Iterated Function System with Affine Maps by José Manuel Dos Santos Dos Santos
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Fractals
This example demonstrates the generation of fractal patterns like triangles and parallelograms through iterated function systems and affine mappings. Key math concepts include affine transformations, iterative function systems, matrix transformations, and coordinate geometry. The construction leverages GeoGebra commands like Sequence for iterative processes, matrix objects (a_{ij}, b_{ij}, c_{ij}) for affine transformations, and elements such as Locus and Slider for dynamic visualization. Suitable for high school and above, exploring fractal geometry, transformation geometry, and matrix applications in mathematics.
GeoGebra Commands & Parameters
Sequence((Element[L_M, i] * (x(A), y(A))) + Element[L_P, i], i, 1, 3, 1) → G_1Sequence(Sequence[(Element[L_M, i] * (x(Element[G_1, j]), y(Element[G_1, j]))) + Element[L_P, i], i, 1, 3, 1], j, 1, 3, 1) → G_2Sequence(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, 3, 1], j, 1, 3, 1], k, 1, 3, 1) → G_3Sequence(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, 3, 1], j, 1, 4, 1], k, 1, 3, 1], l, 1, 3, 1) → G_4Sequence(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, 3, 1], j, 1, 3, 1], k, 1, 3, 1], l, 1, 3, 1], m, 1, 3, 1) → G_5
AI Drawing Prompt Reference
Draw fractal patterns like triangles and parallelograms using affine transformations and iterative function systems, controlled by matrix objects, generating multi-level patterns with sequence commands, and enabling dynamic visualization with sliders to demonstrate principles of coordinate geometry and transformation geometry.
The prompt above was compiled from the original drawing commands and is for reference only.