**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 x^{3}-2x^{2}-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

### From 2015

### Derivatives added 2015

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

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

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

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

### More new discoveries from 2015

#### More order 5 fractional (non-unitary) tiles

#### More order 7 fractional (non-unitary) tiles

### From 2016

### From 2017

© 2015, 2016, 2017 Stewart R. Hinsley