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Penrose Kites and Darts

Penrose Kites and Darts
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2020-06-08 00:58
PolygonTransformationGeometric Construction
This example demonstrates the construction of Penrose kites and darts. Key mathematical concepts include geometric transformations and random experiments. Using GeoGebra commands such as Point, Rotate, Circle, Intersect, and Polygon, these unique aperiodic tiling patterns can be constructed. Suitable for middle school and above students or math enthusiasts learning about geometric transformations and random experiments.

GeoGebra Commands & Parameters

  1. Point(Circle[A, 1 + (1 + sqrt(5)) / 2]) → B
  2. Rotate(B, 72°, A) → BB
  3. Circle(BB, 1 + 1 / (1 + 2 / (1 + sqrt(5)))) → c
  4. Circle(B, 1 + 1 / (1 + 2 / (1 + sqrt(5)))) → d
  5. Intersect(c, d, 1) → C
  6. Polygon(A, B, C, BB) → vierhoek1, a, b, c_1, bb
  7. Circle(C, 1) → e
  8. Intersect(e, c_1, 1) → D
  9. Intersect(e, b, 1) → E
  10. CircleArc(C, D, E) → f
  11. Circle(A, (1 + sqrt(5)) / 2) → g
  12. Intersect(g, bb, 1) → F
  13. Intersect(g, a, 1) → G
  14. CircleArc(A, G, F) → h
  15. Point(Circle[H, 1 + (1 + sqrt(5)) / 2]) → I
  16. Rotate(I, 72°, H) → II
  17. Circle(I, 1 + 2 / (1 + sqrt(5))) → k
  18. Circle(II, 1 + 2 / (1 + sqrt(5))) → p
  19. Intersect(k, p, 1) → J
  20. Polygon(H, I, J, II) → vierhoek2, h_1, i, j, ii
  21. Circle(J, 2 / (1 + sqrt(5))) → q
  22. Intersect(q, i, 1) → K
  23. Intersect(q, j) → L,
  24. CircleArc(J, L, K) → r
  25. Circle(H, 1) → s
  26. Intersect(s, h_1, 1) → M
  27. Intersect(s, ii, 1) → N
  28. CircleArc(H, M, N) → t

AI Drawing Prompt Reference

Draw a Penrose tiling composed of two kites and one dart. Start by drawing a circle and finding a point on it. Rotate this point, then draw two new circles and find their intersections. Connect these points to form two kites and a dart, using curves to connect the intersections. Finally, create arcs between the vertices of the kites and dart to form an aperiodic, non-repeating pattern.
The prompt above was compiled from the original drawing commands and is for reference only.