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

## I. Fractional
Symmetric Tiles

If the modifications of the grouped element technique are applied to one of
the order 7 asymmetric metasymmetric tiles, even though
it has no symmetries, a set of 128 attractors, including the original one, is
found, all of which have a similarity dimension of 2. Some generate partial
postcomposition derivatives of the symmetric and demisymmetric tiles, and several more are disconnected,
leaving 82 novel tiles.

Four of these tiles have similar outlines to the order 5 metasymmetric
tiles, though they contain voids.

Examination of the weight function suggested that 6 of these tiles make up a
symmetric tile. (Three copies - red, blue and cyan
on the left above, and red, blue and gold on the right above - have the outline
of a symmetric tile, and adding three more copies rotated by 180° fills in the
voids.) This was confirmed when the makeup of the unit cell for the wider class
of tiles including these was determined.

The unit cells are shown below. It is not obvious from visual inspection
that the 6 copies make up a unit cell, but this is confirmed by plotting the
weight function for the unit cell.

Next

© 2015, 2016 Stewart R. Hinsley