Example of a Monohedral Regular Heptagonal Polygon 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: < 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.

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.