Uniform tiling

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain.

Each is represented by a set of lines of reflection that divide the plane into fundamental triangles.

A prismatic symmetry group, (2 2 2 2), is represented by two sets of parallel mirrors, which in general can make a rectangular fundamental domain.

It is called the elongated triangular tiling, composed of alternating layers of squares and triangles.

There are infinitely many uniform tilings by convex regular polygons on the hyperbolic plane, each based on a different reflective symmetry group (p q r).

The Coxeter-Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.

Further symmetry groups exist in the hyperbolic plane with quadrilateral fundamental domains — starting with (2 2 2 3), etc.

Right angle fundamental triangles: (p q 2) General fundamental triangles: (p q r) There are several ways the list of uniform tilings can be expanded: Symmetry group triangles with retrogrades include: Symmetry group triangles with infinity include: Branko Grünbaum and G. C. Shephard, in the 1987 book Tilings and patterns, section 12.3, enumerate a list of 25 uniform tilings, including the 11 convex forms, and add 14 more they call hollow tilings, using the first two expansions above: star polygon faces and generalized vertex figures.

[2] Besides the 11 convex solutions, the 28 uniform star tilings listed by Coxeter et al., grouped by shared edge graphs, are shown below, followed by 15 more listed by Grünbaum et al. that meet Coxeter et al.'s definition but were missed by them.

There is also a third tiling for each vertex configuration that is only pseudo-uniform (vertices come in two symmetry orbits).

[2] In the pictures below, the included squares with horizontal and vertical edges are marked with a central dot.

These expansions to the definition for a tiling require corners with only 2 polygons to not be considered vertices — since the vertex configuration for vertices with at least 3 polygons suffices to define such a "uniform" tiling, and so that the latter has one vertex configuration alright (otherwise it would have two) —.