The 120-Cell (Spheres XIII)

Dodeca right

Pentagons do not tile the plane. If you fit three of them around a corner, there will be a gap of 36 degrees.
But, on a sphere, the pentagons can be inflated so that their angles become 120 degrees, and then twelve of them can be used to tile the sphere, creating a spherical version of the dodecahedron.

Dodeca spherical

Likewise, dodecahedra do not tile space. When you fit three around an edge, they leave a gap of about 10.3 degrees.
But again, they can be inflated in the 3-dimensional sphere. This time you will need 120 of them to tile the entire sphere. To visualize this, we start with one dodecahedron, and attach copies at opposite faces. After 10 copies, you will obtain an annulus of dodecahedra, which looks like this, after stereographic projection:

Dodeca

Repeat this with all immediately neighboring dodecahedra to get five more intertwined annuli of dodecahedra. They hide the original annulus from view. All six annuli together form one half of the 120 cell, the rest just being the complement in the 3-sphere of what we already have.

Dodeca 2

Here is an image of just the vertices and edges of the 120-cell. No elephants were harmed in making the 1200 ivory edges.

120cell

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