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Screw Bisection of a Cube

Screw Bisection of a Cube
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2018-12-08 22:42
Solid GeometryTransformationGeometric Construction
This example demonstrates the geometric phenomenon of bisecting a cube through a screw motion. It involves core mathematical concepts such as spatial geometry, helical motion, and bisection principles. The construction utilizes GeoGebra commands/objects like Point, Cube, Surface, CurveCartesian, Rotate, Translate, Polygon, and Segment. Suitable for high school and above, exploring applications in solid geometry, spatial transformations, and parametric equations.

GeoGebra Commands & Parameters

  1. Point(Circle[A, Distance[B, A], Segment[B, A]]) → C
  2. Cube(B, A, C) → a, D, faceCDHG, faceEFGH, edgeBD, edgeAB, edgeAC, edgeCD, edgeBE, edgeDH, edgeEH, edgeEF, E, edgeAF, edgeFG, edgeCG, edgeGH, F, G, H, faceABDC, faceBDHE, faceABEF, faceACGF
  3. Surface((u * cos(t)), (u * sin(t)), (sqrt(2) * t / π), t, 0, π, u, 0, 1.5) → b
  4. CurveCartesian((r(t) * cos(t)), (r(t) * sin(t)), (sqrt(2) * t / π), t, 0, π / 2) → c
  5. CurveCartesian((p(t) * cos(t)), (p(t) * sin(t)), (sqrt(2) * t / π), t, π / 2, π) → d
  6. Rotate(b, π) → b'
  7. Rotate(c, π) → c'
  8. Rotate(d, π) → d'
  9. Surface((u * x(d(t))) + ((1 - u) * x(d'(t))), (u * y(d(t))) + ((1 - u) * y(d'(t))), (u * z(d(t))) + ((1 - u) * z(d'(t))), t, π / 2, π, u, 0, 1) → e
  10. Surface((u * x(c(t))) + ((1 - u) * x(c'(t))), (u * y(c(t))) + ((1 - u) * y(c'(t))), (u * z(c(t))) + ((1 - u) * z(c'(t))), t, 0, π / 2, u, 0, 1) → f
  11. Translate(Rotate[e, n], Vector[(0, 0, (sqrt(2) / π * n))]) → g
  12. Translate(Rotate[f, n], Vector[(0, 0, (sqrt(2) / π * n))]) → h
  13. Translate(Rotate[Polygon[H, F, G], n], Vector[(0, 0, (sqrt(2) / π * n))]) → t1
  14. Translate(Rotate[Polygon[D, B, A], n], Vector[(0, 0, (sqrt(2) / π * n))]) → t2
  15. Translate(Rotate[Surface[x(c(t)), y(c(t)), ((1 - u) * z(c(t))) + (u * sqrt(2)), t, 0, π / 2, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → j
  16. Translate(Rotate[Surface[x(d(t)), y(d(t)), ((1 - u) * z(d(t))) + (u * sqrt(2)), t, π / 2, π, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → i
  17. Translate(Rotate[Surface[x(d'(t)), y(d'(t)), (u * z(d'(t))), t, π / 2, π, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → k
  18. Translate(Rotate[Surface[x(c'(t)), y(c'(t)), ((1 - u) * z(c'(t))), t, 0, π / 2, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → l
  19. Surface(x(c(t)), y(c(t)), ((1 - u) * z(c(t))), t, 0, π / 2, u, 0, 1) → m
  20. Surface(x(d(t)), y(d(t)), ((1 - u) * z(d(t))), t, π / 2, π, u, 0, 1) → o
  21. Surface(x(c'(t)), y(c'(t)), ((1 - u) * z(c'(t))) + (u * sqrt(2)), t, 0, π / 2, u, 0, 1) → q
  22. Surface(x(d'(t)), y(d'(t)), ((1 - u) * z(d'(t))) + (u * sqrt(2)), t, π / 2, π, u, 0, 1) → s
  23. Polygon(F, H, E) → t3, e_1, f_1, h_1
  24. Polygon(A, C, D) → t4, d_1, a_1, c_1
  25. Translate(Rotate[Surface[x(c'(t)), y(c'(t)), ((1 - u) * z(c'(t))), t, 0, π / 2, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → t
  26. Translate(Rotate[c, n], Vector[(0, 0, (sqrt(2) / π * n))]) → u
  27. Translate(Rotate[c', n], Vector[(0, 0, (sqrt(2) / π * n))]) → v
  28. Translate(Rotate[d', n], Vector[(0, 0, (sqrt(2) / π * n))]) → w
  29. Translate(Rotate[d, n], Vector[(0, 0, (sqrt(2) / π * n))]) → b_1
  30. Translate(Segment[Rotate[A, n], Rotate[F, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → g_1
  31. Translate(Segment[Rotate[G, n], Rotate[F, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → i_1
  32. Translate(Segment[Rotate[G, n], Rotate[H, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → j_1
  33. Translate(Segment[Rotate[F, n], Rotate[H, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → k_1
  34. Translate(Rotate[Segment[G, Midpoint[G, C]], n], Vector[(0, 0, (sqrt(2) / π * n))]) → l_1
  35. Translate(Segment[Rotate[A, n], Rotate[B, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → m_1
  36. Translate(Rotate[Segment[B, Midpoint[B, E]], n], Vector[(0, 0, (sqrt(2) / π * n))]) → n_1
  37. Translate(Segment[Rotate[D, n], Rotate[B, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → p_1
  38. Translate(Segment[Rotate[D, n], Rotate[A, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → q_1
  39. Segment(H, F) → r_1
  40. Segment(H, E) → s_1
  41. Segment(E, F) → t_1
  42. Segment(A, D) → f_2
  43. Segment(A, C) → g_2
  44. Segment(D, C) → h_2
  45. Segment(H, D) → i_2
  46. Segment(E, Midpoint[B, E]) → j_2
  47. Segment(C, Midpoint[C, G]) → k_2

AI Drawing Prompt Reference

Draw a cube and demonstrate how to bisect it through a screw motion. Start by creating a cube, then construct a helical surface. Use rotations and translations to combine the helical surface with the cube, forming a bisection effect. The final image should include the cube, helical surface, and related rotated and translated geometric objects, clearly showing the bisection process.
The prompt above was compiled from the original drawing commands and is for reference only.