Integer IFS

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Let [math]A[/math] - invertible integer matrix [math]n\times n[/math], [math]S=\{S_1,S_2,..S_k\}[/math] - finite group of integer matrices [math]n\times n[/math], such that [math]AS=SA[/math] (symmetry group of [math]A[/math]).

Let [math]P[/math] - matrix [math]m\times n[/math] (projector from [math]R^n[/math] to [math]R^m[/math]), [math]B[/math] - matrix [math]m\times m[/math], [math]G=\{G_1,G_2,..G_k\} [/math] - matrices [math]m\times m[/math] such that:

  1. [math]\rho(B^{-1})\lt1[/math] (see Spectral Radius)
  2. [math]PA=BP[/math]
  3. [math]PS_i=G_iP[/math]

Let [math]f_i(x)=G_{k_i}B^{-d_i}(x+P t_i)[/math], where [math]d_i\gt0[/math] and [math]t_i[/math] is an integer vector from [math]R^n[/math].

We call any such IFS [math]\{f_1,f_2,.. f_n\}[/math] an Integer IFS.


It easily to extend this definition to directed graph IFS and rational numbers.


Many famous IFS can be represented as Integer:

Sierpinski triangle, Rauzy fractal, Dragon curve, Golden Bee, Pentadentrite, McWorter's Pentigree

References

Self-Similar Sets 5. Integer Matrices and Fractal Tilings of Rn

Iterated Function System

Aperiodic tiling: Cut-and-project method