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Iterated Function System with Affine Maps

Iterated Function System with Affine Maps
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2018-07-19 08:05
FunctionTransformationGeometric Construction
In collections Fractals
This example demonstrates the generation of fractal patterns through an Iterated Function System (IFS) composed of four affine maps. Key mathematical concepts include affine transformations, IFS, self-similarity, and fractal geometry. The construction leverages GeoGebra commands like Sequence and objects such as d_{23}, d_{13}, c_{23}, c_{13}, b_{13}, b_{23}, a_{23}, a_{13}, and others listed in the summary. It is suitable for high school and advanced learners exploring fractal geometry and IFS applications.

GeoGebra Commands & Parameters

  1. Sequence((Element[L_M, i] * (x(A), y(A))) + Element[L_P, i], i, 1, 4, 1) → G_1
  2. Sequence(Sequence[(Element[L_M, i] * (x(Element[G_1, j]), y(Element[G_1, j]))) + Element[L_P, i], i, 1, 4, 1], j, 1, 4, 1) → G_2
  3. Sequence(Sequence[Sequence[(Element[L_M, i] * (x(Element[Element[G_2, j], k]), y(Element[Element[G_2, j], k]))) + Element[L_P, i], i, 1, 4, 1], j, 1, 4, 1], k, 1, 4, 1) → G_3
  4. Sequence(Sequence[Sequence[Sequence[(Element[L_M, i] * (x(Element[Element[Element[G_3, j], k], l]), y(Element[Element[Element[G_3, j], k], l]))) + Element[L_P, i], i, 1, 4, 1], j, 1, 4, 1], k, 1, 4, 1], l, 1, 4, 1) → G_4
  5. Sequence(Sequence[Sequence[Sequence[Sequence[(Element[L_M, i] * (x(Element[Element[Element[Element[G_4, j], k], l], m]), y(Element[Element[Element[Element[G_4, j], k], l], m]))) + Element[L_P, i], i, 1, 4, 1], j, 1, 4, 1], k, 1, 4, 1], l, 1, 4, 1], m, 1, 4, 1) → G_5