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Cutting the Cube

Cutting the Cube
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2020-04-12 16:23
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates how to visualize the internal geometric structure and spatial relationships of a cube through cutting. It involves core mathematical concepts such as solid geometry, coordinate geometry, polygons, curve sketching, and polynomial functions. Key construction points include using GeoGebra commands like Intersect, Point, Cube, Line, Segment, OrthogonalPlane, IntersectPath, Polygon, and If. Suitable for high school students and above for exploring spatial geometry problems and mathematical teaching demonstrations.

GeoGebra Commands & Parameters

  1. Intersect(zAxis, xAxis) → 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, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. Line(A, G) → f
  6. Segment(A, G) → g
  7. Point(g) → I
  8. OrthogonalPlane(I, g) → b
  9. IntersectPath(b, faceEFGH) → h
  10. Intersect(b, faceEFGH) → J, K
  11. IntersectPath(b, faceABFE) → i
  12. Intersect(b, faceABFE) → L, M
  13. IntersectPath(b, faceBCGF) → j
  14. Intersect(b, faceBCGF) → N, O
  15. IntersectPath(b, faceBCGF) → k
  16. Intersect(b, faceBCGF) → P, Q
  17. IntersectPath(b, faceCDHG) → l
  18. Intersect(b, faceCDHG) → R, S
  19. IntersectPath(b, faceADHE) → m
  20. Intersect(b, faceADHE) → T, U
  21. IntersectPath(b, faceABCD) → n
  22. Intersect(b, faceABCD) → V, W
  23. Polygon(N, S, R, K, J, L) → poly1, n_1, s, r, k_1, j_1, l_1
  24. Polygon(L, U, M) → t1, m_1, l_2, u
  25. Polygon(N, K, J) → t2, j_2, n_2, k_2
  26. If(poly1 ≟ poly1, true, false) → c
  27. Segment(Z, Z1) → p
  28. Point(p) → A_1

AI Drawing Prompt Reference

Draw a cube with its base defined by two points on the coordinate axes and a third point on the base. Construct a space diagonal from one vertex to the opposite vertex, and place a moving point on this diagonal. Through this point, create a plane perpendicular to the diagonal. This plane intersects each face of the cube, producing a set of intersection lines. These lines form a polygonal cross-section, while the intersections with the cube's edges create several triangles. Show how the cross-section changes as the moving point slides along the diagonal, highlighting the polygon and its intersection lines with the faces.
The prompt above was compiled from the original drawing commands and is for reference only.