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Cube Dissection Problem

Cube Dissection Problem
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2015-06-28 11:08
Solid GeometryMatrixGeometric Construction
This example demonstrates the process and properties of dissecting a cube into different parts using planes. Core mathematical concepts include solid geometry, planes, integers, matrices, combinatorics, and congruence. It utilizes GeoGebra commands and objects like Point, Cube, Segment, Polygon, Midpoint, Intersect, Pyramid, If, and Sequence as outlined in the construction summary. Suitable for high school students and math enthusiasts learning about solid geometry dissection, matrix applications, and combinatorial counting.

GeoGebra Commands & Parameters

  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