Simple Fractal Tilings

Method 3

A theorem of Bandt gives another way to construct fractal tiles using matrices.

Suppose M is a 2 x 2 matrix with integer entries and det(M) = m > 1.

Denote by L the set of integer points in the plane. Note L is a group under addition (x1,y1) + (x2,y2) = (x1 + x2,y1 + y2). Then M(L) = {Mv: v belongs to L} is a subgroup of L.

A set of integer points V = {v1, ..., vm} is a residue system for M if v1 = 0 and V contains exactly one point from each coset of M(L).

Then the IFS with r, s, theta, and phi determined by M-1, and e and f the coordinates of the points of V, generates a tile with fractal perimeter.

One way to find a residue system uses the integer points inside the parallelogram spanned by the columns of M, and on the parallelogram edges containing the origin.

Here is one example.

Here is another example.

More symmetric choices of residue systems are constructed by Darst, Palagallo, and Price.

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