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May 14, 2017 at 18:27 comment added Manfred Weis @AdamP.Goucher that was exactly what I had in mind, but "forgot" to restrict conformity to the open disk; the tour would then be the the closure of the open disk's image.
May 14, 2017 at 12:35 comment added Dirk Oops, this was actually not meant as an answer...
May 14, 2017 at 12:32 comment added Adam P. Goucher Wait, what?! The Riemann mapping theorem allows you to conformally map the interior of any Jordan curve (including any polygon; cf. Schwarz-Christoffel transformation) to the interior of the unit disk.
May 14, 2017 at 9:37 history answered Dirk CC BY-SA 3.0