Vector[(s * v)]) → M'
  26. Translate(P, Vector[(s * v)]) → P'
  27. Translate(Q, Vector[(s * v)]) → Q'
  28. Translate(I, Vector[(s * v)]) → I'
  29. Translate(J, Vector[(s * v)]) → J'
  30. Rotate(Prism[Polygon[I', J', F'], Q'], 0°, zAxis) → j
  31. Translate(N, Vector[(s * v)]) → N'
  32. Rotate(Prism[Polygon[Q', N', I'], P'], 0°, zAxis) → k
  33. Vector(B, Q) → u
  34. Sequence(Translate[Dilate[list1, 0.5, B'], Vector[(t * u)]], t, 0, 1) → list1'
  35. Sequence(Translate[Dilate[list2, 0.5, B], Vector[(t * u)]], t, 0, 1) → list2'
  36. Sequence(Translate[Dilate[list1, 0.25, B'], Vector[(t * u / 2)]], t, 0, 3) → list1''
  37. Sequence(Translate[Dilate[list2, 0.25, B], Vector[(t * u / 2)]], t, 0, 3) → list2''
  38. Sequence(Translate[Dilate[list1, 0.125, B'], Vector[(t * u / 4)]], t, 0, 7) → list1'''
  39. Sequence(Translate[Dilate[list2, 0.125, B], Vector[(t * u / 4)]], t, 0, 7) → list2'''
  40. Sequence(Translate[Dilate[list1, 0.0625, B'], Vector[(t * u / 8)]], t, 0, 15) → list1''''
  41. Sequence(Translate[Dilate[list2, 0.0625, B], Vector[(t * u / 8)]], t, 0, 15) → list2''''
  42. Translate(Pyramid[Polygon[B, C, F], E], Vector[(s * v)]) → c'
  43. Translate(Pyramid[Polygon[A, B, C, D], E], Vector[(0 * v)]) → f'
  44. Rotate(Prism[Polygon[D, L, P, H], O], 0°, zAxis) → g'
  45. Rotate(Prism[Polygon[A, O, G], K], 0°, zAxis) → h'
  46. Rotate(Prism[Polygon[Q, R, N], P], 0°, zAxis) → i'
  47. Rotate(Prism[Polygon[I', J', F'], Q'], 0°, zAxis) → j'
  48. Rotate(Prism[Polygon[Q', N', I'], P'], 0°, zAxis) → k'

AI 绘图提示语参考

绘制一个金字塔体积公理的数学证明图，展示第三世纪数学家刘徽的证明方法。图中包含一个底为四边形的棱柱和一个金字塔，通过平移和旋转这些几何体，并利用中点和向量工具，最终形成一系列相似的小几何体，直观说明金字塔体积公式。构造顺序从基本几何体开始，逐步平移、旋转并生成序列，强调截面相似性和体积关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。