  A polyhedral surface~$\mathcal{C}$ in $\mathbb{R}^3$ with convex polygons as
faces is a side-contact representation of a graph~$G$ if there is a bijection
between the vertices of $G$ and the faces of~$\mathcal{C}$ such that the
polygons of adjacent vertices are exactly the polygons sharing an entire common
side in~$\mathcal{C}$.
  We show that $K_{3,8}$ has a side-contact representation but $K_{3,250}$ has
not. The latter result implies that the number of edges of a graph with
side-contact representation and $n$ vertices is bounded by $O(n^{5/3})$.
