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Aperiodic Tiling via a Single Monotile and Translation Structure

Aperiodic Tiling via a Single Monotile and Translation Structure
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2025-08-09 20:20
In collections La fábrica de teselados
This GeoGebra example demonstrates how an aperiodic tiling can be generated from a single prototile (“monotile”) by enforcing a translation-based geometric construction. It highlights key ideas from tiling theory: non-periodicity, symmetry/translation structure, and the geometric constraints that make local pieces fit globally. The construction uses geometric relations involving parallelogram-like

GeoGebra Commands & Parameters

  1. Point(Circle[A, 2]) → B
  2. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-199.9999999999999, 799), (1499, -49.99999999999994)) → l1
  3. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-649.9999999999998, 749), (1049, -99.99999999999989)) → l2
  4. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-49.99999999999997, 799), (1649, -49.99999999999994)) → l3
  5. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-199.9999999999999, 749), (1499, -99.99999999999989)) → l4
  6. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-849.9999999999998, 949), (849, 99.99999999999989)) → l5
  7. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-99.99999999999994, 749), (1599, -99.99999999999989)) → l6
  8. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-50.00000000000003, 849), (1649, 0)) → l7
  9. Point(Circle[E, 1]) → F
  10. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (49.999999999999915, 849), (1749, 0)) → l8
  11. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-199.99999999999994, 899), (1499, 49.99999999999994)) → l11
  12. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-199.99999999999994, 799), (1499, -49.99999999999994)) → l12
  13. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-199.99999999999994, 699), (1499, -149.99999999999983)) → l13
  14. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (149.99999999999986, 649), (1849, -199.99999999999977)) → l15
  15. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (99.99999999999989, 799), (1799, -49.99999999999994)) → l16
  16. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-100, 849), (1599, 0)) → l17
  17. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-499.9999999999998, 699.0000000000002), (1199.0000000000002, -149.99999999999983)) → l18
  18. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-149.99999999999997, 849), (1549, 0)) → l19
  19. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-349.9999999999999, 749), (1349, -99.99999999999989)) → l20
  20. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), ((-5.6843418860808015E-14), 899), (1699, 49.99999999999994)) → l21
  21. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-50.00000000000003, 899), (1649, 49.99999999999994)) → l22
  22. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-449.9999999999999, 849), (1249, 0)) → l25
  23. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-499.9999999999999, 899), (1199, 49.99999999999994)) → l26
  24. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (199.99999999999983, 699), (1899, -149.99999999999983)) → l27
  25. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (299.9999999999998, 799), (1999, -49.99999999999994)) → l28
  26. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (249.9999999999998, 799), (1949, -49.99999999999994)) → l29
  27. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (199.99999999999983, 899), (1899, 49.99999999999994)) → l30
  28. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-549.9999999999999, 949.0000000000002), (1149, 99.99999999999989)) → l32
  29. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (249.9999999999997, 649), (1949, -199.99999999999977)) → l33
  30. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-699.9999999999999, 899), (999, 49.99999999999994)) → l34
  31. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-50.00000000000003, 749), (1649, -99.99999999999989)) → l35
  32. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-549.9999999999999, 749), (1149, -99.99999999999989)) → l36
  33. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-649.9999999999999, 849), (1049, 0)) → l37
  34. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-599.9999999999999, 799), (1099, -49.99999999999994)) → l38
  35. AttachCopyToView(c2, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-399.9999999999999, 799), (1299, -49.99999999999994)) → l39
  36. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-849.9999999999999, 749), (849, -99.99999999999989)) → l40
  37. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-349.9999999999998, 699), (1349, -149.99999999999983)) → l41
  38. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-799.9999999999998, 899), (899, 49.99999999999994)) → l42
  39. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-149.99999999999991, 699), (1549, -149.9999999999999)) → l43
  40. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (49.99999999999997, 749), (1749, -99.99999999999989)) → l44
  41. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-349.9999999999998, 749), (1349, -99.99999999999994)) → l45
  42. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (49.99999999999997, 899), (1749, 49.99999999999994)) → l46
  43. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-699.9999999999998, 799), (999, -49.99999999999994)) → l47
  44. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-99.99999999999994, 899), (1599, 49.99999999999994)) → l48
  45. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-349.9999999999998, 949), (1349, 99.99999999999989)) → l49
  46. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-899.9999999999998, 799), (799, -49.99999999999994)) → l50
  47. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-249.99999999999986, 699), (1449, -149.99999999999983)) → l51
  48. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-749.9999999999998, 849), (949, 0)) → l52
  49. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-949.9999999999998, 849), (749, 0)) → l53
  50. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-349.9999999999998, 799), (1349, -49.99999999999994)) → l54
  51. AttachCopyToView(c1, 1, (-16.99000000000001, -8.490000000000009), (16.99000000000001, 8.490000000000009), (-249.9999999999998, 849), (1449, 0)) → l55