allo-Trianguloids

There are 4 fractals with d3-symmetry that consist of 4 elements. These can be divided into 2 pairs, depending on whether the orientation of the central element is the same as, or rotated by 60°/180° with respect to, the orientation of the other 3 elements. I introduce the term allotriangularoid to cover the 2 fractals were the orientation is different, and related fractals.

The 0-allotriangularoid is the 0-cyclotrimer-60-monomer or equilateral triangle. The 60-isotriangularoid is the w-60-cyclotrimer-60-monomer.

Both these figures have a similarity dimension of 2. Both tile the plane, as shown on the 2D cyclomers page.

The first is the initial member of a countably infinite series of IFSs, the rep-k²triangles. These however all have the same attractor - the equilateral triangle.

The second is also the initial member of a countably infinite series of IFSs. Only 2/3rds of the member of the potential members of the series have a uniform measure. These are those for which n mod 3 <> 2.

In some cases additional fractals can be generated by 'replacing' each element of a fractal with a 'copy' of a second fractal. (This can always be done with a 'copy' of the first fractal, but in this case the result is the original fractal, as is to be expected from the self-similarity property.) Some care, or experimentation, is required to get the scale, position and orientation correct. However if the case of the isotrianguloids this technique is unproblematical. The scale factor is found to be 1.

Denoting the 0-allotrianguloidas T, and the 60-allotrianguloid as W, we can recursively combine these to form a uncountably infinite set. (This cardinality can be generated by a diagonal argument analogous to that that proves the uncountability of R.)

Combining 2 units gives 2 new fractals, corresponding to T.W and W.T (T.T and W.W are the same as T and W). Combining 3 units gives 6 new fractals (T.T.T and W.W.W are the same as T and W). All of these new fractals contain voids, and do not tile the plane.

The number of fractals increases with increasing number of units, in the same manner as with the cis- and trans-fudgeflakes.

Allotrianguloids of different orders can be combined in the same fashion. If we denote the nth order triangle as T_n, and the nth order 60-allotrianguloid as W_n then we can have such fractals as W₁.W₃ and W₃.T₂.

Note that W₁.T₁ is the same as W₃.

Source: All are independently discovered.

References:

© 2000 Stewart R. Hinsley