8th Unit Cubic Pisot Tiles

A construction for tiles with dissection equation nx + x2 + x3 produces an order 4 8th unit cubic Pisot tile, with dissection equation 2x + x2 + x3.

order 4 tile

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 + 3x2 + 2x3 + x4, 2x + 3x3 + x4 +x5, 2x + x2 + 2x4 + x5 + x6 and x + 3x2 + 2x3 + x4 (again).

order 7 tileorder 7 tile order 7 tileorder 7 tile

The order 10 tiles have dissection equations 2x + 2x3 + 3x4 + 2x5 + x6, x + 2x2 + 4x3 + 2x4 + x5, x + 3x2 + x3 + 3x4 + x5 + x6 and 5x2 + 3x3 + 2x4.

order 10 tileorder 10 tileorder 10 tileorder 10 tile

The first group of order 13 tiles contains the following.

order 13 tileorder 13 tileorder 13 tile

The second group of order 13 tiles contain the following.

order 13 tileorder 13 tileorder 13 tileorder 13 tileorder 13 tileorder 13 tileorder 13 tileorder 13 tileorder 13 tileorder 13 tileorder 13 tileorder 13 tile

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.

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2016 Stewart R. Hinsley