I only have a beginners level understanding of hyperbolic geometry, and I am afraid that the following question might be too vague, but here goes.
I know one can make real 3D models of regular tilings that should exist in 2D hyperbolic space. For example, if one keeps gluing together equilateral triangles so that 7 touch at each vertex then one can get a 3D model of the {3,7} tiling of 2D hyperbolic space. Making the triangles out of paper, and taping them together results in a 3D model that looks like crinkled paper. Similar 3D models can be made by hyperbolic crochet. My question is, what if all the edges one uses in the hyperbolic tiling have to be rigid, and of the same length, in 3D. Are there any regular 2D hyperbolic tilings which can be embedded in 3D space to form a simple predictable pattern of rigid regular polygons in 3D ? For example, is there a nicely organized layout of equilateral triangles in 3D space corresponding to an embedding of the aforementioned {3,7} tiling.
To elaborate, I am thinking that the tiling of the sphere with triangles, 5 of which meet at a vertex, corresponds to the regular icosahedron (which is a 3D shape, made of regular, rigid polygons). I know the pseudosphere can be considered to be a 3D model of 2D hyperbolic space, and so I am wondering if there is some simple pattern of tiles on a discretized pseudosphere, which corresponds to a regular tiling of 2D hyperbolic space.