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I had asked Example of a heptagonal polyhedron? when looking for an example of a polyhedron with only heptagonal sides.

The answers revealed that no such convex polyehdron exists; but that if one introduces topological holes it may still be possible.

My question: Is it possible to construct such genus $\ge 1$ polyhedron purely out of regular congruent heptagons? What would an example of such a construction?

Some searching on google with the terms "monohedral regular heptagonal polyhedron" and "monohedral heptagonal torus" revealed nothing. Eventually I found a paper looking at a similar problem: http://archive.bridgesmathart.org/2008/bridges2008-433.pdf but it seems the author wasn't particularly interested in genus $ \ge 1 $

I'm not sure how one even begins proving that such an object CAN exist let alone finding it.

Sidharth Ghoshal
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The Klein quartic may be represented with 56 triangular faces, seven of which meet at each of 24 vertices. The dual of this would have 24 seven-sided faces., and in fact the surface can indeed be tiled by 24 regular heptagons.

The genus of this object is, in fact, greater than one. It is three. Similar any other "hyperbolic" tiling in the form of a closed object would have genus greater than one.

Oscar Lanzi
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  • This was very cool, but it’s not clear to me if it embeds into Euclidean space. The article on wiki assumes we are in the hyperbolic plane before looking at the construction – Sidharth Ghoshal May 24 '18 at 16:18
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    The pretzel representation is in Euclidean space. But then the heptagons are, of course, not planar. – Oscar Lanzi May 24 '18 at 16:41
  • I guess it was omitted in the question, but the regular heptagons I was interested in were assumed to be regular planar ones. Is there some way to use the knowledge of the existence of this object to create something in the standard euclidean world? / would you be able to point me to some resources, as my research efforts felt a bit unfruitful/not very thorough. – Sidharth Ghoshal May 25 '18 at 01:01