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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 isotrianguloid to cover the 2 fractals where the orientation is the same, and related fractals.
The 0-isotrianguloid is the w-0-cyclotrimer-0-monomer (to which I give the trivial name of pseudo-Sierpinski triangle). The 60-isotrianguloid is the z-60-cyclotrimer-60-monomer.
Both these figures have
Unlike several of the other cyclomers these appear not to be the first members of series of many fractals. The equivalents to the rep-k2 triangles have non-uniform measures.
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-isotrianguloid as P, and the 60-isotrianguloid as Z, we can recursively combine these to form a uncountably infinite set. (This cardinality can be demonstrated by a diagonal argument analogous to that that proves the uncountability of R.) All these fractals continue to have a similarity dimension of 2, to tile the plane, and to contain voids.
Combining 2 units gives 2 new fractals, corresponding to P.Z and Z.P (P.P and Z.Z are the same as P and Z).
Combining 3 units gives 6 new fractals (P.P.P and Z.Z.Z are the same as P and Z).
The number of fractals increases with increasing number of units, in the same manner as with the cis- and trans-fudgeflakes.
Source: All are independently discovered. The z-60-cyclotrimer-60-monomer is also documented by Roger Bagula.
References:
© 2000 Stewart R. Hinsley