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Trisecting a Cube into Three Pyramids

Trisecting a Cube into Three Pyramids
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2018-09-14 19:27
VectorSolid GeometryGeometric Construction
This example demonstrates how to trisect a cube into three pyramidal sections. It involves core mathematical concepts such as solid geometry, volume calculations, and spatial transformations. The construction utilizes GeoGebra commands like If, Cube, Pyramid, Rotate, Midpoint, OrthogonalLine, and Segment. It is suitable for high school students and above, as well as for mathematical education and exploration of spatial geometry problems.

GeoGebra Commands & Parameters

  1. If(t < 0.5, (2 * t), 1) → t1
  2. Cube(A, B, xOyPlane) → a, C, faceBCGF, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH_1, edgeEH_1, D, edgeEF, edgeBF, edgeFG, edgeCG, edgeGH_1, E, F, G, H, faceABCD, faceADHE, faceABFE
  3. Pyramid(faceABCD, H) → b, faceADH, faceABH_1, faceBCH_1, faceCDH, edgeDH, edgeAH_1, edgeBH_1, edgeCH_1
  4. Pyramid(faceBCGF, H) → c, faceBCH, faceBFH_1, faceFGH, faceCGH, edgeCH, edgeBH_2, edgeFH_1, edgeGH
  5. Pyramid(faceABFE, H) → d, faceABH, faceAEH, faceEFH, faceBFH, edgeBH, edgeAH, edgeEH, edgeFH
  6. Rotate(E, (90° * t1), edgeAB) → E'
  7. Midpoint(E', B) → I
  8. OrthogonalLine(I, Polygon[A, I, B]) → e
  9. Rotate(G, (90° * t1), edgeBC) → G'
  10. Midpoint(B, G') → J
  11. OrthogonalLine(J, Polygon[B, C, G']) → f
  12. Rotate(Pyramid[faceABCD, H], 0*°, edgeDH) → b'
  13. If(t ≥ 0.5, (2 * (t - 0.5)), 0) → t2
  14. Rotate(Rotate[c, (90° * t1), edgeBC], (-((90° * t2))), f) → c'
  15. Rotate(Rotate[d, (90° * t1), edgeAB], (((-90))° * t2), e) → d'
  16. Segment(A, B) → g
  17. Segment(B, C) → h
  18. Segment(C, D) → i
  19. Segment(D, A) → j
  20. Segment(A, E) → k
  21. Segment(B, F) → l
  22. Segment(C, G) → m
  23. Segment(D, H) → n
  24. Segment(E, F) → p
  25. Segment(F, G) → q
  26. Segment(G, H) → r
  27. Segment(H, E) → s