259 research outputs found
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
Spherical Tiling by 12 Congruent Pentagons
The tilings of the 2-dimensional sphere by congruent triangles have been
extensively studied, and the edge-to-edge tilings have been completely
classified. However, not much is known about the tilings by other congruent
polygons. In this paper, we classify the simplest case, which is the
edge-to-edge tilings of the 2-dimensional sphere by 12 congruent pentagons. We
find one major class allowing two independent continuous parameters and four
classes of isolated examples. The classification is done by first separately
classifying the combinatorial, edge length, and angle aspects, and then
combining the respective classifications together.Comment: 53 pages, 40 figures, spherical geometr
Spherical tilings by congruent quadrangles : forbidden cases and substructures
In this article we show the non-existence of a class of spherical tilings by congruent quadrangles. We also prove several forbidden substructures for spherical tilings by congruent quadrangles. These are results that will help to complete of the classification of spherical tilings by congruent quadrangles
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 , where is the number of packing
disks. Several classes of collectively jammed packings are presented where the
conjecture holds.Comment: 26 pages, 13 figure
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