Planar coincidences for N-fold symmetry

A geometric study of twin and grain boundaries in crystals and quasicrystals is achieved via coincidence site lattices (CSLs) and coincidence site modules (CSMs), respectively. Recently, coincidences of shifted lattices and multilattices (i.e. finite unions of shifted copies of a lattice) have been investigated. Here, we solve the coincidence problem for a shifted hexagonal lattice. This result allows us to analyze the coincidence isometries of the hexagonal packing by viewin...

We survey some principal results and open problems related to colorings of geometric and algebraic objects endowed with symmetries, concentrating the exposition on the maximal symmetry numbers of such objects.

We survey some principal results and open problems related to colorings of algebraic and geometric objects endowed with symmetries.

A coloring of a planar semiregular tiling $\mathcal{T}$ is an assignment of a unique color to each tile of $\mathcal{T}$. If $G$ is the symmetry group of $\mathcal{T}$, we say that the coloring is perfect if every element of $G$ induces a permutation on the finite set of colors. If $\mathcal{T}$ is $k$-valent, then a coloring of $\mathcal{T}$ with $k$ colors is said to be precise if no two tiles of $\mathcal{T}$ sharing the same vertex have the same color. In this work, we ob...

The {\em d} -- dimensional $n$ -- colour lattice ${\bf \Bbb{L}}^d$ with modular sublattices are studied, when the only one crystallographic type of sublattices does exist and the only one of the colours occupies a sublattice, which is still invariant under $k$--fold rotation $C_k$. Such kind of colouring always preserves an equal fractions of the colours composed ${\bf \Bbb{L}}^d$. The $n$ -- colour lattice with modular sublattices allow to exist the crystallographic rotation...

We solve a long-standing problem by enumerating the number of non-degenerate Desargues configurations. We extend the result to the more difficult case involving Desargues blockline structures in Section 8. A transparent proof of Desargues theorem in the plane and in space is presented as a by-product of our methods.

Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice $\Lambda_1$ and to a sublattice $\Lambda_2$ may differ. Here, we examine the coloring of $\Lambda_1$ induced by $\Lambda_2$ to identify how the coincidence indices with respect to $\Lambda_1$ and to $\Lambda_2$ are related. This leads to a generalization of the notion of color symmetries of lattices to what we call ...

This contains Part I of the book: Congruence lattices of finite lattices, which covers about 80 years of research and more than 250 papers.

In this paper, we study the unimodular equivalence of sublattices in an $n$-dimensional lattice. A recursive procedure is given to compute the cardinalities of the unimodular equivalent classes with the indices which are powers of a prime $p$. We also show that these are integral polynomials in $p$. When $n=2$, the explicit formulae of the cardinalities are presented depending on the prime decomposition of the index $m$. We also give an explicit formula on the number of co-cy...

Finite projective geometry underlies the structure of the 35 square patterns in R. T. Curtis's Miracle Octad Generator, and also explains the surprising symmetry properties of some simple graphic designs.