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Fractal Curve 4

Fractal Curve 4
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2019-10-29 07:25
FunctionTrigonometryTransformationGeometric Construction
This example demonstrates a complex fractal curve, generated through iterative transformations and recursive constructions. Core mathematical concepts include fractal geometry, geometric transformations, symmetry, functions, and standard deviation. Key construction steps involve using commands like PolyLine, Rotate, and Translate in GeoGebra, along with objects A, B, C, etc. Suitable for high school and above students to understand the principles of fractal pattern generation and applications of geometric transformations.

GeoGebra Commands & Parameters

  1. PolyLine(A, G, H, F) → f
  2. Rotate(PolyLine[Dilate[A, 1 / sqrt(2), A], Dilate[G, 1 / sqrt(2), A], Dilate[H, 1 / sqrt(2), A], Dilate[F, 1 / sqrt(2), A]], π / 4, A) → A1
  3. Translate(Rotate[PolyLine[Dilate[A, 0.5 / sqrt(2), A], Dilate[G, 0.5 / sqrt(2), A], Dilate[H, 0.5 / sqrt(2), A], Dilate[F, 0.5 / sqrt(2), A]], ((-π)) / 4, A], B - A) → B1
  4. Rotate(B1, π / 2, C) → C1
  5. Rotate(C1, ((-π)) / 2, D) → D1
  6. Rotate(Translate[B1, F - B], ((-π)) / 2, F) → E1
  7. Rotate(Dilate[{A1, B1, C1, D1, E1}, 1 / sqrt(2), A], π / 4, A) → A2
  8. Translate(Rotate[Dilate[{A1, B1, C1, D1, E1}, 0.5 / sqrt(2), A], ((-π)) / 4, A], B - A) → B2
  9. Rotate(B2, π / 2, C) → C2
  10. Rotate(C2, ((-π)) / 2, D) → D2
  11. Rotate(Translate[B2, F - B], ((-π)) / 2, F) → E2
  12. Rotate(Dilate[{A2, B2, C2, D2, E2}, 1 / sqrt(2), A], π / 4, A) → A3
  13. Translate(Rotate[Dilate[{A2, B2, C2, D2, E2}, 0.500000000000002 / sqrt(2), A], ((-π)) / 4, A], B - A) → B3
  14. Rotate(B3, π / 2, C) → C3
  15. Rotate(C3, ((-π)) / 2, D) → D3
  16. Rotate(Translate[B3, F - B], ((-π)) / 2, F) → E3
  17. Rotate(Dilate[{A3, B3, C3, D3, E3}, 1 / sqrt(2), A], π / 4, A) → A4
  18. Translate(Rotate[Dilate[{A3, B3, C3, D3, E3}, 0.500000000000004 / sqrt(2), A], ((-π)) / 4, A], B - A) → B4
  19. Rotate(B4, π / 2, C) → C4
  20. Rotate(C4, ((-π)) / 2, D) → D4
  21. Rotate(Translate[B4, F - B], ((-π)) / 2, F) → E4
  22. Rotate(Dilate[{A4, B4, C4, D4, E4}, 1 / sqrt(2), A], π / 4, A) → A5
  23. Translate(Rotate[Dilate[{A4, B4, C4, D4, E4}, 0.500000000000006 / sqrt(2), A], ((-π)) / 4, A], B - A) → B5
  24. Rotate(B5, π / 2, C) → C5
  25. Rotate(C5, ((-π)) / 2, D) → D5
  26. Rotate(Translate[B5, F - B], ((-π)) / 2, F) → E5