# Dissection (or Full and Partial Pre(Syn)Composition)

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.

#### Nomenclature

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.

#### General Technique

Let `{ S`_{i} } and `{ T`_{i} } 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 `{ U`_{j}
} and `{ V`_{k} }, subject to the
constraints `{ V`_{k} } ≠ { }, `{ U`_{j} } ∪ `{
V`_{k} } ≡ { T_{i} } and `{
U`_{j} } ∩ `{ V`_{k} } ≡ {
}. `{ U`_{j} } = { } corresponds to
full precomposition, and `{ U`_{j} } ≠ {
} to partial precomposition. The precomposed IFS is then `{ U`_{j} } ∪ { V_{k}.S_{i}
}, 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. `{ S`_{i} } = { T_{i} } 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 { T`_{i} } + (card {
S_{i} } - 1) * card { V_{k} }, 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.

#### Partial Pre(allo)composition

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 `{ S`_{i} }
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