Some tiles associated with the 6th unit cubic Pisot number

Summary by Order

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

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+c3 and the minimal signature is variously 0 or 00 (the windowed tile has tilings with signatures 00, 01 and 02). For tiles with greater order identification of the dissection equation and signature serves to divide the tiles into groups, and acts as an aid in confirming that visually similar tiles are distinct. For this reason the dissection equation and signature is given for each group of tiles below.

Order 3 Tiles

symmetric tilewindowed tile1st order 3 demisymmetric tile2nd order 3 demisymmetric tileexternal tileexternal tilecomplex teragoncomplex teragon

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 → ±a3p + v2 } (or { p → ±ap - 1; p → ±ap + 1; p → ±a3p + v2 }). In principle we can perform an exhaustive search by iterating the value of v2, and subsequently closing in from values visually identifiable as close to a tile. However in practice a step size sufficiently small to be certain to find the tiles requires too many candidates to be examined for this to be computationally practicable. It also produces an impracticably high number of candidates for visual inspection. The process would also not necessarily find cryptic tiles (tiles like the Levy curve, where the area of the plane wholly covered by the tile is small compared to the outline of the tile). However for 3 element tiles a heuristic search with v2 = P(a), that is v2 is a polynomial in a, is practicable.

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 ±an and ±1 + ±an turned out to be incomplete (applied to the two alternative sets of IFSs didn't find the same set of tiles). A second survey was performed using aj + ak + al. This also turned out to be incomplete. At this point a survey using (aj + ak + al)/(1-a3) was performed. (Vectors of the form a/(1-a3) turn up in tilings using these tiles.) More tiles were found, but using a table of values of polynomials in a it was found that the corresponding vectors were equivalent to sums of powers of a divided by 2 or 4, supporting the conjecture that the coefficients are ratios of small numbers. (It was already known from the study of rep-tiles that non-integer coefficients occurred.)

The resulting final survey produced 24 IFSs (with values of v2) 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.

The number of values of v2 is lower. With the 1st class of IFSs, the demisymmetric tiles both tiles have v2 = 1, while with the 2nd class, all versions of the symmetric tile have v2 = 0.

Order 5 Tiles

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+2c4+c6 and c+2c2+c3+c4 - both of which can be reached by dissection and by partial postautocomposition. Grouped element derivatives exist for some tiles with the former dissection equation, and co-cell derivatives for some tiles with the latter dissection equation.

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

Order 5 Tiles - dissection equation 2c+2c4+c6

giving a total of 20 tiles with dissection equation 2c+2c4+c6.

Order 5 Tiles - dissection equation c+2c2+c3+c4

giving a total of 36 tiles with dissection equation of c+2c2+c3+c4 and a combined total of 59 known order 5 tiles.

Order 7 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+2c2+3c4+c6 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 4c2+c3+2c4 corresponds to the other double partial postautocomposition derivatives of order 3 tiles, and grouped element derivations of one of these, and also of a dissection of the symmetric tile. The other 4 dissection equations are reached by partial postallocomposition of the order 5 metatiles with their base order 3 tiles, giving “allo(meta)tiles”. There are also grouped element derivatives of a dissection of the symmetric tile for one of these dissection equations, and co-cell derivatives of “allosymmetric” tiles.

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

Order 7 Tiles - dissection equation c+2c2+3c4+c6

giving a total of 91 tiles with dissection equation c+2c2+3c4+c6.

Order 7 Tiles - dissection equation 4c2+c3+2c4

giving a total of 113 tiles with dissection equation 4c2+c3+2c4.

Order 7 Tiles - dissection equation 2c+2c4+2c7+c9

giving a total of 53 tiles with dissection equation 2c+2c4+2c7+c9.

Order 7 Tiles - dissection equation c+c2+3c3+c4+c5

giving a total of 192 tiles with dissection equation c+c2+3c3+c4+c5.

Order 7 Tiles - dissection equation c+2c2+c3+2c5+c7

giving a total of 106 tiles with dissection equation c+2c2+c3+2c5+c7.

Order 7 Tiles - dissection equation 2c+c4+2c5+c6+c7

giving a total of 71 tiles with dissection equation 2c+c4+2c5+c6+c7.

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

© 2016, 2017 Stewart R. Hinsley