arXiv research

Tessellations cover planes without gaps or overlaps.

Up to isomorphism there are six fixed-point free crystallographic groups in Euclidean Space generated by twists (screw motions). In each case, an orientable 3-manifold is obtained as the quotient of E3 by such a group. The cubic tessellation of E3 induces tessellations on each such manifold. These tessellations of the …

By regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we pr…

This paper is the second part of a two-part study of an elementary functorial construction of tesselated surfaces from finite groups. This elementary construction was discussed in the first part and generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to…

We present an explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F(n) embedded in CP3 defined by the equation z0^n + z1^n + z2^n + z3^n = 0 in the standard homogeneous coordinates [z0, z1, z2, z3], where n is any positive integer. Note that F(4) in particular is a K3 surface. Our tessellation…

The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.

A key problem in location-based modeling and forecasting lies in identifying suitable spatial and temporal resolutions. In particular, judicious spatial partitioning can play a significant role in enhancing the performance of location-based forecasting models. In this work, we investigate two widely used tessellation s…

In this paper, we investigate the significance of choosing an appropriate tessellation strategy for a spatio-temporal taxi demand-supply modeling framework. Our study compares (i) the variable-sized polygon based Voronoi tessellation, and (ii) the fixed-sized grid based Geohash tessellation, using taxi demand-supply GP…

This paper is the first part in a 2 part study of an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this construction results in a collection of compact, connected, oriented tesselated smooth su…

Develops a new non-adversarial framework for better generative models.

The paper generalizes Farey tessellation to 3D hyperbolic space.

The Delaunay tessellation of a locally finite subset of hyperbolic space is constructed using convex hulls in Euclidean space of one higher dimension. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily modified from the Euclidean setting) of the empty circumsp…

For every half-translation surface with marked points $(M,Σ)$, we construct an associated tessellation $Π(M,Σ)$ of the Poincaré upper half plane whose tiles have finitely many sides and area at most $π$. The tessellation $Π(M,Σ)$ is equivariant with respect to the action of $\mathrm{PSL}(2,\mathbb{R})$, and invariant w…

Bayesian model captures mean and variance of response variables.

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.

New algorithm optimizes tessellated kernels for larger datasets and improved performance.

To each once-punctured-torus bundle, $T_φ$, over the circle with pseudo-Anosov monodromy $φ$, there are associated two tessellations of the complex plane: one, $Δ(φ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the…

We classify the normal subgroups K of the tetrahedral group Delta=[3,5,3]^+, the even subgroup of the Coxeter group Gamma=[3,5,3], with Delta/K isomorphic to a finite simple group L_2(q). We determine their normalisers N(K) in the isometry group of hyperbolic 3-space H^3, the isometry groups N(K)/K of the associated hy…

A new method learns quantization boundaries in continuous space using tessellation.

A new method for Bayesian optimization uses Voronoi tessellation candidates to reduce search time.

It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting …

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…

Accurate taxi demand-supply forecasting is a challenging application of ITS (Intelligent Transportation Systems), due to the complex spatial and temporal patterns. We investigate the impact of different spatial partitioning techniques on the prediction performance of an LSTM (Long Short-Term Memory) network, in the con…

A new proof shows how to characterize maps using simple geometry.

New knot quandle structure for twist-spun trefoils discovered.

The paper proves actions of lattices in higher rank groups have cost one.

Constructs constant mean curvature surfaces using geometric flow.

With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic $q$-equivelar triangulations of orientable and non-orientable surfaces for every $q=3k$, $k\geq 2$, and every $q=3k+1$, $k\geq 3$. Series of cy…

A Riemannian symmetric space is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. On such spaces, we introduce the notion of a distributional lattice, generalizing the notion of lattice. Distributional lattices exist in any Riemannian symmetric space: th…

Study jigsaw constructions of hyperbolic lattices and answer questions on arithmeticity and pseudomodularity.

The analytic torsion is computed on fixed-point free and non fixed-point free factors (tessellations) of the three--sphere. We repeat the standard computation on spherical space forms (Clifford-Klein spaces) by an improved technique. The transformation to a simpler form of the spectral expression of the torsion on sphe…

Explicitly constructed 5-manifolds tessellated by prisms.

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular pol…

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a "tessellation" if there is a discrete subgroup D of G, such that D acts properly discontinuously on G/H, and the double-coset space D\G/H is compact. Note that if either H or G/H is compact, then…

Hexagonal tilings minimize perimeter with unequal volumes.

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 give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.

New upper bound for Cheeger constant of hyperbolic surfaces.

Generalizes crystallographic properties to all dimensions.

A finite subset S of a closed hyperbolic surface F canonically determines a "centered dual decomposition" of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection …

Quantized Variational Inference improves ELBO optimization with fast convergence.

We describe a general family of curved-crease folding tessellations consisting of a repeating "lens" motif formed by two convex curved arcs. The third author invented the first such design in 1992, when he made both a sketch of the crease pattern and a vinyl model (pictured below). Curve fitting suggests that this init…

Classifies 3-braids from choreographic motions on Lissajous curves, linking them to mapping classes and geodesics.

We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic tessellations. For the octahedra…

Study of circle homeomorphisms with square summable diamond shears.

New random forest variants achieve optimal performance in high dimensions.

Constructs an explicit cycle in arithmetic group cohomology.

Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two different metrics on a surface have no geodesic arcs in common, but in special ca…