A s-polymer is a 2 dimensional fractal in which at least one pair of components have a common centre.
The phase space for potential s-trimers is large, and hasn't been fully investigated. However I conjecture that there are no s-trimers. However, at first sight, there are at least 32 g-trimers (D < 2) in which 2 components aligned parallel to the long axis, and one aligned perpendicular to the long axis.
3 s-tetramers have been identified.
The disquare (s-0,90-tetramer) obviously tiles the plane. The cis- and trans-s-0,60-tetramers contain voids, and as far as I can tell do not tile the plane. (The cis- and trans-s-0,120-tetramers are mirror images of the cis- and trans-s-0,120-tetramers, as follows from their c2 symmetry.)
In addition there is a disconnected s-0,180,90,270-tetramer, which also tiles the plane.
Large numbers of s-polymers with larger numbers of elements exist. Most of the ones I have found have pairs of components combined to form squares (as in the disquare above), rectangles, Mandelbrot quintets and flowsnakes. It is not clear whether this is because there is a greater number of these, or because they are easier to find.
However a countably infinite set of other s-polymers with 9, 25, 49, 81, ... elements has been discovered. All have these have a central element, and the remaining elements arranged in pairs at right angles. Although all are generated by the same algorithm, the attractors with 9 elements differ from those with more elements. There are 8 distinct 2 dimensional attractors with 9 elements, and only 4 with 25, 49, 81, ... elements. The attractors with 9 elements contain voids, and the others do not. The attractors with 9 elements do not tile the plane, and the others do.
In the attractors with 9 elements the central element can take 4 orientations, and the other elements 2 orientations.
As stated above there are 4 25 element attractors.
These tile the plane. When a figure is tiled with a copy of itself rotated 90° the resulting figure has the same shape as the cyclotetramermonomer aka cross fractal aka Mandelbrot quintet. The cis- forms map to the trans- form of the cyclotetramermonomer, and the trans- forms to the cis- form of the cyclotetramermonomer.
And there are also 4 49 element attractors.
These tile the plane. I conjecture, but cannot demonstrate, that the figures consisting of two copies of these figures laid at right angles are also the attractors of IFSs.
