Universal cover of a figure eight?

Question

An example in my lecture notes says, 'draw a simply connected covering space over the figure eight'.

Howerver, after googling, wikipedia tells me that ''The universal cover of the figure eight can be visualized by the Cayley graph'' but I can't see how this graph is simply connected?

NOR, how I am supposed to draw it?

Is there a simple 'draw-able' universal cover of the figure-eight?

Answer 1

The universal covering space is unique up to homeomorphism. Remember the fundamental group of the figure 8 is the non-commutative free group on two elements. Hence the complex appearance of its covering space.

Answer 2

Here is a picture of the universal cover of a figure eight I made for a topology course in 1991 (!), using Macintosh-Pascal. Note the self-similarity of the "ends".