arXiv research

Hexagonal tilings minimize perimeter with unequal volumes.

A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these for…

There is a natural generalization of domino tilings to tilings of a polygon by hexagons, or, dually, configurations of oriented curves that meet in triples. We show exactly when two such tilings can be connected by a series of moves analogous to the domino flip move. The triple diagrams that result have connections to …

The study examines polyhedra with hexagonal and triangular faces, focusing on their 3-regular planar graphs.

The effectiveness of Convolutional Neural Networks stems in large part from their ability to exploit the translation invariance that is inherent in many learning problems. Recently, it was shown that CNNs can exploit other invariances, such as rotation invariance, by using group convolutions instead of planar convoluti…

Hexagonal diagrams link complex curves in $\mathbb{CP}^2$ to minimal genus surfaces.

An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. We prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right, or 120-isosceles triangle; a 120-rhombus; a 60-90-120 kite; or a regular hexagon…

Inspired by the human visual perception system, hexagonal image processing in the context of machine learning deals with the development of image processing systems that combine the advantages of evolutionary motivated structures based on biological models. While conventional state-of-the-art image processing systems o…

New hexagonal circular 3-webs with reducible curves classified.

Classifies hexagonal circular 3-webs with cubic polar curves.

We present a method for scalable and fully 3D magnetic field simultaneous localisation and mapping (SLAM) using local anomalies in the magnetic field as a source of position information. These anomalies are due to the presence of ferromagnetic material in the structure of buildings and in objects such as furniture. We …

Study of graphs from hexagon decompositions of surfaces.

In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulation and all inner angles…

Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.

We study the geometry of oriented right-angled hexagons in H^4, the hyperbolic 4-space, via Clifford numbers or quaternions. We show how to augment alternate sides of such a hexagon so that for the non-augmented sides, we can define quaternion half side-lengths whose angular parts are obtained from half the Euler angle…

Study hexagonal network evolution under curvature flow.

Hexagonal norm double bubble problem solved with minimal configurations.

The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elem…

Improved bounds for knot crossings in different mosaic patterns.

We provide a complete classification of hexagonal singular 3-web germs in the complex plane, satisfying the following two conditions: 1) the Chern connection remains holomorphic at the singular point, 2) the web admits at least one infinitesimal symmetry at this point. As a by-product, a classification of hexagonal wei…

For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components …

This note characterizes monohedral tilings of regular polygons with up to three tiles.

The study of tiling homology on flat surfaces, proving impossibility of certain tilings.

Research finds bounds for knots in hexagonal lattice and classifies 11-stick knots.

We show the rigidity of the hexagonal Delaunay triangulated plane under Luo's PL conformality. As a consequence, we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.

New tiles allow efficient knot mosaics for small knots.

Rep-tiles fill cubes in any dimension.

Study tiling spaces over irrational tori using diffeological classification.

Shellable tilings on simplicial complexes help understand their structure.

In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly constructed highly singular symplectic spaces and we show that they are diffeomorphic…

The study classifies tilings of the sphere by congruent quadrilaterals.

New method constructs tilings of the plane using directed edges and alignments.

New tile types for knots and links reduce complexity.

Study on tilings of the plane with two types of tiles of varying areas.

Proves rigidity of circle packings in the plane, generalizing previous work.

New spectral sequences derived from shellable tilings.

The Phi- relationship also known as Phi-factor appears in a number of lattice structures, mostly considering the lines within several separate circles or polygons. The paper considers a regular hexagonal tessellation as a lattice with the highest specific mechanical stiffness.

The main goal of this paper is to define a 1-1 correspondence between between substitution tilings constructed by inflation and the arithmetic of positional representation in the underlying real vector space. It introduces a generalization of inflationary tessellations to equivalence classes of tiles. Two tiles belong …

We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…

New geometric perspective for optimal learning on hexagonal structures.

We determine the topology of the moduli space of periodic tilings of the plane by parallelograms. To each such tiling, we associate combinatorial data via the zone curves of the tiling. We show that all tilings with the same combinatorial data form an open subset in a suitable Euclidean space that is homotopy equivalen…

Soft cells fill space without gaps, derived from minimal surfaces and deformed using edge bending.

Softens tilings in 3D space, proving conjectures about polyhedral tilings.

Paper proves corner connection tiles can represent knots with fewer tiles.

This paper classifies all 3D rep-tiles up to homeomorphism.

We give an infinite family of knots such that for any given $r \geq 3$, the family contains a knot which can be embedded on a hexagonal $r$-mosaic, but cannot fit on a hexagonal $r$-mosaic in an embedding that achieves its crossing number. This extends the rectangular mosaic result of Ludwig, Evans, and Paat. We also i…

In this note we prove that any monohedral tiling of the closed circular unit disc with $k \leq 3$ topological discs as tiles has a $k$-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain t…

This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.