**Some tiles associated with the 6th unit cubic Pisot number**

## Partial
Postcomposition Tiles

## I. Metasymmetric
Tiles

A 3 element tile potentially has 6 partial postautocomposition
derivatives, 3 of order 5 and 3 of order 7. However on the one hand the symmetric tile is 8-fold degenerate - each of its 3
elements can be rotated by 180° - which increase the number of candidates to
48. But on the other hand the same symmetry results in attractors which differ
only in orientation, or only in dissection, or in degenerate attractors which
occur more than once. The result of this is to produce a total of 3 order 5
tiles (1 symmetric tile and 2 asymmetric tiles) and 5 order 7 tiles (1
symmetric tile and 4 asymmetric tiles.)

The grouped element technique can be applied to the symmetric tiles to
produce order 5 and order 7 demimetasymmetric tiles.
The co-cell technique can be applied to the asymmetric tiles also producing new
order 5 and order 7 tiles.
Repeated partial autocomposition leads to order 9, 13, 17, 19, 21, 25, 31 and
so tiles.

#### Order 5 tiles

The symmetric order 5 tile has the dissection polynomial `2c+2c`^{4}+c^{6}. It is a complex
teragon whose boundary has a countable number of crossing points. It is 32-fold
degenerate (8 of these are among the 24 candidate order 5 tiles). It has two
copies in the minimal unit cell, with signature **03**.

The asymmetric order 5 tiles have the dissection polynomial `c+2c`^{2}+c^{3}+c^{4}. Each
appears 8 times among the 24 candidate order 5 tiles, the 8 instances forming 4
pairs related by inversion in the origin, each pair having a different
dissection. The 4 different dissections - each pair of elements forming a
symmetric tile (red/blue, green/cyan) can be rotated by 180° - can be obtained
by starting with 4 of the 8 degenerate IFSs for the symmetric tile, or by
rotating the red/blue elements around the point `-1`, and the green/cyan elements around the point `0`. The second has an additional 4 dissections
obtained by rotating the orange/green/cyan elements around the point `(a-1)/(a`^{2}-1). (That point is a stationary
point (i.e. `T(x) = x`) of the transform
corresponding to the blue element.)

The symmetric order 7 tile has the dissection polynomial `4c`^{2}+c^{3}+2c^{4}. It is
128-fold degenerate (8 of these are among the 24 candidate order 7 tiles). It
has 3 copies in the minimal unit cell, with signature **011**.

The asymmetric order 7 tiles have the dissection polynomial `c+2c`^{2}+3c^{4}+c^{6}. Each
of these has two dissections - the block of 3 elements making up a symmetric
tile can be rotated by 180?. All have them have 3 copies in the minimal unit
cell, with signature **013**.

The unit cells are shown below.

© 2015, 2016, 2017 Stewart R. Hinsley