Tilings Encyclopedia | Richard Kenyon

Preview Kenyon's non FLC (volume hierarchic)

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 (plus accumulation points). Unlike the Koch curve, for instance, or the fractal Dart and Kite prototiles, here the entire length of the boundary is finite. For more details, see Kenyon non-FLC.

Rep Tiles Self Similar Substitution

Kenyon's non FLC

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 neighboured squares shifted against each other by t, 3t+t, 9t+3t+t,… mod 1. Since t is irrational, these sequence contains infinitely many values (mod 1), thus there are infinitely many pairwise incongruent pairs of tiles.

Polytopal Tiles Parallelogram Tiles Rhomb Tiles

Kenyon 2 Polygonal

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

Kenyon (1,2,1) Polygon

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

Kenyon 2

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

Kenyon (1,2,1)

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. A locally isomorphic version with polygonal tiles is Kenyon (1,2,1) Polygon.

Finite Rotations Self Similar Substitution Kenyons Construction

Richard Kenyon

Discovered Tilings