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Coxeter Tessellation

Coxeter Tessellation
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2025-08-19 02:06
In collections Teselaciones regulares
This example demonstrates the construction of Coxeter tessellation. It involves geometric transformations such as reflection and symmetry. Using commands like Circle, Intersect, Sequence, Flatten, etc., the dynamic generation of the tiling pattern can be shown. Suitable for high school or university geometry courses.

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. Sequence(Polygon[cox0(k)], k, 1, 6) → coxeter01
  8. Sequence(Polygon[cox0(k)], k, 7, 12) → coxeter02
  9. Intersect(hc1, Ray[(V(1) + V(p)) / 2, (0, 0)]) → V0,
  10. CircumcircleArc(V(p), V0, V(1)) → arcoCelda
  11. Join(Sequence[Rotate[Sequence[Point[arcoCelda, s], s, 0, 0.9, 0.1], ((2 * k) * pi / p)], k, 0, p - 1]) → celda0
  12. Sequence(Polygon[cox(k)], k, 1, 6) → coxeter1
  13. Sequence(Polygon[cox(k)], k, 7, 12) → coxeter2
  14. Sequence({Vac(k), q - 1, ncActual}, k, 3, p - 1) → Vplus
  15. If(floor(p / 2) ≠ p / 2 ∧ q ≟ 3, 4, floor(q / 2) ≟ q / 2, 2, 3) → nc
  16. Sequence({V(k), q - 1, 0}, k, 1, p) → vert0
  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'