The cutting produced one outer face that is a $c'\sqrt v$-gon for some small $c'\ge c$.įor $\nu=3,4,\ldots$, let $f_\nu$ be the number of $\nu$-gonal faces apart from that outer face. The cutting will turn about $\sqrt v$ vertices (say $c\sqrt v$ for some small $c$) into degree $2$ vertices.īy counting edge-vertex incidences, we find $3v-c\sqrt v=2e$. Now if we cut out some large but finite portion of this infinite graoh with $v$ vertices, $e$ edges and $f$ faces, then Euler says that $v+f=e+2$. We can also ignore the case of degree $\ge 4$ as so many edges incident with one vertex would be highly coincidental. We can ignore vertices of degree 1 (dead ends) and of degree 2 (not distinguished from a point of an edge). Hexagonal patterns occur in two dimensions essentially.Ĭonsider an infinte set of points (vertices) in the plane joined by edges, forming an infinite graph.
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