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Screwed Bisection of a Cube by Diego Lieban

Screwed Bisection of a Cube by Diego Lieban
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2018-12-18 06:45
Coordinate GeometrySolid GeometryTransformation
In collections Dissecting a cube into equal parts
This example demonstrates the screw bisection of a cube. Key mathematical concepts include solid geometry rotations and plane geometry polygons. Using GeoGebra commands such as Point, Cube, Rotate, Surface, etc., the various faces and edges of the cube can be constructed. Suitable for high school or university solid geometry studies.

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

AI Drawing Prompt Reference

Draw a cube and screw bisect it to show the screw section. Start by creating vertices A and B, then construct the cube's faces and edges. Next, draw a curve e and rotate it to get f. Through rotation and translation, generate a series of surfaces b, d, t1, t2, t3, t4, q1, q2, g, h, i, j, k, and l, ultimately displaying the screw section of the cube.
The prompt above was compiled from the original drawing commands and is for reference only.