This tiling and Nautilus are dual tilings generated by non-PV morphisms. As such they are the first step in a generalisation of the work of G. Rauzy, P. Arnoux, S. Ito and others for PV substitution rules. The work that developed out of G. Rauzy’s seminal paper [Rau82] .

The inflation factor for this substitution rule is either of the expanding roots of:
`$x^{4}-x+1 = 0$`

. Note that this it is related to R. Kenyon’s example
Kenyon 2 that is shown on his homepage: http://www.math.brown.edu/~rkenyon/gallery/gallery.html and has
inflation factor satisfying `$x^{4}-x+1 = 0$`

.

Conch (Volume Hierarchic) is a volume hierarchic, version of this substitution rule with fractal bounded tiles.

[Rau82]

Rauzy, G

**Nombres algébriques et substitutions (French) [Algebraic numbers and substitutions]**

*Bull. Soc. Math. France*
1982,
110, 2,
pp. 147-178,
MR0667748