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We calculate the Bieri-Neumann-Strebel-Renz invariant $\Sigma^1(G)$ for even Artin groups $G$ with underlying graph $\Gamma$ such that if there is a closed reduced path in $\Gamma$ with all labels bigger than 2 then the length of such path is always odd. We show that $\Sigma^1(G)^c$ is a rationally defined spherical polyhedron.

The shielding performance of novel spherical polyhedral structures illuminated by an electromagnetic plane wave is investigated. The spherical shield screens consist of a juxtaposition of wire conductors and planar metallic elements distributed according to the regular pattern of a spherical polyhedron. The considered 3-D screens represent four groups of structures with fundamental elements of design for attaining frequency selectivity. It is shown that at low frequency, the wire frame screen provides high levels of shielding effectiveness, while its counterpart of metallic surfaces has a similar behavior for higher frequency bands. Moreover, the shielding characteristics of a combined wire-surface and wire-loop spherical surface are reported and compared with the cases of a single screen of wire conductors or metallic surfaces. The reported numerical results illustrate the advantage of using each configuration. Detailed physical interpretations of the obtained numerical results are given based on the operation mechanism of spherical shield structures according to structural-based and pattern-based resonances.

We prove the completeness of the list of closed convex polyhedra in E 3 , that are strongly symmetric with respect to the rotation of the faces . Polyhedron is called symmetric if it has at least one non-trivial rotation axis. All axes intersect at a single point called the center of the polyhedron. All considered polyhedra are polyhedra with the center. A convex polyhedron is called a strongly symmetrical with respect to the rotation of the faces, if each of its faces F has an rotation axis L , intersects the relative interior of F , and L is the rotation axis of the polyhedron. It is obvious that the order of rotation axis of L does not necessarily coincide with the order of this axis, if the face of F regarded as a figure separated from the polyhedron. It has previously been shown, that the requirement of global symmetry of the polyhedron faces the rotation axis can be replaced by the weaker condition of symmetry of the star of each face of the polyhedron: to polyhedron was symmetrical with respect to the rotation of the faces, it is necessary and sufficient that some nontrivial rotation axis of each face, regarded as a figure separated from the polyhedron, is the rotation axis of the star of face. Under the star of face F is understood face itself and all faces have at least one common vertex with F . Given this condition, the definition of the polyhedron strongly symmetric with respect to the rotation of the faces is equivalent to the following: the polyhedron is called a strongly symmetrical with respect to the rotation of the faces , if some non-trivial rotation axis of each face, regarded as a figure separated from the polyhedron, is the rotation axis of the star of face. In the proof of the main theorem on the completeness of the list of this class of polyhedra using the result of the complete listing of the so- called polyhedra of 1st and 2nd class [1]. In this paper we show that in addition to the polyhedra of the 1st and 2nd class, listed in [1], only 8 types of polyhedra belongs to the class of polyhedra stronghly symmetric with respect to the rotation of faces. Seven of this eighteen types are not combinatorially equivalent regular or semi-regular (Archimedean). One type of eight is combinatorially equivalent Archimedean polyhedra, but does not belong to polyhedra of 1st or 2nd class. Turning to the polyhedra, dual strongly symmetrical about the rotation of faces, that is, to the polyhedra, stronghly symmetric about the rotation of polyhedral angles, we get their complete listing. It follows that there are 7 types of polyhedra, highly symmetric with respect to the rotation of polyhedral angles which are not combinatorially equivalent to Gessel bodies. Class of polyhedra stronghly symmetric with respect to the rotation of faces, as well as polyhedra 1st and 2nd class mentioned above can be viewed as a generalization of the class of regular (Platonic) polyhedra. Other generalizations of regular polyhedra can be found in [3],[4], [12]-[15].

As a natural generalization of surfaces of Platonic solids, we define a class of polyhedra, called simple regular polyhedral BP-complexes, as a class of 2-dimensional polyhedral metric complexes satisfying certain conditions on their vertex sets, and we give a complete classification of such polyhedra. They are either the surface of a Platonic solid, a p-dodecahedron, a p-icosahedron, an m -covered regular n -gon for some m ≧ 2 or a complete tripartite polygon.

The aim of the present paper is twofold. On the one hand, the paper will contribute to geometry with the introduction of a new type of polyhedra, and on the other hand the paper adds to kinematics with the synthesis of new linkages based on regular or irregular cyclic polyhedra. Cyclic polyhedra are polyhedra whose faces are cyclic polygons, i.e. the vertices of all of their faces are located on circles. By inserting appropriate “streching stars” into the faces of a cyclic polyhedron and interconnecting them at the vertices by triple gussets, a mechanism, mobilizing the cyclic polyhedron is obtained. If the circles in the cyclic polyhedron all are of equal size, we call it a special cyclic polyhedron, which is in fact a new polyhedron. The mechanism derived from such a special cyclic polyhedron transforms the special cyclic polyhedron mechanically into its dual polyhedron, which is again a special cyclic polyhedron. The synthesis of irregular special cyclic polyhedra and the synthesis of linkages derived on the basis of these special polyhedra is demonstrated in detail. For the linkages derived from regular cyclic polyhedra diverse possibilities of combinations to generate complex mechanisms are finally shown.

Omni-directional cameras have many advantages overconventional cameras in that they have a much wider field-of-view (FOV). Accordingly, several approaches have beenproposed recently to apply convolutional neural networks(CNNs) to omni-directional images for various visual tasks.However, most of them use image representations defined inthe Euclidean space after transforming the omni-directionalviews originally formed in the non-Euclidean space. Thistransformation leads to shape distortion due to nonuniformspatial resolving power and the loss of continuity. Theseeffects make existing convolution kernels experience diffi-culties in extracting meaningful information. This paper presents a novel method to resolve such prob-lems of applying CNNs to omni-directional images. Theproposed method utilizes a spherical polyhedron to rep-resent omni-directional views. This method minimizes thevariance of the spatial resolving power on the sphere sur-face, and includes new convolution and pooling methodsfor the proposed representation. The proposed method canalso be adopted by any existing CNN-based methods. Thefeasibility of the proposed method is demonstrated throughclassification, detection, and semantic segmentation taskswith synthetic and real datasets.

Skeletal polyhedra are discrete structures made up of finite, flat or skew, or infinite, helical or zigzag, polygons as faces, with two faces on each edge and a circular vertex-figure at each vertex. When a variant of Wythoff's construction is applied to the forty-eight regular skeletal polyhedra (Grunbaum-Dress polyhedra) in ordinary space, new highly symmetric skeletal polyhedra arise as "truncations" of the original polyhedra. These Wythoffians are vertex-transitive and often feature vertex configurations with an attractive mix of different face shapes. The present paper describes the blueprint for the construction and treats the Wythoffians for distinguished classes of regular polyhedra. The Wythoffians for the remaining classes of regular polyhedra will be discussed in Part II, by the second author. We also examine when the construction produces uniform skeletal polyhedra.

It is common practice to use a metal conducting sphere for radar calibration purposes. The aspect-independence of a sphere allows for a more accurate and repeatable calibration of a radar than using a nonspherical calibration artifact. In addition, the radar cross section (RCS) for scattering spheres is well-known and can be calculated fairly easily using far field approximations. For Doppler radar testing, it is desired to apply these calibration advantages to a dynamic target. To accomplish this, a spherical polyhedron is investigated as the calibration target. This paper analyzes the scattering characteristics for various spherical polyhedral geometries. Each geometry is analyzed at 3.6 GHz in two states: contracted and expanded. For calibration purposes, it is desired that the target have a consistent monostatic RCS over the entirety of its surface. The RCS of each spherical polyhedral is analyzed and an optimized geometry, for calibration purposes, is chosen.

Triangulations of 3-dimensional polyhedron are partitions of the polyhedron with tetrahedra in a face-to-face fashion without introducing new vertices. Schonhardt (Math. Ann. 89:309–312, 1927), Bagemihl (Amer. Math. Mon. 55:411–413, 1948), Kuperberg (Personal communication 2011) and others constructed special polyhedra in such a way that clever one line geometric reasons imply nontriangulability. Rambau (Comb. Comput. Geom. 52:501–516, 2005) proved that twisted prisms over n-gons are nontriangulable. Our approach for proving polyhedra are nontriangulable is to show that partitions with tetrahedra, which we call tilings, do not exist even if the face-to-face-restriction is relaxed. First we construct a polyhedron which is tileable but is not triangulable. Then we revisit Rambau type twisted prisms. In fact we consider a slightly different class of polyhedra, and prove that these new twisted prisms are nontileable, thus are nontriangulable. We also show that one can twist the regular dodecahedron so that it becomes nontileable, which is abstracted to a new family of nontileable polyhedra, called nonconvex twisted pentaprisms.

We study properties of three-dimensional Klein polyhedra. The main result is as follows. Let be the set of integer -dimensional lattices with determinant , and let be the set of edges of Klein polyhedra in the lattice satisfying (that is, the integer length of the edge is ). Then for any , where is a positive constant depending only on , and Bibliography: 39 titles.

The existence of indecomposable polyhedra , that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. However, the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we investigate the structure of some well-known examples, the so-called Schonhardt polyhedron [10] and the Bagemihl's generalization of it [1] , which will be called Bagemihl's polyhedra . We provide a construction of an additional point, so-called Steiner point , which can be used to decompose the Schonhardt and the Bagemihl's polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Schonhardt's and Bagemihl's polyhedra, but they may need more than one Steiner point to be decomposed. Given such a polyhedron with n ≥ 6 vertices, we show that it can be decomposed by adding at most interior Steiner points. We also show that this number is optimal in theworst case.

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper non-unitary c-polyhedra with M\"obius-congruent faces that are consistently oriented are M\"obius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere as well as that of certain hyperideal hyperbolic polyhedra in $\mathbb{H}^{3}$.

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.

In this paper we consider 2 n mass points located at the vertices of two nested regular polyhedra with the same number of vertices and the ( 2 n + 1 ) th mass located at the geometrical center of the nested regular polyhedra. We show the existence of central configurations for any given mass ratios and the size ratio of nested polyhedra.

It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed and endowed with a suitable area, is isometric to a spherical polyhedron. These polygons have an infinite number of vertices, are space-like, contained in the future cone of the origin, and setwise invariant under the action of a linear isometry., Comment: New text, title slightly changed

A thorough study of regular and quasi-regular polyhedra shows that the symmetries of these polyhedra identically describe the quantization of orbital angular momentum, of spin, and of total angular momentum, a fact which permits one to assign quantum states at the vertices of these polyhedra assumed as the average particle positions. Furthermore, if the particles are fermions, their wave function is anti-symmetric and its maxima are identically the same as those of repulsive particles, e.g., on a sphere like the spherical shape of closed shells, which implies equilibrium of these particles having average positions at the aforementioned maxima. Such equilibria on a sphere are solely satisfied at the vertices of regular and quasi-regular polyhedra which can be associated with the most probable forms of shells both in Nuclear Physics and in Atomic Cluster Physics when the constituent atoms possess half integer spins. If the average sizes of the constituent particles are known, then the average sizes of the resulting shells become known as well. This association of Symmetry with Quantum Mechanics leads to many applications and excellent results.

We introduce the notion of integral equivalence and formulate a criterion for the equivalence of two polyhedra having certain special properties. The category of polyhedra under consideration includes Klein polyhedra, which are the convex hulls of nonzero points of the lattice ℂ3 that belong to some 3-dimensional simplicial cone with vertex at the origin, and therefore the criterion enables one to improve some results related to Klein polyhedra. In particular, we suggest a simplified formulation of a geometric analog of Lagrange’s theorem on continued fractions in the three-dimensional case.

A convex polyhedron P is equiprojective (similarly, biprojective) if, for some fixed k (similarly, k"1 and k"2), the orthogonal projection (or ''shadow'') of P in every direction, except directions parallel to faces of P, is a k-gon (similarly, k"1- or k"2-gon). Since 1968, it is an open problem to construct all equiprojective polyhedra, while the only results include a characterization, a recognition algorithm, and some non-trivial examples of equiprojective polyhedra. In this note, we show that simplicial polyhedra cannot be equiprojective. Then, we extend the idea of equiprojectivity to biprojectivity and show that simplicial polyhedra having all faces in parallel pairs are not (k,k+1)-biprojective.

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.

A semi-closed polyhedron is defined as the intersection of finitely many closed and/or open half-spaces, which can be regarded as an extension of a polyhedron. The same as a polyhedron, a semi-closed polyhedron admits both -representation and -representation. In this paper, we introduce the concept of a minimal -representation for semi-closed polyhedra which can be regarded as a natural extension of a minimal -representation for polyhedra. We derive some criteria for refining generators of a semi-closed polyhedron. These refining criteria can be applied to obtain a minimal generator for a semi-closed polyhedron.

A wire mesh conformed to a spherical surface forms a spherical polyhedron that has unique resonator and scattering properties for electromagnetic (EM) waves. Excitation of the spherical polyhedron by an incident plane wave at specified resonant frequencies yields large internal electric fields. Narrowband enhancements in the backscatter radar cross section are found at these same frequencies. These effects are explained with a model where the mesh is treated as an inductive frequency-selective surface that is used to match a shorted-line resonator attached to a transmission line. With this transmission-line model, the impedance of the inductive surface is shown to be an addition to the expression for the spherical harmonic series used to describe solid conducting spheres and spherical cavities. The resonant modes for the internal and external electric fields of the porous sphere are found with this formulation. The surface mesh is used in the low-frequency limit, where the EM wavelength is much larger than the polygon's holes in the mesh. The impedance (inductance and resistance) of the mesh are adjusted by varying the radii of the edges of the polygons and the conductivity of the edge material. Applications of the spherical porous conducting resonator (SPCR) include glow plasma discharges, measurements of dielectric constants of gases, and frequency-selective radar targets.

The problem treated here is: amongst the convex polyhedra that can be circumscribed about the unit sphere and have faces, which has the minimum surface area? A new optimization method based on mechanical analogies is worked out to solve this problem. By using this method, new computer-generated solutions are presented for and . The second of these two conjectured roundest polyhedra has icosahedral symmetry. The relation of the results of this problem to the minimum coverings of the sphere with equal circles is discussed.

We enumerate the regular and chiral polyhedra (in the sense of Grunbaum’s skeletal approach) whose vertex and edge sets are a subset of those of the primitive cubic lattice, the face-centred cubic lattice, or the body-centred cubic lattice.

The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a 6.6.6 tiling of hexagons and pasting them onto the full triangular faces of an icosahedron produces icosahedral fullerene cages. Here we show that pasting cutouts from a 6.6.6 tiling onto the full hexagonal and triangular faces of an Archimedean host polyhedron, the truncated tetrahedron, produces two series of tetrahedral (T-d) fullerene cages. Cages in the first series have 28n(2) vertices (n >= 1). Cages in the second (leapfrog) series have 3 x 28n(2). We can transform all of the cages of the first series and the smallest cage of the second series into geometrically convex equilateral polyhedra. With tetrahedral (T-d) symmetry, these new polyhedra constitute a new class of "convex equilateral polyhedra with polyhedral symmetry". We also show that none of the other Archimedean polyhedra, six with octahedral symmetry and six with icosahedral, can host full-face cutouts from regular tilings to produce cages with the host's polyhedral symmetry.

The shielding performance of new spherical polyhedral structures illuminated by an electromagnetic plane wave is investigated. The spherical shield screens consist of a juxtaposition of wire conductors and metallic screens distributed according to the regular pattern of a spherical polyhedron. It is shown that at low frequency, the wire frame screen provides high levels of shielding effectiveness, while its counterpart of metallic surfaces has a similar behavior for higher frequency bands. Moreover, the shielding characteristics of a combined wire-surface spherical surface are reported and compared with the cases of a single screen of wire conductors or metallic surfaces. The reported numerical results illustrate the advantage of using the combined configuration. It is observed that the combined wire-surface spherical screen can be designed in order to be effective simultaneously against interference at the low and high frequency limits for any polarization of the incident field.

In this paper a method is suggested for naming any convex polyhedron by a numerical code arising from the adjacency matrix of its edge graph. A polyhedron is uniquely fixed by its name and can be built using it. Classes of convexn-acra (i.e.n-vertex polyhedra) are strictly ordered by their names.

Disordered structures such as liquids and glasses, grains and foams, galaxies, etc. are often represented as polyhedral tilings. Characterizing the associated polyhedral tiling is a promising strategy to understand the disordered structure. However, since a variety of polyhedra are arranged in complex ways, it is challenging to describe what polyhedra are tiled in what way. Here, to solve this problem, we create the theory of how the polyhedra are tiled. We first formulate an algorithm to convert a polyhedron into a codeword that instructs how to construct the polyhedron from its building-block polygons. By generalizing the method to polyhedral tilings, we describe the arrangements of polyhedra. Our theory allows us to characterize polyhedral tilings and thereby paves the way to study from short- to long-range order of disordered structures in a systematic way.

Every regular map on a closed surface gives rise to generally six regular maps, its Petrie relatives, that are obtained through iteration of the duality and Petrie operations (taking duals and Petrie-duals). It is shown that the skeletal polyhedra in Euclidean \(3\)-space which realize a Petrie relative of the classical Gordan regular map and have full icosahedral symmetry, comprise precisely four infinite families of polyhedra, as well as four individual polyhedra.

We consider one of the problems of the theory of flexible polyhedra—the problem about the number of the parameters that must be defined additionally to the edge lengths for a polyhedron of a given combinatorial type in order to exclude its possible bendings. We give a description for the combinatorial structure of polyhedra of spherical type for which this number is equal to 1.

Lattice polyhedra, as introduced by Gröflin and Hoffman, form a common framework for various discrete optimization problems. They are specified by a lattice structure on the underlying matrix satisfying certain sub- and supermodularity constraints. Lattice polyhedra provide one of the most general frameworks of total dual integral systems. So far no combinatorial algorithm has been found for the corresponding linear optimization problem. We show that the important class of lattice polyhedra in which the underlying lattice is of modular characteristic can be reduced to the Edmonds–Giles polyhedra. Thus, submodular flow algorithms can be applied to this class of lattice polyhedra. In contrast to a previous result of Schrijver, we do not explicitly require that the lattice is distributive. Moreover, our reduction is very simple in that it only uses an arbitrary maximal chain in the lattice.

The well-known duality and Petrie duality operations on maps have natural analogs for abstract polyhedra. Regular polyhedra that are invariant under both operations have a high degree of both “external” and “internal” symmetry. The mixing operation provides a natural way to build the minimal common cover of two polyhedra, and by mixing a regular polyhedron with its five other images under the duality operations, we are able to construct the minimal self-dual, self-Petrie cover of a regular polyhedron. Determining the full structure of these covers is challenging and generally requires that we use some of the standard algorithms in combinatorial group theory. However, we are able to develop criteria that sometimes yield the full structure without explicit calculations. Using these criteria and other interesting methods, we then calculate the size of the self-dual, self-Petrie covers of several polyhedra, including the regular convex polyhedra.

Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete, in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E3. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E3.

A convex set with nonempty interior is maximal lattice-free if it is inclusion maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision of a rational polyhedron P in ℝd is the smallest natural number s such that sP is an integral polyhedron. In this paper we show that, up to affine mappings preserving ℤd, the number of maximal lattice-free rational polyhedra of a given precision s is finite. Furthermore, we present the complete list of all maximal lattice-free integral polyhedra in dimension three. Our results are motivated by recent research on cutting plane theory in mixed-integer linear optimization.

In this paper we study the problem of reconstructing orthogonal polyhedra from a putative vertex set, i.e., we are given a set of points and want to find an orthogonal polyhedron for which this is the set of vertices. This is well-studied in 2D; we mostly focus on 3D, and on the case where the given set of points may be rotated beforehand. We obtain fast algorithms for reconstruction in the case where the answer must be orthogonally convex.

In 1922, Steinitz’s theorem gave a complete characterization of the topological structure of the vertices, edges, and faces of convex polyhedra as triconnected planar graphs. In this paper, we generalize Steinitz’s theorem to non-convex polyhedra. More specifically, we introduce a new class of polyhedra, wider than convex polyhedra, called upward star-shaped polyhedra and spherical polyhedra, and present graph-theoretic characterization for both polyhedra. Upward star-shaped polyhedra are polyhedra where each face is star-shaped, all faces except the bottom face are visible from a view point, and any two faces sharing two vertices are non-coplanar. Spherical polyhedra are non-singular, non-coplanar polyhedra with no holes. We first present a graph-theoretic characterization of upward star-shaped polyhedra, i.e., characterization of upward star-shaped polyhedral graphs, which are the vertex-edge graphs (or 1-skeleton) of the upward star-shaped polyhedra. Roughly speaking, they correspond to biconnected planar graphs with special conditions. The proof of the characterization leads to an algorithm that constructs an upward star-shaped polyhedron with n vertices in O(n 1.5) time. Moreover, one can test whether a given plane graph is an upward star-shaped polyhedral graph in linear time. We then present a graph-theoretic characterization of spherical polyhedra for planar cubic graphs, and planar graphs with maximum face size 4. We also formally define the Polyhedra Realizability Problem, and discuss its reducibility. Our result is the first graph-theoretic characterization of non-convex polyhedra, which solves an open problem posed by Grunbaum (Discrete Math. 307(3–5), 445–463, 2007), and a generalization of Steinitz’s theorem (Polyeder und Raumeinteilungen, 1922), which characterized convex polyhedra as triconnected planar graphs.

By analogy with the solution of triangles, the solution of polyhedra means a theory and methods for calculating some geometric parameters of polyhedra in terms of other parameters of them. The main content of this paper is a survey of results on calculating the volumes of polyhedra in terms of their metrics and combinatorial structures. It turns out that a far-reaching generalization of Heron's formula for the area of a triangle to the volumes of polyhedra is possible, and it underlies the proof of the conjecture that the volume of a deformed flexible polyhedron remains constant. Bibliography: 110 titles.

A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2 if it is combinatorially regular but "fails geometric regularity by a factor of 2"; its combinatorial automorphism group is flag-transitive but its geometric symmetry group has two flag orbits. The present paper, and its successor by the first author, describe a complete classification of regular polyhedra of index 2 in 3-space. In particular, the present paper enumerates the regular polyhedra of index 2 with vertices on two orbits under the symmetry group. The subsequent paper will enumerate the regular polyhedra of index 2 with vertices on one orbit under the symmetry group., Comment: 34 pages; 6 figures; to appear in "Contributions to Algebra and Geometry"

We propose a new approach to the strict separation of convex polyhedra. This approach is based on the construction of the set of normal vectors for the hyperplanes, such that each one strict separates the polyhedra A and B. We prove the necessary and sufficient conditions of strict separability for convex polyhedra in the Euclidean space and present its applications in optimization.

New polyhedra are derived with symmetry codes and variables.

Schulte classified the discrete chiral polyhedra in Euclidean 3-space and showed that they belong to six families. The polyhedra in three of the families have finite faces and the other three families consist of polyhedra with (infinite) helical faces. We show that all the chiral polyhedra with finite faces are combinatorially chiral. However, the chiral polyhedra with helical faces are combinatorially regular. Moreover, any two such polyhedra with helical faces in the same family are isomorphic.

The directional contact range of two convex polyhedra is the range of positions that one of the polyhedron may locate along a given straight line so that the two polyhedra are in collision. Using the contact range, one can quickly classify the positions along a line for a polyhedron as "safe" for free of collision with another polyhedron, or "unsafe" for the otherwise. This kind of contact detection between two objects is important in CAD, computer graphics and robotics applications. In this paper we propose a robust and efficient computation scheme to determine the directional contact range of two polyhedra. We consider the problem in its dual equivalence by studying the Minkowski difference of the two polyhedra under a duality transformation. The algorithm requires the construction of only a subset of the faces of the Minkowski difference, and resolves the directional range efficiently. It also computes the contact configurations when the boundaries of the polyhedra are in contact. © 2008 Springer-Verlag Berlin Heidelberg.

If a convex polyhedron with regular faces cannot be divided by any plane into two polyhedra with regular faces, then it is said to be noncomposite. We indicate the exact coordinates of the vertices of noncomposite polyhedra that are neither regular (Platonic), nor semiregular (Archimedean), nor their parts cut by no more than three planes. Such a description allows one to obtain a short proof of the existence of each of the eight such polyhedra (denoted by M 8, M 20–M 25, M 28) and to obtain other applications.

Some of the semiregular (Archimedean) polyhedra (1–13 in Table 1) afford on truncation polyhedra that contain vertices where the sum of planar degrees for the faces which meet at those vertices is equal to (for 17, 18, and 23 in Table 3) or higher than 360° (21, 22, 24–26 in Table 3). Therefore such polyhedra are nonconvex.

Let Σ be a set of polyhedra. A set Ω of polyhedra is said to be an element set for Σ if each polyhedron in Σ is the union of a finite number of polyhedra in Ω. We call each polyhedron of the element set Ω an element for Σ. In this paper, we determine one element set for the set Π of the Platonic solids, and prove that this element set is, in fact, best possible; it achieves the minimum in terms of cardinality among all the element sets for Π. We also introduce the notion of indecomposability of a polyhedron and present a conjecture in Sect. 3.

Whenever two nonempty convex polyhedra can be properly separated, a separating hyperplane may be chosen to contain a face of either polyhedron. It is demonstrated that, in fact, one or the other of the polyhedra admits such an exposed face having dimension no smaller than approximately half the larger dimension of the two polyhedra. An example shows that the bound on face dimension is optimal, and a linear programming representation of the problem is given.