相关论文: Representing the Sporadic Archimedean Polyhedra as…

In this article, we found all simple closed geodesics on regular spherical octahedra and spherical cubes. In addition, we estimate the number of simple closed geodesics on regular spherical tetrahedra.

A beautiful result of Br\"ocker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most…

We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover…

Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by…

We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact…

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…

This paper contains a classication of the regular minimal abstract polytopes that act as covers for the convex polyhedral prisms and antiprisms. It includes a detailed discussion of their topological structure, and completes the enumeration…

Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational…

Unlike the situation in the classical theory of convex polytopes, there is a wealth of semi-regular abstract polytopes, including interesting examples exhibiting some unexpected phenomena. We prove that even an equifacetted semi-regular…

In classical differential geometry, a central question has been whether abstract surfaces with given geometric features can be realized as surfaces in Euclidean space. Inspired by the rich theory of embedded triply periodic minimal…

Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates ("in $\mathbb Z^3$"), we look at the problem of characterizing all regular polyhedra (Platonic Solids) with the…

This is a chapter in an upcoming Tamari Festscrift. Permutahedra are a class of convex polytopes arising naturally from the study of finite reflection groups, while generalized associahedra are a class of polytopes indexed by finite…

Peter McMullen has developed a theory of realizations of abstract regular polytopes, and has shown that the realizations up to congruence form a pointed convex cone which is the direct product of certain irreducible subcones. We show that…

We analyze the abstract representations of the groups of rational points of even-dimensional quasi-split special unitary groups associated with quadratic field extensions. We show that, under certain assumptions, such representations have a…

Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these…

We propose a simple formula for the coordinates of the vertices of the Stasheff polytope (associahedron) and we compare it to the permutohedron.

The theory of elliptic pairs, as investigated in a paper by Castravet, Laface, Tevelev, and Ugaglia, provides useful conditions to determine polyhedrality of the pseudo-effective cone, which give rise to interesting arithmetic questions…

Given a connected graph G with p vertices and q edges, the G-graphicahedron is a vertex-transitive simple abstract polytope of rank q whose edge-graph is isomorphic to a Cayley graph of the symmetric group S_p associated with G. The paper…

We give combinatorial models for complex, smooth, non-spherical, generic, irreducible representations of the group G=PGL(2,F), where F is a non-archimedean locally compact field. They use the graphs X_k lying above the tree of G, introduced…

This paper is devoted to the study of the $m$-point homogeneity property for the vertex sets of polytopes in Euclidean spaces. In particular, we present the classifications of $2$-point and $3$-point homogeneous polyhedra in $\mathbb{R}^3$.