Partial postcomposition is a technique for manipulating an Iterated Function System such that if the similarity dimension of the attractor of the original IFS is 2 then the similarity dimension of the attractor of the modified IFS is also 2. (I suspect that similarity dimension is conserved by this technique, regardless of its value, and the technique could be profitably applied to figures such as the Sierpinski triangle.) This technique can therefore be used to find additional self-similar tiles, given a self-similar tile. Note that while disconnectedness is conserved by this technique, connectness isn't necessarily conserved; hence it is necessary to visually or otherwise check that the resulting attractor is connected.

I have no reason to suspect that the technique does not also work with self-affine tiles.

When I discovered the technique (back in the last millenium) this was the
only technique I had for mechanically deriving one tile from another, and I
labelled the technique the meta-figure technique, using the sense of
*meta*- as meaning change, as it produces a changed tile. As I have
subsequently discovered other techniques such as the grouped element technique
and the co-cell technique, which also produce changed tiles, I have since
adopted a more tightly descriptive name for the technique, as in the title. I
retain the use of the prefix *meta*- as a less cumbersome way of referring
to the figures produced by the technique, both in the general (e.g. metafigure,
metatile, metaatractor, metapolygon) and specific (e.g. metaammonite, for the
derivatives of the ammonite tile shown below) sense. As there is usually more
than one (but, since connectedness is not always conserved, sometimes 0 or 1)
derived tile the latter sense usually denotes a set of tiles rather than an
individual tile. In some cases, particular the smaller sets, the members of the
set can be distinguished by short descriptions. For example the 3 order 5
metapseudoterdragons shown below can be distinguished as the simple, complex
and disconnected metapseudoterdragons (one is a simple teragon, one is a
complex teragon, and one is a disconnected figure).

Let `{ T _{i} }` be an IFS. This can be
divided into two disjoint non-empty sets

Clearly the order of the tile is increased by the process. The order of the
derived tile is `card { T _{i} } + (card {
T_{i} } - 1) * card { V_{k} }`, i.e. 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.

The number of derived tiles also increases with the increasing order of the base tile. For asymmetric tiles, an 2-element tile has two order 3 derivatives, a 3-element tile three order 5 and three order 7 derivatives, a 4-element tile four order 7, 6 order 10 and 4 order 13 derivatives, a 5-element tile 5 order 9, 10 order 13, 10 order 17 and 5 order 21 tiles, and so on. For symmetric tiles, the number of derivatives is on the one hand decreased by symmetry, and on the other hand increased by the degeneracy of the attractor, and is considered in more, but not exhaustive, detail below.

The ammonite tile is used as an example of the use of the technique on a
asymmetric tile, using the `x ^{2} +
x^{3} = 1` dissection.

This has two order 3 derivatives. Note that while the ammonite is a simple
teragon, only one of the derivatives is a simple teragon, the other being a
complex teragon, though in this case as simple as a complex teragon can be.
This behaves like connectedness and disconnectedness - a simple teragon can
give rise to a complex teragon, but not vice versa. Note also that the tiles
have different dissection equations (as is obvious from examination of the
construction); the first has the dissection equation `x ^{3} + x^{4} + x^{5} = 1`,
and the second

A three element dissection of the ammonite tile, such as the `x ^{3} + x^{4} + x^{5} = 1`
dissection, can also be used as a starting point.

This has three order 5 derivatives. Note that in this case one of resulting
attractors (the third) is one of the attractors obtained as an order 3 tile.
The general rule is that, if we denote a derived tile as `{ T* _{i} }` and its two subsets as

It also has three order 7 derivatives, including a pair of complex teragons with infinite genus.

One can perform the technique a second time on the order 3 tiles derived
above. As there are two of them, this gives 6 order 5 attractors. Among these
we find a couple of disconnected attractors (here homologous to an infinite
number of separated discs), so the number of tiles is smaller. My working
hypothesis is that all elements adjacent to a particular element `n` in the original corresponding to elements of `{ V _{k} }`, is a sufficient but not
necessary, condition for the element corresponding to

There are also 6 order 7 attractors, which I don't show.

To show the technique applied to an order 4 tile I use the `x ^{4} + 2x^{5} + x^{6} = 1`
dissection of the ammonite tile, which provides an example of two derivatives
(here the 2

This generates 4 order 7 tiles. The first two of these have already been encountered as order 5 tiles.

In this instance, as the ammonite tile has two order 2 dissections the other
two turn up as order 3 (from `x + x ^{5} =
1` dissection) and order 5 (from

The original tile is the union of several copies of each derived tile,
specifically `card { V _{k} } + 1` copies.
If

Thus the ammonite tile is made up of two copies of the order 3 derivatives,
and as the ammonite tile tiles the plane with signature **0**, the order 3
derivatives tile the plane with signatures **02** and **03**
respectively.

The order 5 derivatives of the order 3 dissection tile the plane with 2
copies in the unit cell, and signatures **04**, **05** and **03**
respectively.

And its order 7 derivatives tile the plane with 3 copies in the unit cell,
and signatures **045**, **034** and **035** respectively.

The four tiles among the order 5 derivatives of the order 3 derivatives tile the plane with 4 copies in the unit cell.

To illustrate derivatives of c2-symmetric tiles the tame twindragon (even orders) and pseudoterdragon (odd orders) are presented below.

The tame twindragon (order 2) is 4-fold degenerate. Thus there are 8 derived IFSs. But the result of symmetry is that the derived attractors are 2-fold degenerate, and the attractors come paired with attractors rotated by 180°, which reduces the number of distinct attractors to 2.

The pseudo terdragon is 8 fold-degenerate, giving 32 derived order 5 IFSs. The attractors are at least 4-fold degenerate, and they again come in rotated pairs, reducing the number to 4, which is further reduced to 3 by the additional degeneracy of the symmetric derivative. (This is 32-fold degenerate, but only 8 of the IFSs are produced by the post-composition technique.) In this case one of the attractors is disconnected.

There are also 32 derived order 7 IFS, which are at least 2-fold degenerate, and again come in rotated pairs, which reduces the number to 8, which is further reduced to 5 by the additional degeneracy of the symmetric (and disconnected) derivative.

The order 4 tame twindragon symmetric dissection is 16-fold degenerate, giving 64 derived order 7 IFSs. The attractors are 8 fold degenerate. The number of distinct attractors is therefore reduced to 4 (2 of which are disconnected).

Generalising, degeneracy is reduced by a factor of `2 ^{card { Vk }}`. If n is the
number of distinct derived attractors for an asymmetric tile, for even order
tiles the number of distinct attractors is

d1-symmetric tiles behave like c2-symmetric tiles, except that the attractors come in mirror image pairs, rather than pairs rotated by 180° with respect to each other. However no order-3 d1-symmetric dissections of a d1-symmetric tile are known offhand.

The right isoceles triangle is used as an example of a d1-symmetric tile. As this is a polygon, the derived tiles are countablegons rather than teragons. The Levy curve would be an alternative example, but as it is a rather cryptic complex teragon the derived tiles would be particularly complex and crypytic - not ideal for illustration.

The order 2 right isoceles triangle gives rise to 2 distinct order 3 attractors, one of which is a a simple countablegon (right isoceles triangular spiral) and one of which is a complex countablegon (single-ended right isoceles triangular rectilinear catenocountablegon).

The symmetric dissection of the order 4 right isoceles triangle gives rise to 4 distinct order 7 attractors, one of which is a simple countablegon, two of which are complex countablegons, and one of which is disconnected.

The fudgeflake is used as an example c3-symmetric tile. As the fudgeflake is 27-fold degenerate there are 81 derived order 5 IFSs. Each attractor is 9-fold degenerate, and they come in rotationally equivalent triplets, so there are only 3 distinct attractors, one of which is disconnected.

There are also 81 derived order 7 IFS. Each attractor is 3-fold degenerate, and comes in rotationally equivalent triplets, so there are 9 distinct attractors, composed of 8 complex teragons and a disconnected attractor.

There is no obvious order 6 c3-symmetric dissection of a tile. There is an
order 9 symmetric dissection of the fudgeflake. This has 3^{11}
distinct derived order 17 IFSs. But the attractors are 3^{8}-fold
degenerate, and come in rotationally equivalent triplets, so the number of
distinct attractors in 9.

Similar analyses can be done for d3-symmetric tiles (e.g. the order 4 equilateral triangle), c4-symmetric tiles (e.g. the order 5 Mandelbrot quintet), d4-symmetric tiles (e.g the order 4 square), c6-symmetric tiles (e.g. the order 7 flowsnake) and d6-symmetric tile (e.g. the order 7 Koch snowflake).

The area of the derived tile compared to the original can be calculated from the rule used to ascertain the tiling of the original by the derived tile. For tiles based on quadratic Perron numbers the area is a rational fraction of that of the original. For example the order 3 meta tame twindragons are ²⁄₃ of a tame twindragon.

For tiles based on non-quadratic Perron numbers the area is an irrational fraction of the original. For non-Perron tiles the area is usually an irrational fraction of that of the original, but as of the time of writing I can't exclude the possibility that there are exceptions. (For example when the technique is applied the order 2 right-angled triangle - in which the ratio of the area of the two parts can take any value between 0 and 1 exclusive - the ratio of areas of derived and original tiles is rational if the ratio of the areas of the two parts of the tile is rational. For most values of the ratio the order 2 right-angled triangle is a non-Perron tile, but it may be the case that the instances where the ratio is rational are Perron tiles, so I can't be sure that this is not an example of an exception.)

A number of transition rules, showing the tendency of the process to produce more complex attractors, can be stated, including

- polygon → polygon | countablegon
- countablegon → countablegon

- simple attractor → simple attractor | complex attractor
- complex attractor → complex attractor

- connected attractor → connected attractor | disconnected attractor
- disconnected attractor → disconnected attractor

It is thought that there are other properties (e.g. topological genus) that display similar rules.

Above partial post(auto)composition derivatives of an IFS `{ T _{i} }` divided into two disjoint
non-empty sets

There is one case in which we can, but which does not give us anything new.
Consider a derived IFS `{ T' _{i} } ≡ {
U_{j} } ∪ { V_{k}.T_{i} }`. Rewrite this as

The next simplest case is when `{ S _{i}
}` and

While investigating the preceding experimentally, due to an editing error, it was discovered that using appropriately scaled, positioned and oriented IFSs for the tame twindragon and pseudoterdragon, partial postallocomposition generation some tiles among its outputs, with a higher hit rate than is obtained from a polynomial based vector search for tiles of the same order. Like with composition of IFSs identifying the appropriately scaled, posititioned and oriented IFSs is a non-trivial task.

Intermediate between the last two cases above is the situation where the two IFSs have the same attractor, but correspond to different dissections. An experiment with the two dissections order two dissections of the ammonite tile failed to generate tiles, but in the light of the preceding it seems likely that there are cases where it does.

Thus partial post(allo)composition is a subject requiring further investigation.

© 2017 Stewart R. Hinsley