A construction for tiles with dissection
equation `nx + x ^{2} + x^{3}`
produces an order 4 8th unit cubic Pisot tile, with dissection equation

From this 4 order 7, 4 order 10 tiles, and 15 order 13 tiles can be derived.

There are 6 candidates for order 10 tiles, but two are disconnected, even though they do have a similarity dimension of 2, and tile the plane. There are two groups of candidates for order 13 tiles, one group of 4 being directly derived from the order 4 tile, and a second group of 28 derived indirectly via the order 7 tiles. However one of the first group, and 16 of the second group, are disconnected, although they again have a similarity dimension of 2, and tile the plane.

The order 7 tiles have dissection equations `x +
3x ^{2} + 2x^{3} + x^{4}`,

The order 10 tiles have dissection equations `2x +
2x ^{}^{3} + 3x^{4} + 2x^{5} +
x^{6}`,

The first group of order 13 tiles contains the following.

The second group of order 13 tiles contain the following.

All of these figures tile the plane. The order 7 tiles have 2 copies in the unit cell, the order 10 tiles 3 copies in the unit cell, and the order 13 tiles 4 copies in the unit cell. The tiling vectors are the same as for the order 4 tile.

© 2016 Stewart R. Hinsley