Some tiles associated with the 6th unit cubic Pisot number

Overview

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)

From 2002

symmetric tile1st order 3 demisymmetric tile2nd order 3 demisymmetric tilewindowed tilemetasymmetric tilemetasymmetric tilemetasymmetric tile

From 2015

complex teragonexternal tile

Derivatives added 2015

Order 5 fractional (unitary) tiles (grouped element tiles)

order 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tile

Order 5 fractional (non-unitary) tiles (partial postcomposition tiles)

metademisymmetric tilemetademisymmetric tilemetademisymmetric tilemetademisymmetric tilemetawindowed tilemetademisymmetric tilemetademisymmetric tilemetacomplex tilemetacomplex tilemetaexternal tilemetaexternal tile

Order 7 fractional (unitary) tiles (grouped element tiles)

order 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorderorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demisymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorderorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorderorderorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tileorder 7 demimetasymmetric tile

Order 7 fractional (non-unitary) tiles (partial postcomposition tiles)

metasymmetric tilemetasymmetric tilemetasymmetric tilemetasymmetric tilemetasymmetric tilemetademisymmetric tilemetademisymmetric tilemetademisymmetric tilemetawindowed tilemetademisymmetric tilemetademisymmetric tilemetacomplex tilemetacomplex tilemetaexternal tile

More new discoveries from 2015

More order 5 fractional (non-unitary) tiles

order 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tileorder 5 tile

More order 7 fractional (non-unitary) tiles

order 7 tileorder 5 tileorder 5 tileorder 5 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tile order 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tileorder 7 tile

From 2016

external tileorder 5 meta external tileorder 5 meta external tileorder 5 meta external tileorder 7 meta external tileorder 7 meta external tilecomplex teragon metacomplex tilemetacomplex tilemetacomplex tilemetacomplex tilemetacomplex tile

From 2017

2015, 2016, 2017 Stewart R. Hinsley