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Cubes Illusion II

Cubes Illusion II
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2018-11-19 13:43
VectorTrigonometrySolid GeometryTransformationGeometric Construction
This example demonstrates the visual effect of the cube illusion. It involves cubes, trigonometric functions, means, trapezoids, and standard deviation. Using GeoGebra commands such as Vector, Sequence, Polygon, and Rotate, dynamic cube transformations can be constructed. Suitable for high school mathematics teaching and learning solid geometry transformations.

GeoGebra Commands & Parameters

  1. Vector((0, 0), A) → u
  2. Vector((0, 0), B) → v
  3. Sequence(Rotate[(cos(((i / n * 2) * π)), sin(((i / n * 2) * π))), π / 6, (0, 0)], i, 1, n) → Pts
  4. Sequence(Rotate[(((t - 1) * cos(((i / n * 2) * π))), ((t - 1) * sin(((i / n * 2) * π)))), π / 6, (0, 0)], i, 1, n) → PtsC
  5. Polygon(Element[Pts, 2], Element[Pts, 3], Element[Pts, 4], (0, 0), Element[Pts, 2]) → C1, f, g, h, i, j
  6. Rotate(C1, (2 * π / 3), (0, 0)) → L1
  7. Rotate(C1, ((2 * 2) * π / 3), (0, 0)) → L2
  8. Rotate(C1, ((3 * 2) * π / 3), (0, 0)) → L3
  9. Polygon(Element[PtsC, 2], Element[PtsC, 3], Element[PtsC, 4], (0, 0), Element[PtsC, 2]) → C2, k, l, m, p, q
  10. Rotate(C2, (2 * π / 3), (0, 0)) → M1
  11. Rotate(C2, ((2 * 2) * π / 3), (0, 0)) → M2
  12. Rotate(C2, ((3 * 2) * π / 3), (0, 0)) → M3
  13. Polygon(Element[PtsC, 2], Element[PtsC, 3], Element[PtsC, 4], (0, 0), Element[PtsC, 2]) → C3, r, s, a, b, c
  14. Rotate(C3, (2 * π / 3), (0, 0)) → N1
  15. Rotate(C3, ((2 * 2) * π / 3), (0, 0)) → N2
  16. Rotate(C3, ((3 * 2) * π / 3), (0, 0)) → N3
  17. Sequence(Sequence[Translate[L1, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → l_1
  18. Sequence(Sequence[Translate[L2, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → l_2
  19. Sequence(Sequence[Translate[L3, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → l_3
  20. Sequence(Sequence[Translate[N1, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → n_1
  21. Sequence(Sequence[Translate[N2, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → n_2
  22. Sequence(Sequence[Translate[N3, Vector[(i * u) + (j * v)]], i, 1, size], j, 1, size) → n_3
  23. Vector((0, 0), C) → w
  24. Sequence(Sequence[Translate[Translate[M1, Vector[(i * u) + (j * v)]], Vector[(0, 0), C]], i, 1, size], j, 1, size) → m_1
  25. Sequence(Sequence[Translate[Translate[M2, Vector[(i * u) + (j * v)]], Vector[(0, 0), C]], i, 1, size], j, 1, size) → m_2
  26. Sequence(Sequence[Translate[Translate[M3, Vector[(i * u) + (j * v)]], Vector[(0, 0), C]], i, 1, size], j, 1, size) → m_3