Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings

For a poset P, a subposet A, and an order preserving map F from A into the real numbers, the marked order polytope parametrizes the order preserving extensions of F to P. We show that the function counting integral-valued extensions is a piecewise polynomial in F and we prove a reciprocity statement in terms of order-reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler (2011) and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty (2007).Comment: 17 pages, 10 figures; V2: minor changes (including title); V3: examples included (suggested by referee), to appear in "SIAM Journal on Discrete Mathematics

Computing the bounded subcomplex of an unbounded polyhedron

We study efficient combinatorial algorithms to produce the Hasse diagram of the poset of bounded faces of an unbounded polyhedron, given vertex-facet incidences. We also discuss the special case of simple polyhedra and present computational results.Comment: 16 page

The Lagrange spectrum of a Veech surface has a Hall ray

We study Lagrange spectra of Veech translation surfaces, which are a generalization of the classical Lagrange spectrum. We show that any such Lagrange spectrum contains a Hall ray. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics in the corresponding Teichm\"uller disk and prove a formula which allows to express large values in the Lagrange spectrum as sums of Cantor sets.Comment: 30 pages, 5 figures. Minor revisio

Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in \mathbb {R}^{3}, as hyperelliptic curves, and as \mathbb {CP}^{1} modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller

COMBINATORIAL INSCRIBABILITY OBSTRUCTIONS FOR HIGHER DIMENSIONAL POLYTOPES

For 3-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every f-vector of 3-polytopes, there exists an inscribable polytope with that f-vector. For higher dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the 4-dimensional cyclic polytopes with at least eight vertices - all of whose faces are inscribable - are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic 4-polytopes with up to seven vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that d-dimensional cyclic polytopes with at least d+4 vertices are not circumscribable, and that no dual of a neighborly 4-polytope with eight vertices, that is, no polytope with f-vector (20,40,28,8), is inscribable

Polytopality of simple games

The Bier sphere $Bier(\mathcal{G}) = Bier(K) = K\ast_\Delta K^\circ$ and the canonical fan $Fan(\Gamma) = Fan(K)$ are combinatorial/geometric companions of a simple game $\mathcal{G} = (P,\Gamma)$ (equivalently the associated simplicial complex $K$), where $P$ is the set of players, $\Gamma\subseteq 2^P$ is the set of wining coalitions, and $K = 2^P\setminus \Gamma$ is the simplicial complex of losing coalitions. We characterize roughly weighted majority games as the games $\Gamma$ such that $Bier(\mathcal{G})$ (respectively $Fan(\Gamma)$) is canonically polytopal (canonically pseudo-polytopal) and show, by an experimental/theoretical argument, that all simple games with at most five players are polytopal

Generating high node congruence in freeform structures with Monge's Surfaces

International audienceThe repetition of elements in a free-form structure is an important topic for the cost rationalization process of complex projects. Although nodes are identified as a major cost factor is steel grid shells, little research has been done on node repetition. This paper proposes a family of shapes, called isogonal moulding surfaces, having high node congruence, flat panels and torsion-free nodes. It is shown that their generalization, called Monge's surfaces, can be approximated by surfaces of revolution, yielding high congruence of nodes, panels and members. These shapes are therefore interesting tools for geometrically-constrained design approach