In 2002 I investigated low order metallic (where the ratio of areas is a metallic mean) and cubic (where the ratio of areas is a cubic Pisot number) tiles, including those associated the 6th unit cubic Pisot number (the real root of x3-2x2-1=0, with a value of approximately 2.20556943040059). At the time I identified 4 order 3 tiles (one with 2 alternative dissections) and 3 simple order 5 derivatives of one of those tiles. Having revisited this in 2015 I have identified 2 additional order 3 tiles (and 4 simple order 5 derivatives), and a considerable number of easily generated order 5 and order 7 derivatives of the original tiles that I didn't get round to generating in 2002. Subsequent investigation in 2016 found two more order 3 tiles, with 6 order 5 and 4 order 7 derivatives, resulting in a total of 322 tiles (8 order 3 tiles, 58 order 5 tiles and 256 order 7 tiles). There may well be other tiles. There are predicted to be in excess of 1000 order 9 tiles.
The contraction ratio for the tiles is the square root of the reciprocal of the Pisot number (approximately 0.67334809089831373) and the rotation approximately 81.2196317797693513°.
The tiles are (click on a tile for a larger image and more details)
© 2015, 2016, 2017 Stewart R. Hinsley