Some tiles associated with the 6th unit cubic Pisot number

Some Order 5 Tiles

I. Fractional Symmetric Tiles

If the modifications of the grouped element technique are applied to one of the order 5 asymmetric metasymmetric tiles, even though it has no symmetries, a set of 32 attractors, including the original one, is found, all of which have a similarity dimension of 2. 8 of these generate 6 partial postcomposition derivatives of the symmetric and demisymmetric tiles, and one is disconnected, leaving 23 novel tiles.

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

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Examination of the weight function suggests that 4 of these tiles make up a symmetric tile. (Two copies - red and green on the left above, and red and blue on the right above - have the outline of a symmetric tile, and adding two more copies rotated by 180° fills in the voids. They're not strictly quarters, as the 4 tiles come in two different sizes.) This is confirmed by plotting the weight function for the unit cells.

order 3 tileorder 3 tileorder 3 tileorder 3 tile

The unit cells are as shown below.

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© 2015, 2016 Stewart R. Hinsley