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A quite direct argument is to solve exercise 4.4.17 in page 252 of Thurston's book 3-D Geom & Topo - the published one in 1997. When doing this exercise, one may want to notice two things: 1. On $S^2$, each of the $5$ angles of a spherical pentagon face of a dodecahedron is $120$ degrees, in contrast to $108$ degrees for a pentagon on plane - this is useful when you try to create a local coordinate system on the circle bundle $UTS^2$; 2. The "fundamental domain" Thurston created here does not look like a dodecahedron in the metric sense - it look like a cylinder over a pentagon base; however other "fundamental domains" will mark a line (with quite a slope!) on each of the five side faces and therefore each side face splits into two "pentagons"... With these in mind, exercise 4.4.17 should be very pleasant to work out.