An Approximation of the Hyperbolic Plane
There is a convenient way to model surfaces of constant curvature using equilateral triangles made of paper. If I want to model a space of positive curvature, then I use a complex with 3, 4, or 5 triangles meeting at every vertex, respectively obtaining the regular tetrahedron, octahedron, and icosahedron. If I want to model a space of zero curvature, then I use 6 equilateral triangles at every vertex, obtaining some piece of flat 2-space. Finally, if I want to model a space of negative curvature, then I use 7 or more triangles at every vertex.
For negative curvature, I get some piece of the hyperbolic plane, but the model has some peculiarities not shared by the non-negatively curved models. (If you don't like using paper, then Polydromes work too.) The model is rather "floppy", not at all rigid like the finite complexes with positive curvature. The model looks like a sort of life form one might find on a coral reef somewhere. It bends, twists, and curves in on itself more and more as I add triangles to the complex. The model pictured above has 224 triangles.
Is it possible to build this model indefinitely without having self-intersections? More precisely, does an imbedding of the corresponding infinite complex of piecewise linear cells exist in 3-space? Note that the triangles are all equilateral with the same edge length and that each triangle is flat.
It would be interesting if such an imbedding did indeed exist, although it appears unlikely because, in a way, this model of the hyperbolic plane grows "exponentially". Choose a vertex situated, say, at the origin. Around this vertex there are 7 equilateral triangles. Call this assembly of 7 triangles around a vertex the "first approximation". Notice now that the boundary of the first approximation is a piecewise linear path consisting of 7 segments of equal length, and that each of the vertices of this path has only two triangles meeting there. Obtain the second approximation by adding the smallest number of triangles so that 7 triangles meet at each of these 7 vertices on the boundary of the first approximation. Continue to build the model inductively, obtaining the (n+1)st approximation by adding a collar of triangles to the nth approximation, each collar being obtained by "completing" the vertices on the boundary of the nth approximation to have 7 triangles at a vertex. One can quickly show that the number of triangles in each succeeding approximation follows the linear recurrence,
This problem interests me because of a theorem forbidding the existence of an isometric imbedding of the hyperbolic plane in flat 3-space. Thus, if such an imbedding did exist, it would serve as a sort of compromise because it is "approximately isometric".
Using Schläfli's notation for regular polyhedra, these embeddings may be labeled