Family Of Polyhedra Research Articles

108.35 Analogues of the 12 pentagons theorem for other families of polyhedra

108.35 Analogues of the 12 pentagons theorem for other families of polyhedra

Numerical semigroups, polyhedra, and posets I: the group cone

Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first introduced by Kunz, has integer points in bijection with numerical semigroups of fixed multiplicity, and another, introduced by Hellus and Waldi, has integer points corresponding to oversemigroups of numerical semigroups with two generators. In this paper, we provide a combinatorial framework from which to study both families of polyhedra. We introduce a new family of polyhedra called group cones, each constructed from some finite abelian group, from which both of the aforementioned families of polyhedra are directly determined but that are more natural to study from a standpoint of polyhedral geometry. We prove that the faces of group cones are naturally indexed by a family of finite posets, and illustrate how this combinatorial data relates to semigroups living in the corresponding faces of the other two families of polyhedra.

Fibonacci Numbers, Integer Compositions, and Nets of Antiprisms

All geometrically distinct nets (edge-unfoldings) are explicitly constructed for the infinite family of polyhedra known as uniform antiprisms. The Fibonacci numbers are shown to count the number of nets of the n-antiprism that have point symmetry. (The n-antiprism refers to the antiprism whose surface consists of two regular n-gons separated by a band of 2n equilateral triangles.) Smaller evenly indexed Fibonacci numbers count certain subfamilies of these nets. Well-known formulas for sums of alternately indexed Fibonacci numbers motivate and become accessories to our constructions and proofs. For counting and constructing the symmetric nets, we first use a known result connecting compositions of integers and Fibonacci numbers. In an alternative proof, we illustrate a striking self-similarity among the counts of certain natural subfamilies of the symmetric nets, which is used to generate an efficient recursive construction of all the symmetric nets of the n-antiprism from those of the -antiprism. (This second proof is found in the online supplement to this article.) Finally, each nonsymmetric net of the n-antiprism is constructed by combining a pair of distinct symmetric nets of the n-antiprism.

Convex polyhedra of distributions preserved by operations over a finite field

We construct families of polyhedra in the space of probability distributions over a finite field, which are preserved, i.e., adding or multiplying independent random variables with distributions from the constructed set, the resulting distribution belongs to the same set.

Continuous Flattening of a Regular Tetrahedron with Explicit Mappings

We use the terminology polyhedron for a closed polyhedral surface which is permitted to touch itself but not self-intersect (and so a doubly covered polygon is a polyhedron). A flat folding of a polyhedron is a folding by creases into a multilayered planar shape ([7], [8]). A. Cauchy [4] in 1813 proved that any convex polyhedron is rigid: precisely, if two convex polyhedra P, P ′ are combinatorially equivalent and their corresponding faces are congruent, then P and P ′ are congruent. By removing the condition of convexity, R. Connelly [5] in 1978 gave an example of a (non-convex) flexible polyhedron: precisely, there is a continuous family of polyhedra {Pt : 0 ≤ t ≤ 1} such that for every t 6= 0, the corresponding faces of P0 and Pt are congruent while polyhedra P0 and Pt are not congruent. (See also [6].) After then I. Sabitov [15] in 1998 proved that the volume of any polyhedron is invariant under flexing: precisely, if there is a continuous family of polyhedra {Pt : 0 ≤ t ≤ 1} such that, for every t, the corresponding faces of P0 and Pt are congruent, then the volumes P0 and Pt are equal for all 0 ≤ t ≤ 1. (See also [14].) A. Milka [12] in 1994 showed that any polyhedron admits a continuous (isometric) deformation by using moving edges, and that all Platonic polyhedra can be changed in

Characterizing and recognizing generalized polymatroids

Generalized polymatroids are a family of polyhedra with several nice properties and applications. One property of generalized polymatroids used widely in existing literature is “total dual laminarity;” we make this notion explicit and show that only generalized polymatroids have this property. Using this we give a polynomial-time algorithm to check whether a given linear program defines a generalized polymatroid, and whether it is integral if so. Additionally, whereas it is known that the intersection of two integral generalized polymatroids is integral, we show that no larger class of polyhedra satisfies this property.

Families of polyhedra

Families of polyhedra

A note on the relationship between the graphical traveling salesman polyhedron, the Symmetric Traveling Salesman Polytope, and the metric cone

In this short communication, we observe that the Graphical Traveling Salesman Polyhedron is the intersection of the positive orthant with the Minkowski sum of the Symmetric Traveling Salesman Polytope and the polar of the metric cone. This follows almost trivially from known facts. There are nonetheless two reasons why we find this observation worth communicating: It is very surprising; it helps us understand the relationship between these two important families of polyhedra.

Tight-binding electronic spectra on graphs with spherical topology: I. The effect of amagnetic charge

This is the first of two papers devoted to tight-binding electronic spectra on graphs withthe topology of the sphere. In this work the one-electron spectrum is investigated as afunction of the radial magnetic field produced by a magnetic charge sitting at the centre ofthe sphere. The latter is an integer multiple of the quantized magnetic charge of the Diracmonopole, that integer defining the gauge sector. An analysis of the spectrum is carried out forthe five platonic solids (tetrahedron, cube, octahedron, dodecahedron and icosahedron), theC60 fullerene and two families of polyhedra, the diamonds and the prisms. Except for thefullerene, all the spectra are obtained in closed form. They exhibit a rich pattern ofdegeneracies. The total energy at half-filling is also evaluated in all the examples as afunction of the magnetic charge.

Nesting of fullerenes and Frank–Kasper polyhedra

The duality relationship between fullerenes and Frank-Kasper polyhedra suggests that these two families of polyhedra appear nested in solid state structures. Magic numbers, described by simple mathematical relationships, identify four families of fullerenes with tetrahedral and icosahedral symmetries.

A New Method to Obtain and Define Regular Polyhedra

The construction method established in this working, 'Umbela Manipulation', let us obtain 'Isotropic Polyhedra' with a same metric property. This method sets up 3 parameters to get and define a family of polyhedra.

Real Stable, Stable, And Not-So Stable Polyhedra: All Stemming From A New Family

A description of a Family of Polyhedra is given where the parent forms are the five regular polyhedra. The facial planes of the parent forms are subdivided into right triangles and by a series of rearrangements are allowed to move out of plane thus creating new volumetric forms while maintaining the same surface area. The axes of symmetry of the parent polyhedra are preserved. One new form is less stable than the parent and the other is more stable than the parent; thus giving rise to a family of real stable, stable and not-so stable polyhedra. Illustrations of all fifteen polyhedra in the family are given along with tables describing several of their geometric properties. The influence of precision on geometrical stability will also be demonstrated as related to architectural applications.

A lift-and-project cutting plane algorithm for mixed 0–1 programs

We propose a cutting plane algorithm for mixed 0---1 programs based on a family of polyhedra which strengthen the usual LP relaxation. We show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point. This cut is found through the solution of a linear program that has about twice the size of the usual LP relaxation. A lifting step is used to reduce the size of the LP's needed to generate the cuts. An additional strengthening step suggested by Balas and Jeroslow is then applied. We report our computational experience with a preliminary version of the algorithm. This approach is related to the work of Balas on disjunctive programming, the matrix cone relaxations of Lovasz and Schrijver and the hierarchy of relaxations of Sherali and Adams.

The Symbolism of Polyhedra in Space Structures

The aim of this paper is to give some symbolic elements on Space Structures and more precisely on Polyhedra and Space Structures. This specific, non scientific approach needs to recall some few notions on symbol concept, before developing some aspects of the symbolism of Polyhedra in Space Structures. The Plato's cosmogony which is the subject of the Timaeus is based on the symbolic constitution of regular polyhedra: description of their interrelations, notions on duality, ternary and harmony are given in the second part of this contribution. It can be shown that duality is the basis of the works of Maxwell, Cremona, Le Ricolais when they established their theory for determining forces in reticulated structures. Among the whole family of polyhedra two of them, the cuboctahedron and the icosahedron have been used by the designers: multiplication of the first one and/or its derivatives, division of the faces of the second one are the fundamental operations used by designers for achieving some famous Space Structures.

Polyhedra of three quasilattices associated with the icosahedral group

A systematic approach is presented for the construction of families of polytopes associated with a given group G and subgroup H. This procedure is applied to the icosahedral group and its three dihedral subgroups yielding three families of polyhedra in E3. These families form the cells of three types of quasilattices associated with the icosahedral group, and hence are candidates for modelling quasicrystal structures. Some of the polyhedra are illustrated.

Common secants for families of polyhedra

Common secants for families of polyhedra