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Screwed Bisection of a Cube

Screwed Bisection of a Cube
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2018-12-08 03:53
Solid GeometryTransformationGeometric Construction
This example demonstrates the geometric phenomenon of bisecting a cube along a median line using a screw-like rotation. It involves core mathematical concepts such as solid geometry, rotational transformations, median lines, quadratic functions, and hypergeometric distributions. The construction utilizes GeoGebra commands/objects like Point, Cube, CurveCartesian, Rotate, Surface, Translate, and Polygon. Suitable for high school and above students studying spatial geometry, transformations, and probability statistics, or for mathematical teaching demonstrations.

GeoGebra Commands & Parameters

  1. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  2. 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
  3. CurveCartesian(1, tan(t - π / 4), (4 * t / π), t, 0, π / 2) → e
  4. Rotate(e, π) → f
  5. Surface((u * cos(t - π / 4)), (u * sin(t - π / 4)), (4 * t / π), t, 0, π / 2, u, -2, 2) → b
  6. Surface((u * x(e(t))) + ((1 - u) * x(f(t))), (u * y(e(t))) + ((1 - u) * y(f(t))), (u * z(e(t))) + ((1 - u) * z(f(t))), t, 0, π / 2, u, 0, 1) → d
  7. Translate(Rotate[d, n, zAxis], Vector[(0, 0, (4 * n / π))]) → g
  8. Translate(Rotate[Polygon[E, F, G], n, zAxis], Vector[(0, 0, (4 * n / π))]) → t1
  9. Translate(Rotate[Polygon[D, A, B], n, zAxis], Vector[(0, 0, (4 * n / π))]) → t2
  10. Translate(Rotate[Polygon[A, B, F, E], n, zAxis], Vector[(0, 0, (4 * n / π))]) → q1
  11. Translate(Rotate[Surface[1, (-u) + ((1 - u) * y(e(t))), (2 * u) + ((1 - u) * z(e(t))), t, 0, π / 2, u, 0, 1], n, zAxis], Vector[(0, 0, (4 * n / π))]) → h
  12. Surface(-1, (-u) + ((1 - u) * y(f(t))), ((1 - u) * z(f(t))), t, 0, π / 2, u, 0, 1) → i
  13. Translate(Rotate[i, n, zAxis], Vector[(0, 0, (4 * n / π))]) → j
  14. Surface(1, (u * y(e(t))) + (1 - u), (u * z(e(t))), t, 0, π / 2, u, 0, 1) → k
  15. Surface(-1, (u * y(f(t))) + (1 - u), (u * z(f(t))) + ((1 - u) * 2), t, 0, π / 2, u, 0, 1) → l
  16. Polygon(E, H, G) → t3, g_1, e_1, h_1
  17. Polygon(B, C, D) → t4, d_1, b_1, c_1
  18. Polygon(C, D, H, G) → q2, c_2, d_2, h_2, g_2