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Irregular Mosaic

Irregular Mosaic
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2023-03-13 20:55
PolygonCoordinate GeometryTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates the construction of an irregular mosaic. Key mathematical concepts include transformations (such as homothety) and geometric construction. Using GeoGebra objects like vectors, circles, and lines, an irregular mosaic can be constructed. Suitable for middle school and above students or teachers for teaching geometric construction and transformations.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. Circle(B, A) → c2
  3. Circle(A, B) → c1
  4. Intersect(c1, c2, 1) → C1
  5. Circle(C1, A) → c3
  6. Intersect(c2, c3, 2) → P2
  7. Circle(P2, C1) → c5
  8. Segment(A, B) → l
  9. OrthogonalLine(A, l) → r0
  10. Intersect(c3, r0, 2) → P1
  11. Circle(P1, C1) → c4
  12. Intersect(c3, c4, 2) → P3
  13. Line(P3, P2) → r1
  14. OrthogonalLine(P2, r1) → r2
  15. Intersect(c5, r2, 2) → A2
  16. Intersect(c4, r0, 2) → A1
  17. Vector(A, A1) → v1
  18. OrthogonalLine(P3, r1) → r3
  19. OrthogonalLine(A2, r2) → r4
  20. Intersect(r3, r4) → P4
  21. Circle(A2, P4) → c6
  22. Circle(P4, A2) → c7
  23. Intersect(c6, c7, 2) → C2
  24. Circle(C2, A2) → c8
  25. Intersect(c8, r2, 2) → P5
  26. Line(B2, P5) → r5
  27. OrthogonalLine(B2, r5) → r7
  28. OrthogonalLine(P5, r5) → r8
  29. Circle(B2, P5) → c9
  30. Intersect(c9, r7, 1) → P6
  31. Line(P6, r4) → r6
  32. Circle(P6, B2) → c9b
  33. Intersect(c9b, r6, 1) → A3
  34. Vector(A, A3) → v3
  35. Vector(A, A2) → v2
  36. Hex(A, B) → h1
  37. Hex(A2, B2) → h2
  38. Sequence(Sequence[Hex[A + (n * v1) + (k * v3), B + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list1, {{2.598076211353302}}
  39. Sequence(Sequence[Hex[A2 + (n * v1) + (k * v3), B2 + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list2, {{2.598076211353323}}
  40. Cua(P1, P3) → cua1
  41. Sequence(Sequence[Cua[P1 + (n * v1) + (k * v3), P3 + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list3, {{1.0000000000000284}}
  42. Intersect(c1, c4, 2) → P7
  43. Cua(P7, P1) → cua2
  44. Sequence(Sequence[Cua[P7 + (n * v1) + (k * v3), P1 + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list4, {{0.9999999999999432}}
  45. Cua(A, P7) → cua3
  46. Sequence(Sequence[Cua[A + (n * v1) + (k * v3), P7 + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list5, {{0.9999999999999716}}
  47. Cua(P2, B) → cua4
  48. Sequence(Sequence[Cua[P2 + (n * v1) + (k * v3), B + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list6, {{1.0000000000000284}}
  49. Cua(P3, P2) → cua5
  50. Sequence(Sequence[Cua[P3 + (n * v1) + (k * v3), P2 + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list7, {{1.0000000000000284}}
  51. Cua(B2, A2) → cua6
  52. Sequence(Sequence[Cua[B2 + (n * v1) + (k * v3), A2 + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list8, {{1}}
  53. Sequence(Sequence[Hex[A + (n * v1) + (k * v3), B + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list1b, {{2.598076211353302}}
  54. Sequence(Sequence[Hex[A2 + (n * v1) + (k * v3), B2 + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list2b, {{2.598076211353323}}
  55. Sequence(Sequence[Cua[P1 + (n * v1) + (k * v3), P3 + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list3b, {{1.0000000000000284}}
  56. Sequence(Sequence[Cua[B2 + (n * v1) + (k * v3), A2 + (n * v1) + (k * v3)], n, (-m), m], k, (-m), m) → list8b, {{1}}

AI Drawing Prompt Reference

Start with a segment defined by two endpoints. Construct two circles using each endpoint as center with the other endpoint as radius, then take their intersection. Build additional circles using the intersection point together with the original endpoints, and keep intersecting circles with previously formed lines to obtain more key points. Draw lines through selected points, including at least one pair of perpendicular guiding lines, and use them to locate further intersection points with the circles. Finally, use translation based on direction vectors to repeat the constructed hexagonal unit in a grid-like manner, forming an irregular mosaic pattern.
The prompt above was compiled from the original drawing commands and is for reference only.