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Area of Sierpinski Triangle Fractal

Area of Sierpinski Triangle Fractal
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2020-05-23 19:57
TrianglePolygonGeometric Construction
This example demonstrates the construction and area calculation of the Sierpinski triangle fractal through iterative processes. Key math concepts include fractal geometry, area computation, and recursive construction. GeoGebra commands like Point, Rotate, Sequence, and Polygon are used to create points, rotate shapes, and generate polygon sequences. Suitable for high school mathematics education to understand fractals and area reduction patterns.

GeoGebra Commands & Parameters

  1. Point(Ray[A, A + (1, 0)]) → B
  2. Rotate(B, 1.0471975511965976, A) → C
  3. Sequence(Translate[Dilate[A1, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → A2
  4. Sequence(Translate[Dilate[A2, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → A3
  5. Sequence(Translate[Dilate[A3, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → A4
  6. Sequence(Translate[Dilate[A4, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → A5
  7. Sequence(Translate[Dilate[A5, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → A6
  8. Sequence(Translate[Dilate[A6, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → A7
  9. Sequence(Translate[Dilate[A7, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → A8
  10. Sequence(Translate[Dilate[A1, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → S2
  11. Sequence(Translate[Dilate[A2, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → S3
  12. Sequence(Translate[Dilate[A3, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → S4
  13. Sequence(Translate[Dilate[A4, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → S5
  14. Sequence(Translate[Dilate[A5, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → S6
  15. Sequence(Translate[Dilate[A6, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → S7
  16. Sequence(Translate[Dilate[A7, 0.5, O], Vector[(1 / 2 * Element[Vec, i])]], i, 1, 3) → S8
  17. Polygon(A, B, C) → poly1, c, a, b

AI Drawing Prompt Reference

Draw a Sierpinski triangle fractal. Start by creating point A and a ray from A. Create point B on the ray and rotate it around A to get point C. Then, with A as the center, dilate and translate triangle ABC by half its size three times to form a new sequence of triangles. Repeat this process multiple times, each time dilating and translating each triangle in the previous sequence by half its size three times to generate multiple levels of triangle sequences.
The prompt above was compiled from the original drawing commands and is for reference only.