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Trisecting a Box

Trisecting a Box
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2015-01-22 19:50
VectorSolid GeometryGeometric Construction
This example demonstrates how to trisect a box using geometric constructions. It involves core math concepts like solid geometry, vectors, and polygon construction/transformation. Key GeoGebra commands and objects used include Segment, Line, Intersect, Polygon, Vector, Translate, Prism, Pyramid, Sphere, and PointIn for constructing and manipulating points, lines, faces, and solids. Suitable for high school and above students learning spatial geometry and geometric construction techniques.

GeoGebra Commands & Parameters

  1. Segment(O, A) → a
  2. Segment(O, B) → b
  3. Line(B, a) → c
  4. Line(A, b) → d
  5. Intersect(d, c) → D
  6. Polygon(O, B, D, A) → poly1, o, b_1, d_1, a_1
  7. Vector(O, C) → u
  8. Translate(poly1, u) → poly1'
  9. Translate(A, u) → A'_1
  10. Translate(B, u) → B'
  11. Translate(D, u) → D'_1
  12. Prism(O, A, D, B, C) → e, V, edgeAO, edgeAD, edgeBD, edgeBO, edgeCO, edgeAE, edgeDF, edgeBG, edgeCE, edgeEF, D', edgeFG, edgeCG, G, faceADBO, faceAECO, faceADFE, faceBDFG, faceBGCO, faceCEFG
  13. Pyramid(Polygon[B, D, D', B'], V) → uno, faceBDV_1, faceDD'V, faceB'D'V, faceBB'V_1, edgeBV_1, edgeDV_1, edgeD'V, edgeB'V_1
  14. Pyramid(Polygon[O, B, B', C], V) → due, faceBOV_1, faceBB'V, faceB'CV, faceCOV, edgeOV_1, edgeBV_2, edgeB'V, edgeCV
  15. Pyramid(Polygon[O, B, D, A], V) → tre, faceBOV, faceBDV, faceADV, faceAOV, edgeOV, edgeBV, edgeDV, edgeAV
  16. Sphere(H, Distance[E, F]) → f
  17. PointIn(f) → I
  18. PointIn(f) → J
  19. PointIn(f) → K
  20. Vector(H, I) → v
  21. Vector(H, J) → w
  22. Vector(H, K) → g
  23. Translate(uno, w) → uno'
  24. Translate(due, g) → due'
  25. Translate(tre, v) → tre'
  26. Segment(E, F) → h

AI Drawing Prompt Reference

Draw a cube and demonstrate how to trisect it. Start by sketching three edges of the cube and finding their intersection points. Construct a triangle using these points, then translate it along a vector to create a new shape. Use the translated and original points to build prisms and pyramids. Finally, draw a sphere above the cube and mark three points on its surface, connecting them to the cube's vertices to show the trisection effect.
The prompt above was compiled from the original drawing commands and is for reference only.