Related papers: Polycubes via Dual Loops
A $\textit{polygonal curve}$ is a collection of $m$ connected line segments specified as the linear interpolation of a list of points $\{p_0, p_1, \ldots, p_m\}$. These curves may be obtained by sampling points from an oriented curve in…
Given fixed distinct points $A, B, C, D$, we examine properties of the locus of points $X$ for which $(XA, XC)$, $(XB, XD)$ are isogonal. This locus is a cubic curve circumscribing $ABCD$. We characterize all possible such cubics $\mathcal…
We present a novel, log-radius profile representation for convex curves and define a new operation for combining the shape features of curves. Unlike the standard, angle profile-based methods, this operation accurately combines the shape…
In this paper, several conjectures proposed in [2] are studied, involving the equivalence and duality of polycyclic codes associated with trinomials. According to the results, we give methods to construct isodual and self-dual polycyclic…
Although little can be gleaned about a loop with the property that its squares are, say, left nuclear ($xx\cdot yz = (xx\cdot y)z$), if its squares are also, say, middle nuclear ($(x\cdot yy)z = x(yy\cdot z)$), then the loop exhibits more…
We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other…
In this work we prove constructively that the complement $\R^n\setminus\pol$ of a convex polyhedron $\pol\subset\R^n$ and the complement $\R^n\setminus\Int(\pol)$ of its interior are regular images of $\R^n$. If $\pol$ is moreover bounded,…
The article studies power complexes and generalized power complexes, and investigates the algebraic structure of their automorphism groups. The combinatorial incidence structures involved are cube-like, in the sense that they have many…
Measuring comodules are defined and shown to provide a useful generalization of the set of maps between modules with a broad range of applications. Three applications are described. Connections on bundles are described in terms of measuring…
We investigate the folding problem that asks if a polygon P can be folded to a polyhedron Q for given P and Q. Recently, an efficient algorithm for this problem has been developed when Q is a box. We extend this idea to regular polyhedra,…
Which surfaces can be realized with two-dimensional faces of the five-dimensional cube (the penteract)? How can we visualize them? In recent work, Aveni, Govc, and Roldan, show that there exist 2690 connected closed cubical surfaces up to…
We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, for example, principal coefficients.
An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron…
Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the…
Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with…
An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the…
An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and…
We introduce an algorithm that conjectures the structure of a permutation class in the form of a disjoint cover of "rules"; similar to generalized grid classes. The cover is usually easily verified by a human and translated into an…
Just how many different connected shapes result from slicing a cube along some of its edges and unfolding it into the plane? In this article we answer this question by viewing the cube both as a surface and as a graph of vertices and edges.…
We describe a bijection between oriented cubes and adjoints of cross-polytopes. This correspondence is used to prove that the real affine cube is, up to reorientation in the same class, the unique oriented cube that is realizable. Moreover,…