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Compound Cube–Octahedron Construction

Compound Cube–Octahedron Construction
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2023-12-15 21:09
PolygonVectorSolid GeometryGeometric Construction
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This GeoGebra example demonstrates how to build and visualize a compound solid formed from a cube and an octahedron, highlighting their shared symmetry and spatial relationships. The key math ideas include solid geometry (cube and octahedron), constructing 3D configurations from derived points, using midpoint-based constructions, and using line intersection to determine the critical vertices that make up the compound. Construction-wise, it relies on GeoGebra commands Midpoint and Segment to generate intermediate points and connecting structure, Line to define the required spatial lines, and Intersect to compute where those lines meet, yielding many key points (labeled A, B, C, … plus grouped sets like f, g, h, …) that assemble the final compound outline. Suitable for advanced secondary school and early university audiences, it supports lessons on space geometry, geometric construction workflows, and visual reasoning about symmetry in 3D solids.

GeoGebra Commands & Parameters

  1. Midpoint(B, D) → H
  2. Midpoint(E, F) → I
  3. Midpoint(E, A) → J
  4. Segment(B, A) → f
  5. Segment(A, C) → g
  6. Segment(A, D) → h
  7. Segment(M, P) → i
  8. Segment(M, L) → j
  9. Segment(P, N) → k
  10. Segment(B, F) → l
  11. Segment(P, K) → m
  12. Segment(P, L) → n
  13. Line(B, E) → p
  14. Line(E, C) → q
  15. Line(C, G) → r
  16. Segment(F, D) → s
  17. Segment(D, G) → t
  18. Segment(N, L) → a
  19. Line(N, K) → b
  20. Line(K, M) → c
  21. Intersect(b, s) → R
  22. Segment(N, R) → d
  23. Intersect(b, l) → S
  24. Segment(S, K) → e
  25. Intersect(l, m) → T
  26. Intersect(k, s) → U
  27. Segment(T, U) → f_1
  28. Intersect(n, h) → V
  29. Segment(U, V) → g_1
  30. Intersect(f, i) → W
  31. Segment(W, V) → h_1
  32. Segment(W, T) → i_1
  33. Intersect(g, j) → Z
  34. Segment(Z, V) → j_1
  35. Intersect(t, a) → A_1
  36. Segment(A_1, V) → k_1
  37. Intersect(p, c) → B_1
  38. Segment(B_1, W) → l_1
  39. Segment(W, Z) → m_1
  40. Intersect(c, l) → C_1
  41. Segment(C_1, K) → n_1
  42. Intersect(c, f) → D_1
  43. Segment(B, B_1) → p_1
  44. Segment(B_1, M) → q_1
  45. Intersect(q, j) → E_1
  46. Segment(E_1, C) → r_1
  47. Intersect(r, j) → F_1
  48. Segment(F_1, C) → s_1
  49. Intersect(r, a) → G_1
  50. Segment(G_1, G) → t_1

AI Drawing Prompt Reference

Create a 3D compound solid formed by a cube together with an octahedron, highlighting their shared symmetry and spatial relationships. Start from two sets of defining points, construct intermediate points using midpoints, and then connect these points to form the main wireframe edges. Build several 3D lines and compute their intersection points to obtain the critical vertices that belong to the compound. Finally, connect the intersection-derived vertices to the appropriate original points/segments so that the cube-octahedron compound outline and internal connecting edges are clearly assembled, emphasizing how corresponding faces and edges relate.
The prompt above was compiled from the original drawing commands and is for reference only.