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Dissecting a Cube into Three Parts

Dissecting a Cube into Three Parts
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2019-04-21 22:19
Solid GeometryTransformationGeometric Construction
This example demonstrates how to dissect a cube into three parts using three lines. Key mathematical concepts include spatial division in solid geometry and the application of similar triangles. Using commands such as Intersect, Point, Cube, Line, Pyramid, and Rotate, a model of the cube's dissection can be constructed. Suitable for high school and above students or teachers for geometry teaching and research.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeAD_1, edgeAB_1, edgeBC_1, edgeCD_1, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF_1, edgeFG, edgeCG_1, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. Line(G, A) → f
  6. Pyramid(B, C, D, A, G) → b, face1, edge5, edge6, edge7, edge8, face2, face3, face4, face5, edge1, edge2, edge3, edge4
  7. Rotate(b, (-(120°)), f) → b_2
  8. Rotate(b, (-(240°)), f) → b_3
  9. Line(G, H) → r_1
  10. Line(F, G) → r_3
  11. Rotate(b, (-α), f) → b'

AI Drawing Prompt Reference

Draw a cube and dissect it into three parts using three lines. First, draw the six faces and eight vertices of the cube. Then, draw a line f through points A, B, and C, and construct a tetrahedron b with G as the apex and B, C, D, and A as the base. Next, rotate tetrahedron b around line f by -120° and -240° to get b_2 and b_3. Finally, draw lines GH and FG, and rotate tetrahedron b around line f to get b'.
The prompt above was compiled from the original drawing commands and is for reference only.