Related papers: Patterns and Tracks
We prove that the three-sphere recognition problem lies in the complexity class NP. Our work relies on Thompson's original proof that the problem is decidable [Math. Res. Let., 1994], Casson's version of her algorithm, and recent results of…
We prove the existence of a new algorithm for 3-sphere recognition based on Groebner basis methods applied to the variety of $\text{\em SL}(2,\C)$-representation of the fundamental group. An essential input is a recent result of the second…
Three particles floating on a fluid surface define a triangle. The aim of this paper is to characterise the shape of the triangle, defined by two of its angles, as the three vertices are subject to a complex or turbulent motion. We consider…
We have found some patterns in some triangles.
In this paper the number of ways to glue a surface of genus $g$ has been investigated. We've proven formulas for the number of gluings sphere from three polygons and from two bicolored polygons. Moreover, we've given a new proofs on the…
We introduce an algorithm to determine when a sorting operation, such as stack-sort or bubble-sort, outputs a given pattern. The algorithm provides a new proof of the description of West-2-stack-sortable permutations, that is permutations…
This article presents a model which is capable of learning and abstracting new concepts based on comparing observations and finding the resemblance between the observations. In the model, the new observations are compared with the templates…
The search for universality in random triangulations of manifolds, like those featuring in (Euclidean) Dynamical Triangulations, is central to the random geometry approach to quantum gravity. In case of the 3-sphere, or any other manifold…
We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the…
We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.
A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance…
We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is…
Modeling deformations of a real object is an important task in computer vision, biomedical engineering and biomechanics. In this paper, we focus on a situation where a three-dimensional object is rotationally deformed about a fixed axis,…
This paper lays the foundations of triangulated persistence categories (TPC), which brings together persistence modules with the theory of triangulated categories. As a result we introduce several measurements and metrics on the set of…
We study the double homology associated to triangulated spheres and present two results. First, we explicitly compute the double homology for minimum degree sphere triangulations. Using a spectral sequence argument, we compute the effect of…
Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k…
The paper is devoted to finding the colorings of the edges of the 1-skeleton of triangulations of the 2-sphere in three colors so that for each face all three of its sides have different colors. First, by the method of adding one vertex…
For three-body problem, R.Montgomery [3] proved a reconstruction formula which calculates the overall rotation relating two similar triangle configurations if the initial triangular configuration is similar to the configuration formed at…
Three--dimensional colored triangulations are gluings of tetrahedra whose faces carry the colors 0, 1, 2, 3 and in which the attaching maps between tetrahedra are defined using the colors. This framework makes it possible to generalize the…
In this article we prove some theorems related to the triplets of triangles, homological two by two. These theorems will be used later to build triplets of triangles two by two tri-homological.