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刘徽金字塔体积证明 by Anthony OR 柯志明

刘徽金字塔体积证明 by Anthony OR 柯志明
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2017-07-17 11:47
向量三角函数立体几何微积分几何作图
①本范例演示刘徽关于金字塔体积的证明方法，通过几何构造展示体积公式推导；②涉及核心数学概念包括体积计算、几何变换、极限思想；③作图要点提炼自构造摘要，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及数学爱好者学习立体几何与几何证明方法。

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'