33 - Poincaré homology 3-sphere

The 120-cell is a way of tiling the "3-sphere" with 120 dodecahedra. The mathematician Henri Poincaré considered a space formed by one dodecahedron with its opposite pairs of faces attached.

Is this also a 3-sphere?

No! It's called the Poincaré homology 3-sphere, since it resembles the usual 3-sphere, but it's different: it contains loops that you can't "pull tight". It contains precisely 120 of them!

The rotation group of the dodecahedron is a 60-element subgroup of SO(3); it has a "double cover" which is a 120-element subgroup Γ ⊂ SU(2). SU(2) is a 3-sphere, and the quotient space SU(2)/Γ is the Poincaré homology 3-sphere. If you know algebraic topology, it follows that the "fundamental group" of the Poincaré homology 3-sphere is Γ. This is the precise meaning of my vague claim that there are "precisely 120 loops that you can't pull tight" in the Poincaré homology sphere.

The background picture is from the Wikipedia article on the 120-cell. It was created by Fritz Obermeyer, who released it into the public domain.