discrete geometry

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We study $(n,k)$-scatters, which are regular $n$-gon tiles arranged so that each tile shares edges with at least $k$ others. To measure how much freedom there is in arranging scatters, we ask which shapes are unavoidable. It turns out that for a few choices of $(n,k)$ there are infinite unavoidable shapes, but otherwise they are finite. We discuss the infinite case as an analogue of crystallization. The main result here is that besides the trivial situations when there’s a unique scatter, there are only four instances of this. Scatters crystallize nontrivially just when $(n, k) = (5, 3)$, $(7, 3)$, $(10, 4)$, or $(14, 4)$.

We construct arbitrarily sparse locally-jammed packings of non- overlapping congruent disks in various finite area regions — in particular, we give constructions for the square, hexagon, and for certain flat tori.

An old theorem of Alexander Soifer's is the following: Given five points in a triangle of unit area, there must exist some three of them which form a triangle of area $1/4$ or less. It is easy to check that this is not true if "five" is replaced by "four", but can the theorem be improved in any other way? We discuss in this article two different extensions of the original result. First, we allow the value of "small", $1/4$, to vary. In particular, our main result is to show that given five points in a triangle of unit area, then there must exist some three of them determining a triangle of area $6/25$ or less. Second, we put bounds on the minimum number of small triangles determined by $n$ points in a triangle, and make a conjecture about the asymptotic right answer as $n \to \infty$. [NOTE: Conjecture 2.6 fails --- the optimal value is $3 - 2 \sqrt{2}$ not $1/6$.]