Planar coincidences for N-fold symmetry

This paper proves the following results: Besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane. If a centrally symmetric octagon can form a lattice multiple tiling, then the multiplicity is at least seven. However, there are decagons which can form five-fold $($or six-fold$)$ lattice tilings. Consequently, whenever $n\ge 3$, there are non-parallelohedral polytop...

The aim of this short lecture series is to expose the students to the beautiful theory of lattices by, on one hand, demonstrating various basic ideas that appear in this theory and, on the other hand, formulating some of the celebrated results which, in particular, shows some connections to other fields of mathematics. The time restriction forces us to avoid many important parts of the theory, and the route we have chosen is naturally biased by the individual taste of the spe...

The goal of the work is to take on and study one of the fundamental tasks studying Diophantine n-gons (the author of the paper considers an integral n-gon is Diophantine as far as determination of combinatorial properties of each of them requires solution of a certain Diophantine equation (equation sets)).

The aim of this paper is to determine the elements which are in two pairs of sequences linked to the regular mosaics $\{4,5\}$ and $\{p,q\}$ on the hyperbolic plane. The problem leads to the solution of diophantine equations of certain types.

Zhegalkin zebra motives are tilings of the plane by black and white polygons representing certain ${\mathbb F}_2$-valued functions on ${\mathbb R}^2$. They exhibit a rich geometric structure and provide easy to draw insightful visualizations of many topics in the physics and mathematics literature. The present paper gives some pieces of a general theory and a few explicit examples. Many more examples will be shown in the forthcoming article "Zhegalkin zebra motives: algebra a...

We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.

In a recent paper, G. Cz\'edli and E.\,T. Schmidt present a structure theorem for planar semimodular lattices. In this note, we present an alternative proof.

This article studies a generalization of magic squares to finite projective planes. In traditional magic squares the entries come from the natural numbers. This does not work for finite projective planes, so we instead use Abelian groups. For each finite projective plane we demonstrate a small group over which the plane can labeled magically. In the prime order case we classify all groups over which the projective plane can be made magic.

The Dedekind tessellation -- the regular tessellation of the upper half-plane by the Mobius action of the modular group -- is usually viewed as a system of ideal triangles. We change the focus from triangles to circles and give their complete algebraic characterization with the help of a representation of the modular group acting by Lorentz transformations on Minkowski space. This interesting example of the interplay of geometry, group theory and number theory leads also to c...

We obtain tilings with a singular point by applying conformal maps on regular tilings of the Euclidean plane, and determine its symmetries. The resulting tilings are then symmetrically colored by applying the same conformal maps on colorings of regular tilings arising from sublattice colorings of the centers of its tiles. In addition, we determine conditions so that the coloring of a tiling with singularity that is obtained in this manner is perfect.