Related papers: Skeletal Snub Polyhedra in Ordinary Space, I
While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the…
Spherical $t$-designs are finite point sets on the unit sphere that enable exact integration of polynomials of degree at most $t$ via equal-weight quadrature. This concept has recently been extended to spherical $t$-design curves by the use…
A convex polyhedron $P$ is $k$-equiprojective if all of its orthogonal projections, i.e., shadows, except those parallel to the faces of $P$ are $k$-gon for some fixed value of $k$. Since 1968, it is an open problem to construct all…
Consider an orthogonal polyhedron, i.e., a polyhedron where (at least after a suitable rotation) all faces are perpendicular to a coordinate axis, and hence all edges are parallel to a coordinate axis. Clearly, any facial angle and any…
An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, in particular convex geometry, optimization, and algebraic…
We prove that any finite polyhedral manifold in 3D can be continuously flattened into 2D while preserving intrinsic distances and avoiding crossings, answering a 19-year-old open problem, if we extend standard folding models to allow for…
In view of solving problems of geometric realizability of polyhedra with given geometric constraints, we describe the space of geometric realizations of a simply-connected triangulated euclidean polyhedron in $\mathbb{R}^3$ up to similarity…
An arrangement of pseudocircles is a finite set of oriented closed Jordan curves each two of which cross each other in exactly two points. To describe the combinatorial structure of arrangements on closed orientable surfaces, in (Linhart,…
Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries.…
A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local…
We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry,…
The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the…
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…
In this article we establish the relation between the spines of 3-manifolds and the polyhedra with identified faces. We do this by showing that the spines of the closed, connected, orientable 3-manifolds can be presented through polyhedra…
Given a polytopal complex $X$, we examine the topological complement of its $k$-skeleton. We construct a long exact sequence relating the homologies of the skeleton complements and links of faces in $X$, and using this long exact sequence,…
It is shown that every orthogonal terrain, i.e., an orthogonal (right-angled) polyhedron based on a rectangle that meets every vertical line in a segment, has a grid unfolding: its surface may be unfolded to a single non-overlapping piece…
A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather…
Polynomial splines are ubiquitous in the fields of computer aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient…
A new parametric surface representation is proposed that interpolates the vertices of a given closed mesh of arbitrary topology. Smoothly connecting quadrilateral patches are created by blending local, multi-sided quadratic interpolants. In…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…