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Answered second part of question.
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Igor Rivin
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A picture is in the comments to John Baez' Blog Post. EDIT For the second part of the question, no there is no obstruction, as long as the implied angle ($2\pi/v$) is smaller than the angle of a Euclidean regular $n$-gon. This is true by an obvious argument I had originally heard from Bill Thurston: a very small hyperbolic regular $n$-gon has angles close to Euclidean, a very large regular hyperbolic $n$-gon has angles $0$ so, by continuity, somewhere in between you will have the angle you want.

A picture is in the comments to John Baez' Blog Post.

A picture is in the comments to John Baez' Blog Post. EDIT For the second part of the question, no there is no obstruction, as long as the implied angle ($2\pi/v$) is smaller than the angle of a Euclidean regular $n$-gon. This is true by an obvious argument I had originally heard from Bill Thurston: a very small hyperbolic regular $n$-gon has angles close to Euclidean, a very large regular hyperbolic $n$-gon has angles $0$ so, by continuity, somewhere in between you will have the angle you want.

Source Link
Igor Rivin
  • 96.4k
  • 11
  • 153
  • 366

A picture is in the comments to John Baez' Blog Post.