The lower order dissection polynomials for this number (polynomials in bold include c2-symmetric tiles) are

- order 3
**2c+c**^{3}

- order 5
- c+2c
^{2}+c^{3}+c^{4}= c+c(2c+c^{3})+c^{3} **2c+2c**= 2c+c^{4}+c^{6}^{3}(2c+c^{3})

- c+2c
- order 7
- c+c
^{2}+3c^{3}+c^{4}+c^{5}= c+c^{2}+c^{2}(2c+c^{3})+c^{3}+c^{4} - c+2c
^{2}+c^{3}+2c^{5}+c^{7}= c+2c^{2}+c^{3}+c^{4}(2c+c^{3}) - c+2c
^{2}+3c^{4}+c^{6}= c+c(2c+c^{3})+c^{3}(2c+c^{3}) - 2c+c
^{4}+2c^{5}+c^{6}+c^{7}= 2c+c^{4}+c^{4}(2c+c^{3})+c^{6} **2c+2c**= 2c+2c^{4}+2c^{7}+c^{9}^{4}+c^{6}(2c+c^{3})**4c**= 2c(2c+c^{2}+c^{3}+2c^{4}^{3})+c^{3}

- c+c
- order 9
- c+c
^{2}+2c^{3}+3c^{4}+c^{5}+c^{6}= c+c^{2}(2c+c^{3})+c^{2}+3c^{4}+c^{6} - c+c
^{2}+3c^{3}+c^{4}+2c^{6}+c^{8}= c+c^{2}+3c^{3}+c^{4}+c^{5}(2c+c^{3}) - c+c
^{2}+3c^{3}+3c^{5}+c^{7}= c+c^{2}+3c^{3}+c^{4}(2c + c^{3}) +c^{5} - c+2c
^{2}+2c^{4}+2c^{5}+c^{6}+c^{7}= c+2c^{2}+c^{4}(2c+c^{3})+2c^{4}+c^{6} - c+2c
^{2}+3c^{4}+2c^{7}+c^{9}= c+2c^{2}+3c^{4}+c^{6}(2c+c^{3}) = c+c(2c+c^{3})+2c^{4}+2c^{7}+c^{9} - c+2c
^{2}+c^{3}+c^{5}+2c^{6}+c^{7}+c^{8}= c+2c^{2}+c^{3}+c^{5}+c^{5}(2c+c^{3})+c^{7} - c+2c
^{2}+c^{3}+2c^{5}+2c^{8}+c^{10}= c+2c^{2}+c^{3}+2c^{5}+c^{7}(2c+c^{3}) - c+5c
^{3}+c^{4}+2c^{5}= c+c^{2}(2c+c^{3})+3c^{3}+c^{4}+c^{5} - 2c+c
^{4}+c^{5}+3c^{6}+c^{7}+c^{8}=2c+c^{4}+c^{5}+c^{5}(2c+c^{3}) + 2c^{6}+c^{7} - 2c+c
^{4}+2c^{5}+3c^{7}+c^{9}= 2c+c^{4}+c^{4}(2c+c^{3})+2c^{7}+c^{9}= 2c+c^{4}+2c^{5}+ c^{6}(2c+c^{3})+c^{7} - 2c+c
^{4}+2c^{5}+c^{6}+2c^{8}+c^{10}= 2c+c^{4}+2c^{5}+c^{6}+c^{7}(2c+c^{3}) - 2c+2c
^{4}+c^{7}+2c^{8}+c^{9}+c^{10}= 2c+2c^{4}+c^{7}+c^{7}(2c+c^{3})+c^{9} **2c+2c**= 2c+2c^{4}+2c^{7}+2c^{10}+c^{12}^{}^{4}+2c^{7}+c^{9}(2c+c^{3})**2c+4c**= 2c+2c^{5}+c^{6}+2c^{7}^{4}(2c+c^{3})+c^{6}- 3c
^{2}+3c^{3}+2c^{4}+c^{5}= c^{2}(2c+c^{3})+3c^{2}+c^{3}+2c^{4} - 3c
^{2}+4c^{3}+c^{6}+c^{8} - 4c
^{2}+c^{3}+c^{4}+2c^{5}+c^{7}= 4c^{2}+c^{3}+c^{4}+c^{4}(2c+c^{3}) **4c**= 2c(2c+c^{2}+4c^{4}+c^{6}^{3})+2c^{4}+c^{6}= 4c^{2}+c^{3}(2c+c^{3})+2c^{4}= c(2c+c^{3})+2c^{2}+3c^{4}+c^{6}

- c+c

The simplest dissection polynomial for the 6th unit cubic Pisot number corresponds to a 3 element tile, so we can deduce that there are no 2 element self similar tiles associated with this number. A search for dissection equations where the positive real root is the inverse of the Pisot number finds no dissection equations with an even number of elements less than 10 and a smaller power also less that 10. It may be that there are no even element tiles at all for this Pisot number.

The dissection equation for all order 3 tiles is `2c+c ^{3}` and the minimal signature is
variously

Restricting consideration to tiles with all elements directly similar to the
tile, we can arbitarily restrict the IFSs for 3 element tiles to { ` p → ±ap; p → ±ap + 1;
p → ±a^{3}p + v_{2}` } (or {

The problem is to select a practical but effective range of polynomials. What values of the exponent should be used? How many non-zero coefficients should be allowed? What values should be allowed for the coefficients?

Investigation of other Perron number tilings led to the hypothesis is that
the maximum exponent is not large compared to the maximum exponent of the
contraction elements of the transforms. Values beween -2 and 6 were surveyed.
An initial survey using `±a ^{n}` and

The resulting final survey produced 24 IFSs (with values of

) which produce tiles, composed of 8 versions
of the symmetric tile, and 8 pairs of rotationally
equivalent tiles, composed of two different dissections of the windowed (or spiral) tile, two demi-symmetric tiles, two external tiles and two complex
teragons, giving 24 IFSs producing 8 distinct tiles. This is pleasingly
symmetric (even if the symmetry is broken in the case of the redundancy of the
windowed tile) and gives some hope that the search found all tiles. Overlooked
cryptic tiles cannot however be excluded.**v _{2}**

The number of values of ` v_{2}`
is lower. With the 1

A considerable number of order 5 tiles can be mechanically derived from the order 3 tiles.

Order 3 tiles are also tiles for any higher odd order, so the 8 order 3 tiles are also order 5 tiles. The symmetric tile has 2 order 5 dissections (because of its symmetry), the windowed tile has 6 order 5 dissections (because it has two order 3 dissections), and the remaining order 3 tiles have 3 order 5 dissections each. As the attractor is the same as for the order 3 tiles I don't count these as new tiles. However the new dissections are intermediate steps in the production of new tiles.

There are two possible dissection equations - `2c+2c ^{4}+c^{6}` and

The known order 5 tiles distinct from the order 3 tiles are

- 1 (symmetric) partial postcomposition derivative of the symmetric tile
(“metasymmetric” tile) [unit cell signature
**03**] - 2 partial postcomposition derivatives of the windowed tiles (“metawindowed” tiles) [unit cell signature
**0033**,**0134**or**0235**] - 1 partial postcomposition derivative of the external tiles (“metaexternal” tile) [unit cell signature
**03**] - 1 partial postcomposition derivative of a complex teragons (“metacomplex” tile) [unit cell signature
**0033**] - 10 grouped element derivatives of the symmetric order 5 dissection of the
symmetric tile (“demisymmetric” tiles) [unit
cell signature
**01**] - 5 grouped element derivatives of the symmetric order 5 metasymmetric tile
(“demimetasymmetric” tiles) [unit cell
signature
**0033**]

giving a total of 20 tiles with dissection equation `2c+2c ^{4}+c^{6}`.

- 2 partial postcomposition derivatives of the symmetric tile (“metasymmetric” tiles) [unit cell signature
**01**] - 4 partial postcomposition derivatives of the order 3 demisymmetric tiles
(“metademisymmetric” tiles) [unit cell
signature
**0011**] - 1 partial postcomposition derivatives of the windowed tiles (“metawindowed” tiles) [unit cell signature
**0011**,**0112**or**0123**] - 4 partial postcomposition derivatives of the external tiles (“metaexternal” tiles) [unit cell signature
**01**] - 4 partial postcomposition derivatives of the complex teragons (“metacomplex” tiles) [unit cell signature
**0011**] - 23 co-cell derivatives of the asymmetric order 5 metasymmetric tiles
(“parametasymmetric” tiles) [unit cell
signature
**0011**] - 1 additional order 5 tile (not a derivative of
an order 3 tile) [unit cell signature
**0123**]

giving a total of 36 tiles with dissection equation of `c+2c ^{2}+c^{3}+c^{4}` and a
combined total of 59 known order 5 tiles.

Order 7 tiles can be mechanically derived, both directly and indirectly,
from the order 3 tiles. There are six dissection equations, all of which can be
reached by partial presyncomposition, giving multiple order 7 dissections of
the order 3 tiles. The dissection equation `c+2c ^{2}+3c^{4}+c^{6}`
corresponds to some double partial postcomposition derivatives of order 3 tiles
- “(di)metatiles)” and to the co-cell derivatives of the metasymmetric
tiles (“para(di)metasymmetric” tiles). The
dissection equation

The known order 7 tiles distinct from the order 3 and order 5 tiles are

- 4 partial postcomposition derivatives of the symmetric tile (“metasymmetric” tiles) [ unit cell signature
**013**] - 1 partial postcomposition derivative of the order 3 demisymmetric tiles
(“metademisymmetric” tile) [ unit cell
signature
**001133**] - 1 partial postcomposition derivative of the windowed tiles (“metawindowed” tile) [ unit cell signature
**012335**] - 1 partial postcomposition derivative of the external tiles (“metaexternal” tiles) [ unit cell signature
**013**] - 2 partial postcomposition derivatives of the complex teragons (“metacomplex“ tiles) [unit cell signature
**001133**] - 82 co-cell derivatives of the asymmetric order 7 metasymmetric tiles
(“parametasymmetric” tiles) [unit cell
signature
**001133**]

giving a total of 91 tiles with dissection equation `c+2c ^{2}+3c^{4}+c^{6}`.

- 1 (symmetric) partial postcomposition derivatives of the symmetric tile
(“metasymmetric” tiles) [ unit cell signature
**011**] - 2 partial postcomposition derivatives of the order 3 demisymmetric tiles
(“metademisymmetric” tiles) [unit cell
signature
**001111**] - 2 partial postcomposition derivatives of the windowed tiles (“metawindowed” tiles) [ unit cell signature
**011223**] - 2 partial postcomposition derivatives of the external tiles (“metaexternal” tiles) [unit cell signature
**011**] - 2 partial postcomposition derivatives of the complex teragons (“metacomplex” tiles)
- 48 grouped element derivatives of the symmetric tile with dissection
equation 4c
^{2}+c^{3}+2c^{4}(“demisymmetric” tiles). - 56 grouped element derivatives of the symmetric order 7 metasymmetric tile (“demimetasymmetric” tiles)

giving a total of 113 tiles with dissection equation `4c ^{2}+c^{3}+2c^{4}`.

- 52 grouped element derivatives of the symmetric tile with dissection
equation 2c+2c
^{4}+2c^{7}+c^{9}(“demisymmetric” tiles) - 1 partial postallocomposition derivative of the 2nd external tile (“alloexternal” tile) [unit cell signature
**036**]

giving a total of 53 tiles with dissection equation `2c+2c ^{4}+2c^{7}+c^{9}`.

- 5 partial postallocomposition derivatives of the symmetric tile (“allosymmetric” tiles) [unit cell signature
**012**] - 7 partial postallocomposition derivatives of the external tiles (“alloexternal” tiles) [unit cell signature
**012**] - 7 partial postallocomposition derivatives of the order 3demisymmetric
tiles (“allodemisymmetric” tiles) [unit cell
signature
**001122**] - 6 partial postallocomposition derivatives of the windowed tile (“allowindowed” tiles) [unit cell signature
**001122, 011223**or**012234**] - 5 partial postallocomposition derivatives of the complex teragons (“allocomplex” tiles) [unit cell signature
**001122**] - 88 co-cell derivatives of an allosymmetric tile (“paraallosymmetric” tiles) [unit cell signature
**001122**] - 74 co-cell derivatives of a second allosymmetric tile (“paraallosymmetric” tiles) [unit cell signature
**001122**]

giving a total of 192 tiles with dissection equation `c+c ^{2}+3c^{3}+c^{4}+c^{5}`.

- 8 partial postallocomposition derivatives of the symmetric tile (“allosymmetric” tiles) [unit cell signature
**014**] - 1 partial postallocomposition derivative of the 2nd external tile (“alloexternal” tile) [unit cell signature
**014**] - 4 partial postallocomposition derivatives of the order 3 demisymmetric
tiles (“allodemisymmetric” tiles) [unit cell
signature
**001144**] - 1 partial postallocomposition derivative of the windowed tile (“allowindowed” tiles) [unit cell signature
**001144, 011245**or**012345**] - 4 partial postallocomposition derivatives of the complex teragons (“allocomplex” tiles) [unit cell signature
**001144**] - 88 co-cell derivatives of an allosymmetric tile (“paraallosymmetric” tiles) [unit cell signature
**001144**]

giving a total of 106 tiles with dissection equation `c+2c ^{2}+c^{3}+2c^{5}+c^{7}`.

- 4 partial postallocomposition derivatives of the symmetric tile (“allosymmetric” tiles) [unit cell signature
**034**] - 4 partial postallocomposition derivative of the windowed tile (“allowindowed” tiles) [unit cell signature
**003344, 013445**or**023456**] - 3 partial postallocomposition derivatives of the complex teragons (“allocomplex” tiles) [unit cell signature
**003344**] - 60 co-cell derivatives of an allosymmetric tile (“paraallosymmetric” tiles) [unit cell signature
**003344**]

giving a total of 71 tiles with dissection equation `2c+c ^{4}+2c^{5}+c^{6}+c^{7}`.

Thus there is a combined total of 626 known order 7 tiles.

© 2016, 2017 Stewart R. Hinsley