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Autoduality

Autoduality
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2025-08-22 20:29
In collections Teselaciones regulares
This example demonstrates the application of autoduality in geometry. It involves the autoduality of geometric shapes, core mathematical concepts include symmetry, properties of geometric shapes and autoduality in geometry. The construction uses GeoGebra commands such as Circle, Intersect, Sequence, Flatten, and objects c, p, q, P0, V. Suitable for middle school math, geometry and math exploration

GeoGebra Commands & Parameters

  1. Circle((0, 0), 1) → c
  2. Intersect(Circle[((1 - (sin(pi / p) / cos(pi / q))^(2))^(-1 / 2), 0), ((cos(pi / q) / sin(pi / p))^(2) - 1)^(-1 / 2)], Ray[(0, 0), Rotate[(1, 0), pi / p]], 1) → P0
  3. Sequence(Rotate[P0, ((k * 2) * pi / p)], k, 0, p - 1) → V
  4. Circle(Mirror[V(p), c], V(p), V(1)) → hc1
  5. Flatten(Join[{{Vac(p)}, {Vac \ {Vac(p)}}}]) → Vacp
  6. Flatten(Join[{{Vac \ {Vac(1)}}, {Vac(1)}}]) → Vac2
  7. Intersect(hc1, Ray[(V(1) + V(p)) / 2, (0, 0)]) → V0,
  8. CircumcircleArc(V(p), V0, V(1)) → arcoCelda
  9. Join(Sequence[Rotate[Sequence[Point[arcoCelda, s], s, 0, 0.9, 0.1], ((2 * k) * pi / p)], k, 0, p - 1]) → celda0
  10. Sequence({Vac(k), q - 1, ncActual}, k, 3, p - 1) → Vplus
  11. If(floor(p / 2) ≠ p / 2 ∧ q ≟ 3, 4, floor(q / 2) ≟ q / 2, 2, 3) → nc
  12. Sequence({V(k), q - 1, 0}, k, 1, p) → vert0
  13. Sequence(Polygon[auto(1, k)], k, 1, 5) → autodual1
  14. Sequence(Polygon[auto(2, k)], k, 1, 5) → autodual2
  15. Sequence(Polygon[auto0(1, k)], k, 1, 5) → autodual01
  16. Sequence(Polygon[auto0(2, k)], k, 1, 5) → autodual02
  17. Flatten(Join[{{Vac \ {Vac(1), Vac(2)}}, {Vac(1), Vac(2)}}]) → Vac2'
  18. Flatten(Join[{{Vac(p - 1), Vac(p)}, {Vac \ {Vac(p - 1), Vac(p)}}}]) → Vacp'