An Approximation of the Hyperbolic Plane

Introduction.

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,

F(1)=7, F(2)=35, F(n+2)=3F(n+1)-F(n)+14, (n>0)

establishing exponential growth. The problem is that, assuming that the edge-length of the triangle is 1 unit, the nth approximation must fit inside a sphere of radius n. The "density" of triangles, say, to unit volume approaches infinity, as the exponential function of n in the numerator overpowers the cubic polynomial in the denominator.

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".

Subsequent Questions.


Using Schläfli's notation for regular polyhedra, these embeddings may be labeled

{3,3}, {3,4}, and {3,5}

for the sphere,
{3,6}

for the flat plane, and
{3,7}, {3,8}, {3,9},

and so on for the hyperbolic plane. Here is the meaning of the Schläfli symbol: The symbol {m,n} designates the regular polyhedron consisting of m-gons attached in such a way that n edges meet at every vertex. This opens the door to slightly more general types of embeddings. For example, {4,3} represents the sphere, {4,4} represents the plane, and {4,n} for n>=5 represents the hyperbolic plane. Which of these can be imbedded in 3-space?

If the polyhedron {3,7} can be imbedded, how about {3,8}? For that matter, if {3,7} and/or {3,8} can be imbedded, is there an upper bound on the number n such that {3,n} can be imbedded? The same questions may be asked about the other polyhedra {m,n}.

Suppose {3,7} cannot be imbedded. Then one naturally wonders about the size of the largest approximation in existence. Recall that the approximations to {3,7} are defined inductively by adding collars of triangles as described above. How many collars can be added without obtaining an intersection. Again, the same questions may be asked more generally about the polyhedra {m,n}.

Exercise. Define what is meant by a "collar" for the polyhedron {m,n}. Define what is meant by the "kth approximation" of {m,n}.

In general, there is a function N{m,n} defined according to the maximum number of collars of m-gons that can be attached to approximate {m,n} without self-intersections; i.e. N{m,n} is the integer k such that the kth approximation can be imbedded but the (k+1)st cannot. Thus, I would find it interesting if N{3,7} turned out to be infinity while N(3,n) were finite for some n>7. Some of the values of N are easy to compute. For example, N{7,3}=0 because there is no way to attach 3 rigid heptagons around a single vertex; one of the heptagons must bend if you want to attach them.

An Aside

The above image shows a model of a simplicial complex which possess some of the critical properties of the hyperbolic imbedding/immersion described above. Namely, it is two dimensional, made from triangles, and there are seven triangles at every vertex. The model represents a lattice of icosahedra and octahedra attached after removing 4 faces from each of the icosahedra and 2 opposite faces of each octahedron. Since each octahedron is attached to two icosahedra and each icosahedron to four octahedra, it follows that there are 7 triangles at each vertex. The arrangement of icosahedra and octahedra corresponds to the "diamond lattice", a superposition of two face-centered cubic lattices. The icosahedra occupy the location of the carbon atoms and the octahedra correspond to valence bonds, 4 per carbon atom. Since this is a lattice, one may extend this model indefinitely. However, this infinite complex is far from being simply connected, so it does not represent an imbedding of the hyperbolic plane into 3-space.

Here are some links to related investigations. Don Hatch has investigated hyperbolic analogues of the Archimedean solids. Melinda Green has a page on infinitely regular polyhedra. David Joyce has a page on hyperbolic tessellations.

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