Related papers: Meissner Polyhedra
A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid…
Mesh models are a promising approach for encoding the structure of 3D objects. Current mesh reconstruction systems predict uniformly distributed vertex locations of a predetermined graph through a series of graph convolutions, leading to…
For any finite set $\A$ of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set $\A$, where a marked graph is defined as a…
We suggest a geometric visualization of the process of constructing a triangle with prescribed bisectors that makes the existence of such a triangle geometrically evident.
Certain semi-Riemannian metrics may be decomposed into a Riemannian part and an isochronal part. We use this idea and an idea of Kasner to construct a manifold in 6+1 Minkowski space with a well known metric. The full embedding we display…
We construct a hyperbolic three-manifold with trivial finite type invariants up to a given degree.
We show that on a generic curve, a bundle obtained by successive extensions is stable. We compute the dimension of the set of such extensions. We use degeneration methods specializing the curve to a chain of elliptic components
The isometric embedding of surfaces in three-dimensional space is fundamental to various physical systems, from elastic sheets to programmable materials. While continuous surfaces typically admit unique solutions under suitable boundary…
By taking quotients of a certain tiling of hyperbolic plane / space by certain group actions, we obtain geometric polyhedra / cellulations with interesting symmetries and incidence structure.
We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by…
For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of…
We present a complete 3-dimensional Blaschke-Santal\'o diagram for planar convex bodies with respect to the four classical magnitudes inner and outer radius, diameter and (minimal) width in euclidean spaces.
The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the…
This paper constructs hyperbolic polyhedral metrics via circle packings. We introduce the curvature of circles as a parameter to include all three types of constant curvature curves in the hyperbolic geometry. This provides a unified…
Superintegrable systems in two- and three-dimensional spaces of constant curvature have been extensively studied. From these, superintegrable systems in conformally flat spaces can be constructed by Staeckel transform. In this paper a…
An N=1 supersymmetric extension of the Ruijsenaars-Schneider three-body model is constructed and its integrability is established. In particular, three functionally independent Grassmann-odd constants of the motion are given and their…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
In this work, we will explore some polygons that individually are capable of filling the plane in an aperiodic way. These polygons were recently discovered by some researchers and constitute a great discovery for Mathematics. We will…
We establish that every embedding of a Grassmann graph in a polar Grassmann graph can be reduced to an embedding in a Grassmann graph or to an embedding in the collinearity graph of a polar space. Also, we consider $3$-embeddings, i.e.…
Bourgain and Yehudayoff recently constructed $O(1)$-monotone bipartite expanders. By combining this result with a generalisation of the unraveling method of Kannan, we construct 3-monotone bipartite expanders, which is best possible. We…