Richard Kenyon
Discovered Tilings
A simple substitution rule, generating tilings which don’t possess flc. The fractally shaped tiles make it a selfsimilar-substitution. Despite the fractal apperance, the dimension of the boundary of the prototile is one almost everywhere: the boundary of the tile consists of lines almost everywhere …
Rep Tiles Self Similar Substitution
A simple substitution, yielding a tiling which is not of finite local complexity (flc). The substitution maps the single prototile, a unit square, to three columns of three squares each, where the third column is shifted by an irrational amount t. In higher iterates of the substitution, there are …
Polytopal Tiles Parallelogram Tiles Rhomb Tiles
A polygonal version of Kenyon 2. The edges are generated by the morphism: a->b, b->c, c->d, d-> b’a’ (where x’ is the inverse of x).
Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Parallelogram Tiles Kenyon's Construction
A polygonal version of Kenyon (1,2,1). The boundary is generated by the morphism $a \to b, b \to c, c \to c a' b' b'$ (where $x'$ is the inverse of $x$).
Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Kenyons Construction
A substitution rule shown on R. Kenyon’s homepage: http://www.math.brown.edu/~rkenyon/gallery/gallery.html with inflation factor that satisfies: $x^4+x+1 = 0$.
Finite Rotations Self Similar Substitution
As well as showing that there are substitution rules with any Perron
inflation factor, in [Ken96]
, R. Kenyon
gives an explicit construction for the Perron numbers that satsify:
$xn - a xn-1 + b x + c$, where $a, b$, and $c$ are natural numbers.
This is an example of that method given in that paper. …
Finite Rotations Self Similar Substitution Kenyons Construction