Is a "nice" plane tiling possible where each tile has 7 (8, 9, ...) neighbors?
With "nice" I mean:
- The tiling is (preferably) periodic.
- The tiles are from a finite set
- The tiles themselves are "nice" (non-degenerate, no holes, connected). It's OK if the tiles are not convex.
This seem to be a simple question, but I lack the terminology to do a proper search.
Are there general results of tiling possibilities in terms of number of neighbors that I can look at? (For example, if I want to know whether a tiling exist where each cell as $m$, $n$, ..., or $p$ neighbors.)
(I have seen this question Why a tesselation of the plane by a convex polygon of 7 or more sides is not possible?, but this is not quite what I am interested in).
(Background: I am the author of a Grids package that allows programmers to set up various types of grids for game programming. Once customer asked whether we will support octagonal grids in the future, and I wondered whether such a grid is even possible).
Suppose it is possible. Take a large region of tiles. Put a frame around it, and then make a new framing polygon that touches all the outer tiles.
Map this map onto a sphere, and add a point to the framing polygon so that's it's unpunctured.
From there, we have a polyhedron. If all the tiles have 5 or 6 sides, then there will be exactly 12 pentagons via Euler's V+F-E=2. The Fullerenes enumerate varying numbers of hexagons.
Sadly, it's not possible for all the polygons to touch 7 or more others on this sphere. That overloads V+F-E=2. It's possible to use only heptagons on surfaces of higher genus. For example, the klein quartic uses 24 heptagons on a three holed torus.