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Another Fractal Curve

Another Fractal Curve
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2019-11-07 18:45
FunctionTransformation
This example demonstrates the generation process of another fractal curve. Fractal geometry is a branch of mathematics that studies complex structures, involving self-similarity, iteration, and infinite detail. Using GeoGebra commands such as Segment, Midpoint, and PolyLine, a fractal curve can be constructed. Suitable for high school students or math enthusiasts to learn the basic concepts of fractal geometry.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(C, D) → h
  4. Segment(D, E) → i
  5. Midpoint(h) → F
  6. Segment(C, F) → j
  7. Segment(F, D) → k
  8. Segment(A, E) → l
  9. PolyLine(A, G, H, I, J, E) → m
  10. Rotate(PolyLine[Dilate[A, f / l, A], Dilate[G, f / l, A], Dilate[H, f / l, A], Dilate[I, f / l, A], Dilate[J, f / l, A], Dilate[E, f / l, A]], Angle[l, f], A) → A1
  11. Rotate(Translate[PolyLine[Dilate[A, g / l, A], Dilate[G, g / l, A], Dilate[H, g / l, A], Dilate[I, g / l, A], Dilate[J, g / l, A], Dilate[E, g / l, A]], B - A], Angle[l, g], B) → B1
  12. Rotate(B1, π, Midpoint[g]) → C1
  13. Mirror(Rotate[Translate[Rotate[PolyLine[Dilate[A, j / l, A], Dilate[G, j / l, A], Dilate[H, j / l, A], Dilate[I, j / l, A], Dilate[J, j / l, A], Dilate[E, j / l, A]], Angle[l, f], A], F - A], Angle[f, j] + π, F], j) → D1
  14. Rotate(D1, π, F) → E1
  15. Mirror(Rotate[PolyLine[Dilate[A, i / l, E], Dilate[G, i / l, E], Dilate[H, i / l, E], Dilate[I, i / l, E], Dilate[J, i / l, E], Dilate[E, i / l, E]], Angle[l, i], E], i) → F1
  16. Rotate(F1, π, Midpoint[i]) → G1
  17. Rotate(Dilate[{A1, B1, C1, D1, E1, F1, G1}, f / l, A], Angle[l, f], A) → A2
  18. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1, G1}, g / l, A], B - A], Angle[l, g], B) → B2
  19. Rotate(B2, π, Midpoint[g]) → C2
  20. Rotate(Translate[Rotate[Dilate[{A1, B1, C1, D1, E1, F1, G1}, j / l, A], Angle[l, f], A], F - A], Angle[f, j] + π, F) → D2
  21. Rotate(D2, π, F) → E2
  22. Mirror(Rotate[Dilate[{A1, B1, C1, D1, E1, F1, G1}, i / l, E], Angle[l, i], E], i) → F2
  23. Rotate(F2, π, Midpoint[i]) → G2
  24. Rotate(Dilate[{A2, B2, C2, D2, E2, F2, G2}, f / l, A], Angle[l, f], A) → A3
  25. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2, G2}, g / l, A], B - A], Angle[l, g], B) → B3
  26. Rotate(B3, π, Midpoint[g]) → C3
  27. Rotate(Translate[Rotate[Dilate[{A2, B2, C2, D2, E2, F2, G2}, j / l, A], Angle[l, f], A], F - A], Angle[f, j] + π, F) → D3
  28. Rotate(D3, π, F) → E3
  29. Mirror(Rotate[Dilate[{A2, B2, C2, D2, E2, F2, G2}, i / l, E], Angle[l, i], E], i) → F3
  30. Rotate(F3, π, Midpoint[i]) → G3
  31. Rotate(Dilate[{A3, B3, C3, D3, E3, F3, G3}, f / l, A], Angle[l, f], A) → A4
  32. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3, G3}, g / l, A], B - A], Angle[l, g], B) → B4
  33. Rotate(B4, π, Midpoint[g]) → C4
  34. Rotate(Translate[Rotate[Dilate[{A3, B3, C3, D3, E3, F3, G3}, j / l, A], Angle[l, f], A], F - A], Angle[f, j] + π, F) → D4
  35. Rotate(D4, π, F) → E4
  36. Mirror(Rotate[Dilate[{A3, B3, C3, D3, E3, F3, G3}, i / l, E], Angle[l, i], E], i) → F4
  37. Rotate(F4, π, Midpoint[i]) → G4
  38. Rotate(Dilate[{A4, B4, C4, D4, E4, F4, G4}, f / l, A], Angle[l, f], A) → A5
  39. Rotate(Translate[Dilate[{A4, B4, C4, D4, E4, F4, G4}, g / l, A], B - A], Angle[l, g], B) → B5
  40. Rotate(B5, π, Midpoint[g]) → C5
  41. Rotate(Translate[Rotate[Dilate[{A4, B4, C4, D4, E4, F4, G4}, j / l, A], Angle[l, f], A], F - A], Angle[f, j] + π, F) → D5
  42. Rotate(D5, π, F) → E5
  43. Mirror(Rotate[Dilate[{A4, B4, C4, D4, E4, F4, G4}, i / l, E], Angle[l, i], E], i) → F5
  44. Rotate(F5, π, Midpoint[i]) → G5