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Non Periodic Tiling (Extended Version)

Non Periodic Tiling (Extended Version)
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2013-06-04 23:04
SequenceTransformationGeometric Construction
This example demonstrates the construction of non-periodic tilings. Key mathematical concepts include geometric transformations and sequences. Using GeoGebra commands such as Intersect, Sequence, and Polygon, the dynamic generation of tiling patterns is showcased. Suitable for high school students and math enthusiasts.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, xAxis) → An
  2. Sequence(Rotate[(Zoom; 0), 2 * π / 10 * i, An], i, 0, 10) → l1
  3. Sequence(Rotate[Element[l1, 1], 2 * π / 5 * i, An1], i, 0, 5) → lPV5
  4. Polygon(lPV5) → V51, d, e, g, h, i, j
  5. Sequence(Dilate[An1, 1 / f^(i), An], i, 0, n_{ext}) → lrP1
  6. Sequence(Dilate[Element[lrP1, i], 1 + f^(3), An], i, 3, n_{ext} + 1) → lrP2
  7. Translate(V51, Element[lrP2, 1] - An1) → V51a
  8. Polygon(Element[lPV5, 1], Element[lPV5, 3], Element[lPV5, 5], Element[lPV5, 2], Element[lPV5, 4], Element[lPV5, 1]) → VZ1, l, m, n, q, s, t
  9. Sequence(Polygon[Element[l1, i + 1], Element[l1, i], 5], i, 1, 10) → R01
  10. Polygon(An, Element[l1, 1], Element[l1, 2]) → V3, a, o, b
  11. Ray(An, (Element[l1, 1] + Element[l1, 2]) / 2) → Str01
  12. Sequence(Dilate[lPV5, 1 / f^(i), An], i, 0, n_{ext}) → lrPV
  13. Sequence(Dilate[V51, 1 / f^(i), An], i, 0, n_{ext}) → lrV
  14. Sequence(Translate[Rotate[V51, (i + 1) * π, An1], Element[lrP1, 2 * i - 1] - An1], i, 1, n_{ext} - 2) → lrV2
  15. Sequence(Dilate[VZ1, 1 / f^(i), An], i, 0, n_{ext}) → lrZ
  16. Sequence(Rotate[V51a, 2 * π / 5 * i, Element[lrP1, 3]], i, 0, 4) → Liste1
  17. Element(lrZ, 2) → ZV2
  18. Element(lrP1, 2) → An2
  19. Sequence(Rotate[Translate[ZV2, Element[lrP2, 2] - An2], 2 * π / 5 * i, Element[lrP1, 4]], i, 0, 4) → Liste2
  20. Sequence(Translate[Rotate[ZV2, (i + 1) * π, An2], Element[lrP1, 2 * i] - An2], i, 1, n_{ext} - 2) → lrZ2
  21. Sequence(Rotate[Translate[Liste1, Element[lrP2, 3] - Element[lrP1, 3]], 2 * π / 5 * i, Element[lrP1, 5]], i, 0, 4) → Liste3
  22. Translate(Rotate[Liste1, 3.141592653589793, Element[lrP1, 3]], Element[lrP1, 5] - Element[lrP1, 3]) → Liste4
  23. Sequence(Rotate[V51a, 2 * π / 5 * i, Element[lrP1, 4]], i, 0, 4) → Liste5
  24. Sequence(Rotate[Element[Liste2, 1], 2 * π / 5 * i, Element[lrP1, 5]], i, 0, 4) → Liste6
  25. Sequence(Rotate[Translate[Element[lrV2, 2], Element[lrP2, 3] - Element[lrP1, 3]], 2 * π / 5 * i, Element[lrP1, 5]], i, 0, 4) → Liste7
  26. Translate(Rotate[Liste5, 3.141592653589793, Element[lrP1, 4]], Element[lrP1, 6] - Element[lrP1, 4]) → Liste41
  27. Translate(Rotate[Liste2, 3.141592653589793, Element[lrP1, 4]], Element[lrP1, 6] - Element[lrP1, 4]) → Liste42
  28. Sequence(Rotate[Translate[Liste2, Element[lrP2, 4] - Element[lrP1, 4]], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4) → Liste43
  29. Sequence(Rotate[Translate[Liste5, Element[lrP2, 4] - Element[lrP1, 4]], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4) → Liste44
  30. Sequence(Rotate[Translate[Element[lrZ2, 2], Element[lrP2, 4] - Element[lrP1, 4]], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4) → Liste45
  31. Sequence(Rotate[Element[Liste7, 1], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4) → Liste46
  32. Sequence(Rotate[Element[Liste3, 1], 2 * π / 5 * i, Element[lrP1, 6]], i, 0, 4) → Liste47
  33. Dilate(V51, 1 + f, (0, 0)) → V52
  34. Circle(An, Element[Element[lrPV, 6], 3]) → p
  35. Sequence(Dilate[VZ1, 1 / f^(i), An], i, 0, n_{ext}) → lrZz
  36. Translate(Rotate[n01, 3.141592653589793, Element[lrP1, 5]], Element[lrP1, 7] - Element[lrP1, 5]) → n02
  37. Sequence(Rotate[Element[Liste43, 1], 2 * π / 5 * i, Element[lrP1, 7]], i, 1, 5) → n03a
  38. Sequence(Rotate[Element[Liste45, 1], 2 * π / 5 * i, Element[lrP1, 7]], i, 1, 5) → n04
  39. Sequence(Rotate[Translate[Rotate[Element[lrV2, 4], 3.141592653589793, Element[lrP1, 7]], Element[lrP2, 5] - Element[lrP1, 7]], 2 * π / 5 * i, Element[lrP1, 7]], i, 1, 5) → n08
  40. Sequence(Rotate[Element[Liste44, 1], 2 * π / 5 * i, Element[lrP1, 7]], i, 1, 5) → n03b
  41. Sequence(Rotate[Translate[Rotate[n02, 3.141592653589793, Element[lrP1, 7]], Element[lrP2, 5] - Element[lrP1, 7]], 2 * π / 5 * i, Element[lrP1, 7]], i, 1, 5) → n09
  42. Sequence(Translate[Rotate[V51a, (i + 1) * π, Element[lrP2, 1]], Element[lrP2, 2 * i - 1] - Element[lrP2, 1]], i, 1, n_{ext}) → n10
  43. Translate(ZV2, Element[lrP2, 2] - An2) → ZV2a
  44. Sequence(Translate[Rotate[ZV2a, i * π, Element[lrP2, 2]], Element[lrP2, 2 * i - 2] - Element[lrP2, 2]], i, 2, n_{ext}) → n11
  45. Translate(Rotate[nn01, 3.141592653589793, Element[lrP1, 6]], Element[lrP1, 8] - Element[lrP1, 6]) → nn02
  46. Sequence(Rotate[Translate[Rotate[nn02, 3.141592653589793, Element[lrP1, 8]], Element[lrP2, 6] - Element[lrP1, 8]], 2 * π / 5 * i, Element[lrP1, 8]], i, 1, 5) → nn09
  47. Sequence(Rotate[Element[n11, 3], 2 * π / 5 * i, Element[lrP1, 8]], i, 1, 5) → nn08
  48. Sequence(Rotate[Element[n09, 5], 2 * π / 5 * i, Element[lrP1, 8]], i, 1, 5) → nn10
  49. Sequence(Rotate[Element[n08, 5], 2 * π / 5 * i, Element[lrP1, 8]], i, 1, 5) → nn11
  50. Sequence(Rotate[n02, π / 5 * i, An], i, 1, 10) → R02
  51. Sequence(Rotate[nn09, π / 5 * i, An], i, 1, 10) → R03
  52. Sequence(Rotate[nn02, π / 5 * i, An], i, 1, 10) → R04
  53. Sequence(Rotate[Rr03, π / 5 * i, An], i, 1, 10) → R06
  54. Sequence(Rotate[Rr04, π / 5 * i, An], i, 1, 10) → R07
  55. Sequence(Rotate[Rr01, π / 5 * i, An], i, 1, 10) → R08
  56. Sequence(Rotate[lrV2, π / 5 * i, An], i, 1, 10) → R09
  57. Sequence(Rotate[lrZ2, π / 5 * i, An], i, 1, 10) → R10
  58. Sequence(Rotate[Rr02, π / 5 * i, An], i, 1, 10) → R11
  59. Translate(Element[lrZ2, 2], An - Element[lrP1, c]) → lrZ2a

AI Drawing Prompt Reference

Create a non-periodic tiling pattern with multiple rotated and scaled polygons. Start by creating a central point, then rotate and scale a series of polygons around it to form a complex tiling pattern. Next, use translations and rotations to further build the tiling pattern, ultimately forming a non-periodic tiling structure.
The prompt above was compiled from the original drawing commands and is for reference only.