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立方体分割问题

立方体分割问题
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2015-06-28 11:08
立体几何矩阵几何作图
①本范例演示立方体通过平面分割成不同部分的过程与性质；②涉及的核心数学概念包括立体几何、平面、整数、矩阵、组合数学、同余等；③作图要点提炼自构造摘要，使用了Point, Cube, Segment, Polygon, Midpoint, Intersect, Pyramid, If, Sequence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分割、矩阵应用及组合计数等知识。

GeoGebra 命令及参数

  1. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  2. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF_1, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  3. Segment(A, F) → b
  4. Segment(B, E) → c
  5. Segment(B, G) → d
  6. Segment(C, F) → e
  7. Segment(C, H) → f
  8. Segment(D, G) → g
  9. Segment(A, H) → h
  10. Segment(D, E) → i
  11. Segment(E, G) → j
  12. Segment(F, H) → k
  13. Segment(A, C) → l
  14. Segment(B, D) → m
  15. Polygon(A, C, F) → poly1, f_1, a_1, c_1
  16. Polygon(D, G, E) → poly2, e_1, d_1, g_1
  17. Polygon(A, F, H) → poly3, h_1, a_2, f_2
  18. Polygon(B, D, G) → poly4, g_2, b_1, d_2
  19. Polygon(D, B, E) → poly5, e_2, d_3, b_2
  20. Polygon(A, C, H) → poly6, h_2, a_3, c_2
  21. Polygon(A, B, G, H) → poly7, a_4, b_3, g_3, h_3
  22. Polygon(B, C, H, E) → poly8, b_4, c_3, h_4, e_3
  23. Polygon(C, D, E, F) → poly9, c_4, d_4, e_4, f_3
  24. Polygon(D, A, F, G) → poly10, d_5, a_5, f_4, g_4
  25. Polygon(B, D, H, F) → poly11, b_5, d_6, h_5, f_5
  26. Polygon(C, A, E, G) → poly12, c_5, a_6, e_5, g_5
  27. Polygon(B, G, E) → poly13, e_6, b_6, g_6
  28. Polygon(C, F, H) → poly14, h_6, c_6, f_6
  29. Midpoint(A, F) → I
  30. Midpoint(B, G) → J
  31. Midpoint(C, H) → K
  32. Midpoint(E, D) → L
  33. Midpoint(F, H) → M
  34. Midpoint(B, D) → N
  35. Intersect(IntersectPath[Plane[B, E, G], Plane[A, F, H]], Plane[C, D, E]) → O
  36. Intersect(IntersectPath[Plane[B, E, G], Plane[A, F, D]], Plane[F, D, E]) → P
  37. Intersect(IntersectPath[Plane[B, E, H], Plane[A, F, D]], Plane[F, D, E]) → Q
  38. Intersect(IntersectPath[Plane[A, F, H], Plane[B, E, H]], Plane[F, D, E]) → R
  39. Pyramid(P, O, R, Q, I) → o_1, faceOPQR, edgeIP_1, edgeIO_1, edgeIR_1, edgeIQ, faceIOP_1, faceIOR_1, faceIQR, faceIPQ, edgeOP_1, edgeOR_1, edgeQR, edgePQ
  40. Pyramid(P, O, I, F) → p_1, faceIOP, edgeFI_1, faceFOP, faceFIO_1, faceFIP, edgeOP, edgeIO_2, edgeIP, edgeFP, edgeFO_1
  41. Pyramid(O, R, I, E) → q_1, faceIOR, edgeEI_1, faceEOR, faceEIR, faceEIO_1, edgeOR, edgeIR, edgeIO_3, edgeEO_1, edgeER
  42. Pyramid(E, I, F, O) → r_1, faceEFI, edgeFO, faceEIO, faceFIO, faceEFO, edgeEI, edgeFI, edgeEF, edgeEO, edgeIO
  43. If(t < 1, t, 1) → t1
  44. If((t ≥ 1) ∧ (t < 2), t - 1, If[t < 1, 0, 1]) → t2
  45. If((t ≥ 2) ∧ (t < 3), t - 2, If[t < 2, 0, 1]) → t3
  46. Sequence(Rotate[Translate[r_1, Vector[(t1 * u_1) + (t1 * v_1)]], (90° * i), Segment[I, K]], i, 0, 3) → list4
  47. Sequence(Rotate[Translate[p_1, Vector[(t2 * u_1) + (t2 * v_1)]], (90° * i), Segment[I, K]], i, 0, 3) → list2
  48. Sequence(Rotate[Translate[q_1, Vector[(t2 * u_1) + (t2 * v_1)]], (90° * i), Segment[I, K]], i, 0, 3) → list3
  49. Sequence(Rotate[Translate[o_1, Vector[(t3 * u_1) + (t3 * v_1)]], (90° * i), Segment[I, K]], i, 0, 3) → list1