Question

Why does Atlas beg Perseus to show him Medusa's head?

Answer 1

Some of the Roman writers tried to rationalize the story of Atlas holding up the sky by equating Atlas to mountains — specifically, the Atlas Mountains. They accomplish this using Medusa’s head.

The gist of the tale is that Perseus passed near Atlas’ location — at the far west of the known world (i.e. continental Europe and Northern Africa) — after slaying Medusa and asked for lodging and protection for the night. Atlas was rude and demanded Perseus to leave, prompting Perseus to expose the Titan to Medusa’s head and turn him into the mountain range. At no point, at least in any source that I know of, did Atlas explicitly ask to be turned to stone.