As presented in previous posts, inversion in three touching circles makes a Poincaré disc representation of a hyperbolic plane tiled by a triangle. Adding a fourth circle touching the other three results in a fractal decoration of the hyperbolic disc with discs. Actually, this is an Apollonian gasket. I want to do essentially the same thing, but in three dimensions: Inversion in four circles can generate a Poincaré ball representation of hyperbolic space and an additional fifth sphere could create a fractal packing of spheres in this space. In analogy to the two-dimensional case, I am using four spheres that have their centers on the corners of a tetrahedron and add a fifth sphere at the center of the tetrahedron.
It is important to realize that the limit set of the inversion in these five spheres has to cover completely the surfaces of all the spheres of the sphere packing. We can only get this if the limit set of the inversions in four of them covers the entire surface of the Poincaré ball that they generate. Observing that the space lying outside the inverting spheres does not belong to the limit set, we conclude that the inverting spheres have to cover the entire surface of the Poincaré ball. This can definitely not be achieved with touching spheres, as I naively tried first.
Instead, we need intersecting spheres. To see how large they have to be, we simply look at the surfaces of the tetrahedron that has four spheres at its corners. They are equilateral triangles and should lie inside these spheres. We achieve this if the spheres reach the center of the triangles. This corresponds to an intersection angle of 60 degrees between the spheres, see Figure 1. Multiple inversion in all four spheres creates a Poincaré ball representation of a tiled hyperbolic space. The surface of the ball touches the centers of the sides of the tetrahedron. The four spheres can be seen as curved sides of another tetrahedron. This tetrahedron tiles the hyperbolic space. Its corners lie on the centers of the sides of the tetrahedron that has the spheres at its corners. Thus the tiling tetrahedron is its dual tetrahedron. The spatial angle of its corners vanishes. Note that this is similar to the planar angle at the corners of a triangle made with three touching spheres, which vanishes too. Also, the corners of the tiling tetrahedron lie on the surface of the Poincaré ball. Thus this tetrahedron is an ideal tetrahedron.
The space outside the ball is an inverted image of its inside. On the triangle surfaces we get an inverted Euclidean tiling of equilateral triangles, see Figure 1. This is simply a particular cross section of this inverted hyperbolic space. The limit set of the inversions in the four spheres covers the entire surface of the Poincaré ball, see Figures 2 and 3.
To get a fractal packing of Poincaré balls we add a fifth inverting sphere at the center of the tetrahedron. It too intersects the other spheres at an angle of 60 degrees. This creates new groups of four intersecting spheres, always three of the first four spheres together with the new one. Each of these groups creates by multiple inversion a similar hyperbolic space as presented in the first part of this post. Thus we get touching Poincaré balls covered by the limit set of the inversions. Inversion in all five spheres multiplies them and fills in the gaps between the spheres. Thus the entire hyperbolic space gets covered by Poincaré balls. We see this in Figures 4 and 5. They are planar cross-sections of this sphere packing. Instead of spheres we see circles as their cross-section. The sphere packing appears thus as a fractal packing of circles in these figures.
Thus, five intersecting spheres really can make a fractal packing of spheres in three dimensions similarly as four circles generate an Apollonian gasket. Further, we proceed as in the last post and think of the three-dimensional space as being the surface of a Poincaré half-“hyper”-space representation of a four-dimensional hyperbolic space. Using inversion in a four-dimensional sphere, we can get five intersecting spheres of equal radius. Their centers lie on the corners of a 4-simplex, which is a “hyper”-tetrahedron in four-dimensional space. They generate a Poincaré “hyper”-ball representation of hyperbolic space. The fractal sphere packing then lies in its surface, a spherical three-dimensional space. It has the symmetries of the 4-simplex. Thus the packing of spheres is also a fractal decoration of three-dimensional spherical space. Both views are related by a stereographic projection from the surface of the four dimensional sphere to three-dimensional Euclidean space. Unfortunately, it is not possible to extend this scheme one dimension higher.