Dissection is a technique for manipulating an Iteration Function System to produce an IFS of higher order with the same attractor. When applied to an IFS whose attractor is a tile, the attractor of the resulting IFS is a version of the tile in which one or more elements is dissected into smaller copies.
Dissection in itself doesn't produce new tiles, but it does produce new starting points for the production of new tiles.
Elsewhere (postcomposition) it is found useful to distinguish between autocomposition (using the same IFS) and allocomposition (using a different IFS). However with precomposition the significant difference is between using an IFS with the same attractor versus one with a different attractor, so the term syncomposition is introduced to include composition with an IFS with the same attractor.
Let { Si } and { Ti } be IFSs, with the same attractor (including position, rotation, size, but not necessarily dissection or in the case of suitably symmetric attractors orientation or mirror parity). The latter IFS can be divided into two subsets { Uj } and { Vk }, subject to the constraints { Vk } ≠ { }, { Uj } ∪ { Vk } ≡ { Ti } and { Uj } ∩ { Vk } ≡ { }. { Uj } = { } corresponds to full precomposition, and { Uj } ≠ { } to partial precomposition. The precomposed IFS is then { Uj } ∪ { Vk.Si }, i.e. each transform in a subset of the IFS is replaced by the product of itself with each of the transforms of an IFS with the same attractor. { Si } = { Ti } is both allowed, and commonly found to be useful.
Clearly the order of the tile is increased by the process. The order of the derived tile is card { Ti } + (card { Si } - 1) * card { Vk }, i.e. with autocomposition, and with alternative dissections of the same order) a 2-element tile gives order 3 tiles, a 3-element tile order 5 and 7 tiles, a 4-element tile order 7, 10 and 13 tiles, a 5-element tile order 9, 13, 17 and 21 tiles, and so on. By repeating the process we can get any order > 2 from a 2-element tile, any odd order > 2 from a 3-element tile, any order equal to 1 modulo 3 from a 4-element tile, and generally any order equal to 1 modulo n - 1 from an n-element tile.
As the application and results of the process is easily understand examples are not (as yet) provided.
Above partial pre(syn)composition of an IFS was defined. This leads to the question as to whether there are any circumstances in which substituting an IFS with a different attractor for { Si } generates a new tile. As there as instance in which partial post(allo)composition does this, it seems possible that there are, and therefore partial pre(allo)composition is a subject worth further investigation.
© 2017 Stewart R. Hinsley