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Decorated Polygons

Decorated Polygons
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2016-05-23 10:09
PolygonTransformationGeometric Construction
This example demonstrates the construction of decorative polygon patterns by transforming a triangle into a regular 2n-gon and applying alternating flips and rotations. It involves core mathematical concepts like polygon transformations, symmetry, and geometric constructions. Key GeoGebra commands and objects used include Polygon, Intersect, Point, PointIn, Sequence, Mirror, Midpoint, Rotate, PolyLine, and Translate. Suitable for high school mathematics education to understand transformations and tessellations.

GeoGebra Commands & Parameters

  1. Polygon(A, B, n) → poly1, f, q, s, t, C, D, F, G, H, I, J, g, K, S, T, h, i, j, k, l, m, p
  2. Intersect(Ray[Rotate[A, (180° - (360 * ° / n)) / 2, A], Rotate[B, (180° - (360 * ° / n)) / 2, A]], Ray[Rotate[B, ((-(180° - (360 * ° / n)))) / 2, B], Rotate[A, ((-(180° - (360 * ° / n)))) / 2, B]]) → E
  3. Point(Polygon[A, B, E]) → L
  4. Point(Polygon[A, B, E]) → M
  5. Point(Polygon[A, B, E]) → N
  6. Polygon(L, M, N) → poly2, n_1, l_1, m_1
  7. PointIn(Polygon[A, B, E]) → O
  8. PointIn(Polygon[A, B, E]) → P
  9. PointIn(Polygon[A, B, E]) → Q
  10. Polygon(O, P, Q) → poly3, q_1, o, p_1
  11. Sequence(Polygon[Rotate[L, ((i * 720) * ° / n), E], Rotate[M, ((i * 720) * ° / n), E], Rotate[N, ((i * 720) * ° / n), E]], i, 1, n / 2) → list1
  12. Sequence(Polygon[Rotate[O, ((i * 720) * ° / n), E], Rotate[P, ((i * 720) * ° / n), E], Rotate[Q, ((i * 720) * ° / n), E]], i, 1, n / 2) → list2
  13. Mirror(list1, Line[A, E]) → list1'
  14. Mirror(list2, Line[A, E]) → list2'
  15. Midpoint(A, B) → R
  16. Rotate(L, 180°, R) → L'
  17. Rotate(M, 180°, R) → M'
  18. Rotate(N, 180°, R) → N'
  19. Rotate(O, 180°, R) → O'
  20. Rotate(P, 180°, R) → P'
  21. Rotate(Q, 180°, R) → Q'
  22. Mirror(list1', Line[C, B]) → list1''
  23. Mirror(list2', Line[C, B]) → list2''
  24. Rotate(list1, 180°, R) → list1'_1
  25. Rotate(list2, 180°, R) → list2'_1
  26. PolyLine(A, B, E, A) → r
  27. Mirror(E, Line[A, B]) → E'
  28. Mirror(list1'_1, Line[E', B]) → list5
  29. Mirror(list2'_1, Line[E', B]) → list7
  30. Translate(Rotate[{list1, list1'}, (-360 * ° / n), A], A - B) → list9
  31. Translate(Rotate[{list2, list2'}, (-360 * ° / n), A], A - B) → list10
  32. Sequence(Rotate[list9, (((i * 360))° * (1 + a) / n), E], i, 1, n / (1 + a)) → list11
  33. Sequence(Rotate[list10, (((i * 360))° * (1 + a) / n), E], i, 1, n / (1 + a)) → list12
  34. Sequence(Rotate[list1'_1, (((i * (1 + a)) * 360) * ° / n), E], i, 1, n / (1 + a)) → list3
  35. Sequence(Rotate[list5, (((i * (1 + a)) * 360) * ° / n), E], i, 1, n / (1 + a)) → list6
  36. Sequence(Rotate[list2'_1, (((i * (1 + a)) * 360) * ° / n), E], i, 1, n / (1 + a)) → list4
  37. Sequence(Rotate[list7, (((i * (1 + a)) * 360) * ° / n), E], i, 1, n / (1 + a)) → list8
  38. Mirror(list2, U) → list2'_2
  39. Mirror(list2', U) → list2''_1
  40. Mirror(list1', U) → list1''_1
  41. Mirror(list1, U) → list1'_2
  42. Sequence(Rotate[{list2''_1, list2'_2}, (((i * (1 + a)) * 360) * ° / n), E], i, 1, n / (1 + a)) → list13
  43. Sequence(Rotate[{list1''_1, list1'_2}, (((i * (1 + a)) * 360) * ° / n), E], i, 1, n / (1 + a)) → list14