In hyperbolic space, a horosphere is both convex and intrinsically flat. So Euclidean plane tilings with regular tiles can be made convex. I suppose these should be excluded from CRFs, simply because they're infinite. Not only does each tiling have infinitely many tiles, but there are (uncountably!*) infinitely many different types of tilings.

Hyperbolic n-space can be embedded in (n,1) psEuc space (as the hyperboloid model); then a horosphere becomes a circular paraboloid. And the square tiling becomes a paraboloidal polyhedron: take z = w - 1 = (x

^{2}+ y

^{2})/2 with x and y integers, to generate the vertices, then take the convex hull. So these Euclidean tilings can be made convex in flat space, though the angles and symmetries are non-Euclidean. In particular, a translation becomes a parabolic rotation.

We could also consider tilings with vertices on the other unit hyperboloid (the single-sheet one), but those would not be convex.

I'm also interested in more general psEuc spaces like (2,2). This is a 4-dimensional space with a null cone made of rays from the origin through a Clifford torus: x

^{2}+ y

^{2}- z

^{2}- w

^{2}= 0. I wonder what types of symmetries a polytope could have here. The orthogonal group O(2,2) contains the general linear group GL(2) which is represented by all invertible 2x2 matrices.

*A Euclidean tiling can be made by stacking rows on each other, where each row is a strip of triangles or squares with the same edge lengths. A strip of squares represents a 0, and a strip of triangles represents a 1; the whole tiling represents a number in binary, with infinitely many digits. There are uncountably many such numbers, thus uncountably many such tilings. See https://en.wikipedia.org/wiki/Cantor%27 ... l_argument .