In addition to tilings of the plane by rep-tiles these are several other classes of tilings of the plane by the attractors of IFSs.
1. Trivially, the plane may be tiled by copies of a rep-n rep-tile differing in area by nm. This can be represented as an IFS as follows. Take the IFS for a rep-tile, with transforms Ti. Divide the transforms into disjoint non-empty sets Uj and Vk. A new IFS can be constructed from the transforms Uj and Ti.Vk.
2. Trivially, there are classes of rep-tiles that occupy an integral fraction of another tile, and which can be fitted together to recreate the original rep-tile. Commonly there is more than one such rep-tile for each symmetric tile, and hence tilings of the plane are possible in which not all the copies of the original symmetric rep-tile are dissected in the same way. Both periodic and non-periodic tilings are possible.
3. Trivially, given a IFS for a rep-tile, with transforms Ti, a new IFS with transforms Si=A-1.Ti.A can be created. The attractor of the new IFS tiles the plane, but in the majority of cases is self-affine, rather than self-similar.
4. There are other self-affine IFSs which have attractors which tile the plane. For example the sunburst, sunrise and starburst triangles are not self-similar constructions, but tile the plane.
5. Given a rep-tile, whose IFS has transforms Ti we can create a new IFS by dividing the transforms into disjoint non-empty sets Uj and Vk, and constructing a new IFS from the transforms Uj and Vk.Ti. The resulting attractor tiles the plane, and is self-similar, but has elements of two different sizes. This technique can be applied recursively, and also to non-rep-tile tiles. Note that not all the resulting attractors are simply connected, so it cannot be trivially demonstrated that the number of simply connected tiling figures derivable from a single rep-tile is infinite. On the other hand it cannot be trivially demonstrated that the number is finite, and I consider the former conjecture more likely to be true.
Documenting these figures is of lower priority, and at the moment only the metatrapezia (derived from the triamond) are available on these web pages.
6. Fractals can be constructed as one or many armed spirals
Grouped element derivatives of symmetric members of the class can be created
as can metafigures.
and grouped element derivatives of metafigures
R. William Gosper has published another derivative of the Koch Snowflake
7. There are other attractors which tile the plane, but are not rep-tiles, and which are self-similar with copies of different sizes, which are to the best of my knowledge not reachable by the algorithm described in 5, and which are neither composed of spirally arranged elements, nor derivable from such fractals.
© 2001, 2002, 2005 Stewart R. Hinsley