Ten colours in quasiperiodic and regular hyperbolic tilings

Counting perfect colourings of plane regular tilings

A first step in investigating colour symmetries of periodic and nonperiodic patterns is determining the number of colours which allow perfect colourings of the pattern under consideration. A perfect colouring is one where each symmetry of the uncoloured pattern induces a global permutation of the colours. Two cases are distinguished: Either perfect colourings with respect to all symmetries, or with respect to orientation preserving symmetries only (no reflections). For the important class of colourings of regular tilings (and some Laves tilings) of the Euclidean or hyperbolic plane, this mainl…

Planar Rosa：一类具有 $2n$ 阶旋转对称性的准周期替换离散平面平铺

Planar Rosa : a family of quasiperiodic substitution discrete plane tilings with $2n$-fold rotational symmetry

提出 Planar Rosa 替换平铺族，证明其具有 $2n$ 阶旋转对称性且满足离散平面条件，并指出 Sub Rosa 平铺不满足该条件，完善了奇偶 $n$ 的结果。

Colourings of aperiodic tilings

We find explicit optimal vertex, edge and face coulourings for the chair tiling, the Ammann--Beenker tiling, the rational pinwheel tiling and the pinwheel tiling.

单面体平铺的着色：缺陷与晶界

Colouring monohedral tilings: defects and grain boundaries

本文证明了存在周期性的单面体平铺和有限有色种子，迫使整个平面产生非周期性着色。

Tiling the plane with hexagons: improved separations for $k$-colourings

It has been common knowledge since 1950 that seven colours can be assigned to tiles of an infinite honeycomb with cells of unit diameter such that no two tiles of the same colour are closer than $d(7)=\frac{\sqrt{7}}{2}$ apart. Various authors have described tilings using $k>7$ colours, giving corresponding values for $d(k)$, but it is generally unknown whether these are the largest possible for a given $k$. Here, for many $k$, we describe tilings with larger values of $d(k)$ than previously reported.

Quasiperiodic bobbin lace patterns

Bobbin lace is a fibre art form in which threads are braided together to form a fabric, often with a very detailed and complex design. In traditional practice, each region of the fabric is filled with a periodic texture. We establish the groundwork for non-periodic lace patterns and present three new quasiperiodic families based on Sturmian words, the Penrose tiling by thick and thin rhombs and the Ammann-bar decoration of the Penrose tiling.