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Cube and Octahedron

Cube and Octahedron
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2020-02-23 06:00
Coordinate GeometrySolid Geometry
This example demonstrates the relationship between a cube and an octahedron. Key mathematical concepts include solid geometry. Using GeoGebra commands such as Point, Cube, Rotate, etc., the faces and edges of the cube can be constructed. Suitable for students at the junior high school level and above.

GeoGebra Commands & Parameters

  1. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  2. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, arêteAD, arêteAB, arêteBC, arêteCD, arêteAE, arêteDH, arêteEH, arêteEF, E, arêteBF, arêteFG, arêteCG, arêteGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  3. Point(arêteAE, 1 - p) → Ka
  4. Point(arêteAB, p) → Ia
  5. Point(arêteAD, 1 - p) → Ja
  6. Rotate(Ja, 90°, zAxis) → Jb
  7. Rotate(Ka, 90°, zAxis) → Kb
  8. Pyramid(Polygon[Ia, Ja, Ka], A) → pyrA, faceAIaJa, faceAJaKa, faceAIaKa, arêteAIa, arêteAJa, arêteAKa
  9. Polygon(Ia, Ja, Ka) → triA, ka, ia, ja
  10. Plane(xAxis, yAxis) → o
  11. Mirror(Ka, o) → Ke
  12. Mirror(Ia, o) → Ie
  13. Mirror(Jb, o) → Jf
  14. Mirror(Kb, o) → Kf
  15. Polygon(Ia, Jb, Kb, Kf, Jf, Ie, Ke, Ka) → fab, ia_1, jb_1, kb_1, kf, jf, ie, ke_1, ka_2
  16. Rotate(fab, 90°, zAxis) → fbc
  17. Rotate(fab, 180°, zAxis) → fcd
  18. Rotate(fab, (-(90°)), zAxis) → fda
  19. Rotate(fab, (-(90°)), arêteAB) → fsous
  20. Rotate(fab, 90°, arêteEF) → fsur
  21. CircumcircleArc(A, I, J) → c
  22. Translate(Rotate[pyrA, (Answer * α), arêteAE], Vector[A, Point[c, Answer]]) → pyramA
  23. Rotate(triA, 90°, zAxis) → triB
  24. Rotate(triA, 180°, zAxis) → triC
  25. Rotate(triA, (-(90°)), zAxis) → triD
  26. Mirror(triA, o) → triE
  27. Rotate(triE, 90°, zAxis) → triF
  28. Rotate(triE, 180°, zAxis) → triG
  29. Rotate(triE, (-(90°)), zAxis) → triH
  30. CircumcircleArc(B, K, J) → d
  31. Translate(Rotate[Rotate[pyrA, 90°, zAxis], (Answer * α), arêteBF], Vector[B, Point[d, Answer]]) → pyramB
  32. CircumcircleArc(C, L, J) → e
  33. CircumcircleArc(D, M, J) → f
  34. Translate(Rotate[Rotate[pyrA, 180°, zAxis], (Answer * α), arêteCG], Vector[C, Point[e, Answer]]) → pyramC
  35. Translate(Rotate[Rotate[pyrA, (-(90°)), zAxis], ((-Answer) * α), arêteDH], Vector[D, Point[f, Answer]]) → pyramD
  36. Mirror(pyramD, o) → pyramH
  37. Mirror(pyramA, o) → pyramE
  38. Mirror(pyramB, o) → pyramF
  39. Mirror(pyramC, o) → pyramG
  40. Midpoint(J, Kb) → N
  41. Mirror(N, O) → N'
  42. Mirror(N, O) → N'_1
  43. Mirror(A, O) → A'
  44. Mirror(N, O) → N'_2

AI Drawing Prompt Reference

Draw a cube and an octahedron that share a plane. Start by creating a cube, then generate multiple triangles by rotating around the cube's edges, which will form the faces of the octahedron. Use mirroring and rotating operations to create the remaining parts of the octahedron, resulting in a structure where the cube and octahedron share a plane.
The prompt above was compiled from the original drawing commands and is for reference only.