No tiling by convex heptagons
09 Mar 2018Fourth post of the GT CoA series, see here for the introduction.
The talk related to this post was given by Michaël Rao on his recent result about pentagonal tilings. This topic has been covered in many places (see for example Quanta magazine^{1} and this video. This post will focus on another aspect: what about heptagonal tilings? More precisely, how to show that there is no tiling of the plane with a convex heptagonal tile?
We first present a simple heurisitic proof that gives the intuition about why this holds, and then a more detailed proof. The first one uses simple geometry, the second is graph theoretical.
First (heuristic) proof via doublecounting of the angles
Let a contact point be a place where three or more tiles meet. Without loss of generality wa can assume that tiles only meet at vertices, that is there is no contact point in the interior of an edge, it is always at an endpoint.^{2}
As the sum of the interior angles of an $n$gon is $(n2)\times 180$ degrees, for an heptagon it is $5\times 180=900$ degrees. The average degree is $900/7\approx 129$ degrees. Which means that at one contact point there are in average $360/129\approx 2.8 $ heptagons meeting. This does not make sense because at every contact point there should be at least three tiles.
Second proof with planar graphs.
First attempt via the degree
Take an infinite tiling, and draw the following graph. In the middle of each tile draw a vertex, and between two vertices that represent adjacent tiles, draw an edge. This graph is infinite and planar. As the tiles are convex, there are at least seven different tiles adjacent to it, and then the minimum degree of a vertex in the graph is seven.
If you are used to planar graphs, you may think that this is enough to conclude. Indeed, it is known that the average degree of panar graphs is strictly smaller than six (see for example here), and the contradiction follows. Actually this is not enough because this average degree upper bound holds only in finite graph. As a counterexample to this in infinite graphs, consider an infinite regular tree with arbitrary large degree $d$ ; it is a planar graph and it has average degree $d$.
Euler formula on the dual
The degree bound comes form the classic tool for planar graphs, Euler’s formula. It states that $VE+F=2$ for finite planar graphs, where $V$ is the set of nodes, $E$ is the set of edges, and $F$ is the set of faces. The problem is that it works only for finite graphs. We will consider a large enough part of the infinite graph and use Euler formula to prove the result.
This can probably be achieved in the graph we have defined, but it is more convenient on its dual. Given a tiling, consider every place where three or more tiles meet as a vertex, and the edges of the tiles are the edges of the graph. Basically drawing the tiling is drawing the graph.
Now consider a ball of radius $r$ much larger than the size of the tile, and the tiles that are at least partilly in this ball. There is order of $r^2$ tiles strictly inside the ball and $O(r)$ tiles at the boundary. Note that the tiles are the faces of the planar graph (except the outer face that is not a tile). A similar thing holds for vertices: there are $\Theta(r)$ vertices that are at the boundary, call them boundary nodes ($V_b$), and order of $r^2$ that are strictly inside the graph, called interior nodes ($V_i$).
We doublecount edges. For every edge, divide it lengthwise into two halfedges. There are clearly $2E$ halfedges. Also every face except the outerface is a tile, so contributes seven halfedges. The number of halfedges of the outerface is in $O(r)$. Thus $2E=7F+O(r)$.
We now count the corners (or interior angles) of tiles. Every tile has seven corners, thus this number is exactly $7F$. As every interior node contributes at least three corners, $7F\geq 3V_i$.
Now using that $V=V_i+V_b$, then $E=\frac{7}{2}F+O(r)$ and finally $F\geq \frac{3}{7}V_i$ one gets: $\frac{15}{14}V_i+V_bO(r)\geq 2$. This is not possible because $V_i$ has a higher order of magnitude than the two other terms, thus the asymptotics should go to $\infty$.
Notes et footnotes
Thanks to Michaël Rao for discussions.

There exists a translation in French, in Pour la science. ↩

This would need a proof, but roughly, having a contact point in the middle of an edge only make the argument stronger, as it forces the other angles to be small. ↩