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Trisecting a Rectangular Prism by Joel Friedman

Trisecting a Rectangular Prism by Joel Friedman
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2015-03-26 03:07
VectorSolid GeometryGeometric Construction
This example demonstrates the geometric construction of trisecting the volume of a rectangular prism. It involves core concepts in solid geometry, volume calculation, vector operations, and polygonal/triangular pyramid formation. GeoGebra commands and objects like Pyramid, Plane, IntersectPath, PointIn, Volume, and Point are utilized in the construction. Suitable for high school and above students learning about spatial geometry, volume partitioning, and geometric constructions.

GeoGebra Commands & Parameters

  1. Pyramid(E, B, D, C, A) → a, faceBDCE, edgeAE, edgeAB_1, edgeAD_1, edgeAC_1, faceABE, faceABD_1, faceACD_1, faceACE, edgeBE, edgeBD_1, edgeCD_1, edgeCE
  2. Pyramid(G, F, D, C, A) → b, faceCDFG, edgeAG, edgeAF_1, edgeAD_2, edgeAC, faceAFG, faceADF_1, faceACD, faceACG, edgeFG, edgeDF_1, edgeCD, edgeCG
  3. Pyramid(faceADF_1, F) → c, faceDFF, faceADF_2, faceAFF, edgeFF, edgeDF_1, edgeAF_1
  4. Pyramid(H, B, D, F, A) → d, faceBDFH, edgeAH, edgeAB, edgeAD, edgeAF, faceABH, faceABD, faceADF, faceAFH, edgeBH, edgeBD, edgeDF, edgeFH
  5. Plane(I, faceAFG) → k
  6. Plane(J, faceACE) → l
  7. Plane(K, faceCDFG) → m
  8. IntersectPath(k, b) → poly1, L, M, N, O, q, r, s, t
  9. IntersectPath(k, d) → poly2, poly3, c_1, d_1, e_1, f_1, P, Q, R, S, T, U, a_1, b_1
  10. IntersectPath(l, a) → poly4, poly6, i_1, j_1, p_1, q_1, V, W, Z, A_1, F_1, G_1, g_1, h_1
  11. IntersectPath(l, b) → poly5, poly7, m_1, n_1, r_1, s_1, B_1, C_1, D_1, E_1, H_1, I_1, k_1, l_1
  12. IntersectPath(m, a) → poly8, poly10, b_2, c_2, h_2, i_2, J_1, K_1, L_1, M_1, R_1, S_1, t_1, a_2
  13. IntersectPath(m, d) → poly9, poly11, f_2, g_2, j_2, k_2, N_1, O_1, P_1, Q_1, T_1, U_1, d_2, e_2
  14. PointIn(faceABH) → PointfaceABH
  15. PointIn(faceABE) → PointfaceABE
  16. PointIn(poly9) → Pointpoly9_1
  17. PointIn(poly8) → Pointpoly8_1
  18. PointIn(faceACG) → PointfaceACG
  19. PointIn(faceACE) → PointfaceACE
  20. PointIn(poly5) → Pointpoly5_1
  21. PointIn(poly4) → Pointpoly4_1
  22. PointIn(poly2) → Pointpoly2_1
  23. PointIn(poly1) → Pointpoly1_1
  24. PointIn(poly5) → Pointpoly5_2
  25. PointIn(poly4) → Pointpoly4_2
  26. PointIn(poly9) → Pointpoly9_2
  27. PointIn(poly8) → Pointpoly8_2
  28. Volume(d) → volumed
  29. Point(d) → Pointd_1
  30. Volume(b) → volumeb
  31. Point(b) → Pointb_1
  32. Volume(d) → o_1
  33. Point(d) → Pointd_2
  34. PointIn(poly2) → Pointpoly2_2
  35. PointIn(poly1) → Pointpoly1_2
  36. Volume(d) → l_2
  37. Point(d) → Pointd_3
  38. Volume(b) → m_2
  39. Point(b) → Pointb
  40. Volume(a) → volumea
  41. Point(a) → Pointa
  42. Volume(d) → n_2
  43. Point(d) → Pointd
  44. PointIn(poly2) → Pointpoly2_3
  45. PointIn(poly1) → Pointpoly1_3
  46. PointIn(poly2) → Pointpoly2_4
  47. PointIn(poly1) → Pointpoly1_4
  48. PointIn(poly5) → Pointpoly5
  49. PointIn(poly4) → Pointpoly4
  50. PointIn(poly9) → Pointpoly9_3
  51. PointIn(poly8) → Pointpoly8
  52. PointIn(poly9) → Pointpoly9
  53. PointIn(poly2) → Pointpoly2
  54. PointIn(poly1) → Pointpoly1

AI Drawing Prompt Reference

Draw a rectangular prism and trisect its volume. Construct four triangular pyramids using different vertices of the prism as apexes and faces or diagonal planes as bases. Then, create three cutting planes, each passing through a specific face of a pyramid, intersecting the pyramids to form multiple intersection paths and polygonal cross-sections. Finally, place random points inside the resulting faces and polygons, and compute the volumes of the pyramids to verify the trisection. The figure illustrates spatial volume partitioning and cross-section construction.
The prompt above was compiled from the original drawing commands and is for reference only.