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).

The contraction ratio for such tiles is an integral power of the square root of the reciprocal of the Pisot number (approximately 0.67334809089831373) and the rotation the corresponding multiple of approximately 81.2196317797693513.

This gives a total of 694 tiles (8 order 3 tiles, 60 order 5 tiles and 626 order 7 tiles). There may well be other tiles. There are predicted to be several thousand order 9 tiles.

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

non-derived order 5 tileallosymmetric tileallosymmetric tileallosymmetric tileallosymmetric tileallosymmetric tileallosymmetric tileallosymmetric tileallosymmetric tileallosymmetric tileallosymmetric tileallosymmetric tilealloexternal tilealloexternal tilealloexternal tilealloexternal tilealloexternal tilealloexternal tilealloexternal tilealloexternal tilealloexternal tileallodemisymmetric tileallodemisymmetric tileallodemisymmetric tileallodemisymmetric tileallodemisymmetric tileallodemisymmetric tileallodemisymmetric tileallodemisymmetric tileallodemisymmetric tileallodemisymmetric tileallodemisymmetric tileallowindowed tileallowindowed tileallowindowed tileallowindowed tileallowindowed tileallowindowed tileallowindowed tileallowindowed tileallowindowed tileallowindowed tileallowindowed tileallocomplex tileallocomplex tileallocomplex tileallocomplex tileallocomplex tileallocomplex tileallocomplex tileallocomplex tileallocomplex tileallocomplex tileallocomplex tileallocomplex tile

2015, 2016, 2017 Stewart R. Hinsley