Az'glqssk the alien lives on a platonic planet whose shape is that of a perfect regular dodecahedron. Being extremely xenophobic, she checks every day that no-one else has arrived on her planet. In order to do this she has to travel a closed path (from her house back to her house) with the property that every point on the planet's surface is visible from some point of this path. If the edge length of the dodecahedron is 1 unit, what is the shortest path she can take?

Az'glqssk has built her house so it is on the shortest route and she is afraid of heights as well as strangers so she confines her path to the planet's flat surface.