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Is a Cloud a Polygonal Fractal?

Is a Cloud a Polygonal Fractal?
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2020-05-24 15:51
PolygonCoordinate GeometryGeometric Construction
This example demonstrates how to construct and observe a polygonal fractal resembling a cloud using GeoGebra. It involves core mathematical concepts like fractal geometry, geometric transformations, linear equations, rotation, tangent lines, logarithms, and rhombuses. Key GeoGebra commands used include Sequence, Union, Line, and Mirror, along with objects like z_1 to z_6, list1 to list6, s1 to s6, and A to D. Suitable for high school and above, exploring the application of fractal geometry in natural phenomena.

GeoGebra Commands & Parameters

  1. Sequence((Element[list1, k] - Element[list1, 1]) / (Element[list1, Length[list1]] - Element[list1, 1]), k, 1, Length[list1]) → l
  2. Union(Flatten[Sequence[Sequence[(Element[list1, k] * (1 - Element[l, j])) + (Element[l, j] * Element[list1, k + 1]), j, 1, Length[l] - 1], k, 0, Length[list1] - 1]], {Element[list1, Length[list1]]}) → list2
  3. Union(Flatten[Sequence[Sequence[(Element[list2, k] * (1 - Element[l, j])) + (Element[l, j] * Element[list2, k + 1]), j, 1, Length[l] - 1], k, 0, Length[list2] - 1]], {Element[list2, Length[list2]]}) → list3
  4. Union(Flatten[Sequence[Sequence[(Element[list3, k] * (1 - Element[l, j])) + (Element[l, j] * Element[list3, k + 1]), j, 1, Length[l] - 1], k, 0, Length[list3] - 1]], {Element[list3, Length[list3]]}) → list4
  5. Union(Flatten[Sequence[Sequence[(Element[list4, k] * (1 - Element[l, j])) + (Element[l, j] * Element[list4, k + 1]), j, 1, Length[l] - 1], k, 0, Length[list4] - 1]], {Element[list4, Length[list4]]}) → list5
  6. Union(Flatten[Sequence[Sequence[(Element[list5, k] * (1 - Element[l, j])) + (Element[l, j] * Element[list5, k + 1]), j, 1, Length[l] - 1], k, 0, Length[list5] - 1]], {Element[list5, Length[list5]]}) → list6
  7. Sequence(Segment[Element[list6, k], Element[list6, k + 1]], k, 1, Length[list6] - 1) → s6
  8. Sequence(Segment[Element[list5, k], Element[list5, k + 1]], k, 1, Length[list5] - 1) → s5
  9. Sequence(Segment[Element[list4, k], Element[list4, k + 1]], k, 1, Length[list4] - 1) → s4
  10. Sequence(Segment[Element[list3, k], Element[list3, k + 1]], k, 1, Length[list3] - 1) → s3
  11. Sequence(Segment[Element[list2, k], Element[list2, k + 1]], k, 1, Length[list2] - 1) → s2
  12. Sequence(Segment[Element[list1, k], Element[list1, k + 1]], k, 1, Length[list1] - 1) → s1
  13. Line(z_1, z_4) → a
  14. Mirror(s1, a) → s1'
  15. Mirror(s2, a) → s2'
  16. Mirror(s3, a) → s3'
  17. Mirror(s4, a) → s4'
  18. Mirror(s5, a) → s5'
  19. Mirror(s6, a) → s6'
  20. Mirror(list1, a) → list1'
  21. Sequence(Polygon[Element[list1, k], Element[list1, k + 1], Element[list1', k + 1], Element[list1', k]], k, 1, Length[list1] - 1) → l2
  22. Mirror(list2, a) → list2'
  23. Sequence(Polygon[Element[list2, k], Element[list2, k + 1], Element[list2', k + 1], Element[list2', k]], k, 1, Length[list2] - 1) → l3
  24. Mirror(list3, a) → list3'
  25. Sequence(Polygon[Element[list3, k], Element[list3, k + 1], Element[list3', k + 1], Element[list3', k]], k, 1, Length[list3] - 1) → l4
  26. Mirror(list4, a) → list4'
  27. Sequence(Polygon[Element[list4, k], Element[list4, k + 1], Element[list4', k + 1], Element[list4', k]], k, 1, Length[list4] - 1) → l5
  28. Mirror(list5, a) → list5'
  29. Sequence(Polygon[Element[list5, k], Element[list5, k + 1], Element[list5', k + 1], Element[list5', k]], k, 1, Length[list5] - 1) → l6
  30. Mirror(list6, a) → list6'
  31. Sequence(Polygon[Element[list6, k], Element[list6, k + 1], Element[list6', k + 1], Element[list6', k]], k, 1, Length[list6] - 1) → l7