Related papers: The net created from the Penrose tiling is biLipsc…

We prove that if a Bessel sequence in a Hilbert space, that is indexed by a countably infinite group in an invariant manner, can be partitioned into finitely many Riesz basic sequences, then each of the sets in the partition can be chosen…

A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the…

Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent…

We consider incomplete tilings of the equilateral triangle of edge length n that is subdivided into n^2 regular equilateral smaller unit triangles. Pairs of the unit triangles that share a side may be converted into lozenges, leaving some…

This paper presents a new bi-Lipschitz invertible neural network, the BiLipNet, which has the ability to smoothly control both its Lipschitzness (output sensitivity to input perturbations) and inverse Lipschitzness (input distinguishability…

We present an alternative geometric representation for the eleven Archimedean lattices, in which each site and bond is uniquely labeled by an ordered pair of integers and characterized via a modular function. This structured labeling…

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…

In algebraic number theory, the finiteness of the Picard group of an order in a number field is generally proved via a lattice argument: the order forms a lattice and every ideal class contains an integral ideal with a small enough non-zero…

We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained as a result of tile…

We prove that universal differentiability sets in Euclidean spaces possess distinctive structural properties. Namely, we show that any universal differentiability set contains a `kernel' in which the points of differentiability of each…

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each…

Incorporating designs into the tiles that form tessellations presents an interesting challenge for artists. Creating a viable MC Escher like image that works esthetically as well as functionally requires resolving incongruencies at a tile's…

Every lattice is isomorphic to a lattice whose elements are sets of sets, and whose operations are intersection and an operation extending the union of two sets of sets A and B by the set of all sets in which the intersection of an element…

A particular discrete rhombohedral lattice consisting of four symmetrically interlaced cuboctahedral point lattices is described that is interesting because of the high degree of symmetry it exhibits. The four constituent cuboctahedral…

We consider Euclidean lattices spanned by images of algebraic conjugates of an algebraic number under Minkowski embedding, investigating their rank, properties of their automorphism groups and sets of minimal vectors. We are especially…

The purpose of this article is to view Penrose rhombus tilings from the perspective of symplectic geometry. We show that each thick rhombus in such a tiling can be naturally associated to a highly singular 4-dimensional compact symplectic…

We comment on the set of visible points of a lattice and its Fourier transform, thus continuing and generalizing previous work by Schroeder and Mosseri. A closed formula in terms of Dirichlet series is obtained for the Bragg part of the…

In a recent article by C. Porrier, A. Blondin Mass\'e and A. Goupil, a first bi-infinite fully leafed induced subcaterpillar of Penrose P2 tilings is presented. In this paper, we formally construct this caterpillar for the first time. We…

We consider the direction set determined by various subsets $E$ of Euclidean space and show that there is a trichotomy: Either (i) The subset is the graph of a Lipschitz function and the direction set is not dense in the sphere, (ii) The…

A lattice in Euclidean $d$-space is called well-rounded if it contains $d$ linearly independent vectors of minimal length. This class of lattices is important for various questions, including sphere packing or homology computations. The…