The 600-Cell (Spheres III)

Various arrangements of touching spheres, with a fair amount of color, reflections, and light, can lead to startling views, like this one:

600cellbsmall

So, what are we seeing here? In short, this is the stereographic image of the 600 cell, with its vertices being represented by spheres so large that they touch in the 3-dimensional sphere.

As usual, an analogy helps. Let’s start with the ordinary cube in space. This appears to be a 3-dimensional object. We can also think of it as a tiling of the 2-dimensional sphere by spherical squares, of which one fell off here:

Cubespherical

Now, still working in the 2-sphere, place a spherical disk at each vertex of the cube with a radius so large that all the disks just touch:

Cubecaps

To view this in the plane instead of in the 2-sphere, we can apply a stereographic projection, and get a rather boring looking collection of eight touching disks.

Octapack2

Now we repeat the same procedure in one dimension higher. The cube is one of the five platonic solids in 3-space. In 4-space, there are six regular polytopes, and one of them is the 600-cell. It consist of 600 tetrahedra that we can use to tile the 3-sphere. It also has 120 vertices. Placing a small 2-sphere at each vertex and connecting adjacent vertices by thin tori in the 3-sphere, results (after stereographic projection) in the following model.

600cellc

Now make the 120 spheres so large that they just touch. The first image shows a partial view of these spheres. The spheres are all reflective, and we are standing inside the 600 cell, so we see mostly reflections of (reflections of) spheres.

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