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Snowflake Fractal

Snowflake Fractal
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2019-11-02 20:43
FunctionTrigonometryGeometric Construction
This example demonstrates the iterative construction of a snowflake fractal pattern. Key math concepts include fractal geometry, tangent functions, logarithmic functions, polygon construction, and rotation. The construction uses GeoGebra commands like Segment, PolyLine, and Rotate, along with objects such as points and curves. Suitable for high school and above, it helps learners understand the generation process and mathematical principles of fractals.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(C, D) → h
  4. Segment(D, E) → i
  5. Segment(E, F) → j
  6. Segment(F, G) → k
  7. Segment(A, G) → l
  8. Segment(G, H) → m
  9. Segment(H, A) → n
  10. PolyLine(A, B, C, D, E, F, G) → p
  11. Rotate(PolyLine[Dilate[A, f / l, A], Dilate[B, f / l, A], Dilate[C, f / l, A], Dilate[D, f / l, A], Dilate[E, f / l, A], Dilate[F, f / l, A], Dilate[G, f / l, A]], Angle[l, f], A) → A1
  12. Rotate(Translate[PolyLine[Dilate[A, g / l, A], Dilate[B, g / l, A], Dilate[C, g / l, A], Dilate[D, g / l, A], Dilate[E, g / l, A], Dilate[F, g / l, A], Dilate[G, g / l, A]], B - A], Angle[l, g], B) → B1
  13. Rotate(Translate[PolyLine[Dilate[A, h / l, A], Dilate[B, h / l, A], Dilate[C, h / l, A], Dilate[D, h / l, A], Dilate[E, h / l, A], Dilate[F, h / l, A], Dilate[G, h / l, A]], C - A], Angle[l, h], C) → C1
  14. Rotate(Translate[PolyLine[Dilate[A, i / l, A], Dilate[B, i / l, A], Dilate[C, i / l, A], Dilate[D, i / l, A], Dilate[E, i / l, A], Dilate[F, i / l, A], Dilate[G, i / l, A]], D - A], Angle[l, i], D) → D1
  15. Rotate(Translate[PolyLine[Dilate[A, j / l, A], Dilate[B, j / l, A], Dilate[C, j / l, A], Dilate[D, j / l, A], Dilate[E, j / l, A], Dilate[F, j / l, A], Dilate[G, j / l, A]], E - A], Angle[l, j], E) → E1
  16. Rotate(Translate[PolyLine[Dilate[A, k / l, A], Dilate[B, k / l, A], Dilate[C, k / l, A], Dilate[D, k / l, A], Dilate[E, k / l, A], Dilate[F, k / l, A], Dilate[G, k / l, A]], F - A], Angle[l, k], F) → F1
  17. Rotate(Dilate[{A1, B1, C1, D1, E1, F1}, f / l, A], Angle[l, f], A) → A2
  18. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, g / l, A], B - A], Angle[l, g], B) → B2
  19. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, h / l, A], C - A], Angle[l, h], C) → C2
  20. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, i / l, A], D - A], Angle[l, i], D) → D2
  21. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, j / l, A], E - A], Angle[l, j], E) → E2
  22. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, k / l, A], F - A], Angle[l, k], F) → F2
  23. Rotate(Dilate[{A2, B2, C2, D2, E2, F2}, f / l, A], Angle[l, f], A) → A3
  24. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, g / l, A], B - A], Angle[l, g], B) → B3
  25. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, h / l, A], C - A], Angle[l, h], C) → C3
  26. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, i / l, A], D - A], Angle[l, i], D) → D3
  27. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, j / l, A], E - A], Angle[l, j], E) → E3
  28. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, k / l, A], F - A], Angle[l, k], F) → F3
  29. Rotate(Dilate[{A3, B3, C3, D3, E3, F3}, f / l, A], Angle[l, f], A) → A4
  30. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, g / l, A], B - A], Angle[l, g], B) → B4
  31. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, h / l, A], C - A], Angle[l, h], C) → C4
  32. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, i / l, A], D - A], Angle[l, i], D) → D4
  33. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, j / l, A], E - A], Angle[l, j], E) → E4
  34. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, k / l, A], F - A], Angle[l, k], F) → F4
  35. Rotate(Translate[PolyLine[Dilate[A, m / l], Dilate[B, m / l], Dilate[C, m / l], Dilate[D, m / l], Dilate[E, m / l], Dilate[F, m / l], Dilate[G, m / l]], G - A], Angle[l, m], G) → q
  36. Rotate(Translate[PolyLine[Dilate[A, n / l, G], Dilate[B, n / l, G], Dilate[C, n / l, G], Dilate[D, n / l, G], Dilate[E, n / l, G], Dilate[F, n / l, G], Dilate[G, n / l, G]], A - G], Angle[l, n], A) → r
  37. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, m / l], G - A], Angle[l, m], G) → A8
  38. Rotate(Translate[Dilate[{A1, B1, C1, D1, E1, F1}, n / l, G], A - G], Angle[l, n], A) → B8
  39. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, m / l], G - A], Angle[l, m], G) → A9
  40. Rotate(Translate[Dilate[{A2, B2, C2, D2, E2, F2}, n / l, G], A - G], Angle[l, n], A) → B9
  41. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, m / l], G - A], Angle[l, m], G) → A10
  42. Rotate(Translate[Dilate[{A3, B3, C3, D3, E3, F3}, n / l, G], A - G], Angle[l, n], A) → B10
  43. Rotate(Translate[Dilate[{A4, B4, C4, D4, E4, F4}, m / l], G - A], Angle[l, m], G) → A11
  44. Rotate(Translate[Dilate[{A4, B4, C4, D4, E4, F4}, n / l, G], A - G], Angle[l, n], A) → B11