Symmetries of Monocoronal Tilings

The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.Comment: 26 pages, 66 figure

Periodic Planar Disk Packings

Several conditions are given when a packing of equal disks in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of any strictly jammed packings, whose graph does not consist of all triangles and the torus lattice is the standard triangular lattice, is at most $\frac{n}{n+1}\frac{\pi}{\sqrt{12}}$, where $n$ is the number of packing disks. Several classes of collectively jammed packings are presented where the conjecture holds.Comment: 26 pages, 13 figure

Optimal Monohedral Tilings of Hyperbolic Surfaces

The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this isoperimetric problem differs for every given area, as solutions do not scale. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric. For area π/3, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima. On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular 7-gon is the optimal n-gonal tile of area π/3 for 3≤n≤10. However, for n\u3e10, it is difficult to rule out non-convex n-gons that tile irregularly