Convex polyhedra whose faces are equiangular or composed of such

Depending on the type, considering only the topological structure of the network of faces, and the angles of corresponding faces at corresponding vertices, convex polyhedra in R³, each face of which is equiangular or composed of such, constitute four infinite series (prism, antiprism, and two types of truncated antiprisms); outside of this series, there are only a finite number of types. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 111–112, 1974.     

Chiral Polyhedra in Ordinary Space, I

Chiral polyhedra in ordinary euclidean space E³ are nearly regular polyhedra; their geometric symmetry groups have two orbits on the flags, such that adjacent flags are in distinct orbits. This paper completely enumerates the discrete infinite chiral polyhedra in E³ with finite skew faces and finite skew vertex-figures. There are several families of such polyhedra of types {4,6}, {6,4} and {6,6}. Their geometry and combinatorics are discussed in detail. It is also proved that a chiral polyhedron in E³ cannot be finite. Part II of the paper will complete the classification of all chiral polyhedra in E³. All chiral polyhedra not described in Part I have infinite, helical faces and again occur in families. So, in effect, Part I enumerates all chiral polyhedra in E³ with finite faces.     

Chiral Polyhedra in Ordinary Space, II

A chiral polyhedron has a geometric symmetry group with two orbits on the flags, such that adjacent flags are in distinct orbits. Part I of the paper described the discrete chiral polyhedra in ordinary euclidean space E³ with finite skew faces and finite skew vertex-figures; they occur in infinite families and are of types {4,6}, {6,4} and {6,6}. Part II completes the enumeration of all discrete chiral polyhedra in E³. There exist several families of chiral polyhedra of types {∞,3} and {∞,4} with infinite, helical faces. In particular, there are no discrete chiral polyhedra with finite faces in addition to those described in Part I.     

The realization of networks of convex polyhedra with equiangular vertices. 4

A fundamental theorem on closed polyhedra with equiangular vertices is presented. The proof of the theorem was begun in parts 1–3 of this paper. Here, in part 4, we find the polyhedra containing faces of type (4, 4,n) and (4, 5,n)-a total of 44 polyhedra and one infinite series of polyhedra dual to prisms. One table. Seven figures.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 32–47.     

Decompositions and boundary coverings of non-convex fat polyhedra

We show that any locally-fat (or (α,β)-covered) polyhedron with convex fat faces can be decomposed into O(n) tetrahedra, where n is the number of vertices of the polyhedron. We also show that the restriction that the faces are fat is necessary: there are locally-fat polyhedra with non-fat faces that require Ω(n²) pieces in any convex decomposition. Furthermore, we show that if we want the tetrahedra in the decomposition to be fat themselves, then their number cannot be bounded as a function of n in the worst case. Finally, we obtain several results on the problem where we want to only cover the boundary of the polyhedron, and not its entire interior.     

On the trace of polyhedra

We consider a convex polyhedronP standing with one of its faces on a fixed plane. We rotateP into another similar position around any of its edges lying on. We call the trace ofP the set of all pointsA of for whichA coincides with some vertex ofP in some position ofP.We investigate the traces of the regular polyhedra, the rectangular parallelepiped. Furthermore we give a sufficient condition for which the trace of a convex polyhedron is everywhere dense.Supported by CNRS Laboratoire de Mathématiques Discrètes, Marseille.     

Zur Eulerschen Charakteristik allgemeiner,insbesondere konvexer Polyeder

We establish the existence of Eulers characteristic χ for general polyhedra P ?R ⁿ and prove, that for a convex polyhedron it takes the value \(\Sigma^{n}_{i=0}(-1)^{i}s_{i}\) , where s _i is the number of those i-dimensional faces of P, for which S ? P. The main reason for this representation is the non-trivial fact, that if S ? P for some face S of a convex polyhedron P, then S is convex. Furthermore we extend the Euler-Schläfli theorem to include all closed and convex (but not necessarily bounded) polyhedra.     

Realization of nets of convex polyhedra with equiangular vertices. 5

A complete list of convex polyhedra with equiangular vertices up to combinatorial equivalence is found. In the list are 104 closed polyhedra, 26 infinite polyhedra, and three infinite series — cones and polyhedra dual to prisms and antiprisms. One table.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 22–36.     

Calculus of convex polyhedra and polyhedral convex functions by utilizing a multiple objective linear programming solver

ABSTRACT

The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for one convex polyhedron are, for example, the polar, the conical hull and the image under affine transformation. The concept of a P-representation of a convex polyhedron is introduced. It is shown that many polyhedral calculus operations can be expressed explicitly in terms of P-representations. We point out that all the relevant computational effort for polyhedral calculus consists in computing projections of convex polyhedra. In order to compute projections we use a recent result saying that multiple objective linear programming (MOLP) is equivalent to the polyhedral projection problem. Based on the MOLP solver bensolve a polyhedral calculus toolbox for Matlab and GNU Octave is developed. Some numerical experiments are discussed.     

On the space-filling enneahedra

A space-filling polyhedron is one whose replications can be packed to fill three-space completely. The space-filling polyhedra of four to eight faces have been previously reported. The search is here extended to the convex space-fillers of nine faces. The number of types is found to be at least 40.     

Reprint of: Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.     

Polyhedral hyperbolic metrics on surfaces

Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space and a group G of isometries of such that the induced metric on the quotient P/G is isometric to g. Moreover, the pair (P, G) is unique among a particular class of convex polyhedra.      

Positivity of curvature and convexity of faces

Two-dimensional polyhedra homeomorphic to closed two-dimensional surfaces are considered in the three-dimensional Euclidean space. While studying the structure of an arbitrary face of a polyhedron, an interesting particular case is revealed when the magnitude of only one plane angle determines the sign of the curvature of the polyhedron at the vertex of this angle. Due to this observation, the following main theorem of the paper is obtained: If a two-dimensional polyhedron in the three-dimensional Euclidean space is isometric to the surface of a closed convex three-dimensional polyhedron, then all faces of the polyhedron are convex polygons.     

Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.     

Five-valent convex polyhedra with prescribed faces

Euler's relation when applied to 5-valent convex polyhedra leads to an equation for the number of triangular faces in terms of the number of other faces. Below we prove that conversely, if a finite sequence (p₃, p₄, …, p_k) of non-negative integers satisfies that equation, and if p₄ is large enough, then there exists a 5-valent convex polyhedron with p_kk-gonal faces for each k.     

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.     

Determination of the element numbers of the regular polytopes

Let ?? be a set of n-dimensional polytopes. A set ?? of n-dimensional polytopes is said to be an element set for ?? if each polytope in ?? is the union of a finite number of polytopes in ?? identified along (n ? 1)-dimensional faces. The element number of the set ?? of polyhedra, denoted by e(??), is the minimum cardinality of the element sets for ??, where the minimum is taken over all possible element sets ${\Omega \in \mathcal{E}(\Sigma)}$ . It is proved in Theorem 1 that the element number of the convex regular 4-dimensional polytopes is 4, and in Theorem 2 that the element numbers of the convex regular n-dimensional polytopes is 3 for n ?? 5. The results in this paper together with our previous papers determine completely the element numbers of the convex regular n-dimensional polytopes for all n ?? 2.