A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal surfaces (under an appropriate smoothing flow, keeping their symmetries). Here we give a complete classification of the cubic polyhedra. Among these are five new infinite uniform polyhedra and an uncountable collection of new infinite semi-regular polyhedra. We also consider the somewhat larger class of all discrete minimal surfaces in the cubic lattice.
A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex-edge-face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron one can assign an inner dihedral angle and say that the given ball-polyhedro…
We study the properties of Kokotsakis polyhedra of orthodiagonal anti-involutive type. Stachel conjectured that a certain resultant connected to a polynomial system describing flexion of a Kokotsakis polyhedron must be reducible. Izmestiev \cite{izmestiev2016classification} showed that a polyhedron of the orthodiagonal anti-involutive type is the only possible candidate to disprove Stachel's conjecture. We show that the corresponding resultant is reducible, thereby confirming the conjecture. We do it in two ways: by factorization of the corresponding resultant and providing a simple geometric …
An orthant polyhedron is a polyhedron with $m$ hyperfaces, that could be realized as a section of the $m$-dimensional non-negative orthant. We classify all 2-dimensional orthant polyhedra and provide some partial results towards the classification of higher dimensional orthant polyhedra. As a consequence of our results we show that every polytope could be realized as a section of the $n$-dimensional non-negative orthant with $n$ being big enough.
In this article we define an algebraic vertex of a generalized polyhedron and show that it is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope $P$ is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of $P$. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier…
A convex polyhedron, that is, a compact convex subset of $\mathbb{R}^3$ which is the intersection of finitely many closed half-spaces, can be rectified by taking the convex hull of the midpoints of the edges of the polyhedron. We derive expressions for the side lengths and areas of rectifications of regular polygons in plane, and use these results to compute surface areas and volumes of various convex polyhedra. We introduce rectification sequences and show that there are exactly two disjoint pure rectification sequences generated by the platonic solids.
On a convex surface $S$, the antipodal map $F$ associates to a point $p$ the set of farthest points from $p$, with respect to the intrinsic metric. $S$ is called a Steinhaus surface if $F$ is a single-valued involution. We prove that any convex polyhedron has an open and dense set of points $p$ admitting a unique antipode $F_p$, which in turn admits a unique antipode $F_{F_p}$, distinct from $p$. In particular, no convex polyhedron is Steinhaus.
How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and "simplicial," i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex-facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circul…
We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed continuously by changing its dihedral angles only. We prove that the polyhedron is not flexible if for each of its edges the following holds true: the length of this edge is not a linear combination with rational coefficients of the lengths of the remaining edges. We prove also that if a polyhedron is flexible, then some linear combinations of its dihedral angles re…
A Kokotsakis polyhedron with quadrangular base is a neighborhood of a quadrilateral in a quad surface. Generically, a Kokotsakis polyhedron is rigid. Up to now, several flexible classes were known, but a complete classification was missing. In the present paper, we provide such a classification. The analysis is based on the fact that the dihedral angles of a Kokotsakis polyhedron are related by an Euler-Chasles correspondence. It results in a diagram of elliptic curves covering complex projective planes. A polyhedron is flexible if and only if this diagram commutes.
We define the mechanical complexity $C(P)$ of a convex polyhedron $P,$ interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria, and the mechanical complexity $C(S,U)$ of primary equilibrium classes $(S,U)^E$ with $S$ stable and $U$ unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class $(S,U)^E$ with $S, U > 1$ is the minimum of $2(f+v-S-U)$ over all polyhedral pairs $(f,v )$, where a pair of integers is call…
In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Unti…
A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic metric with conical singularities of positive singular curvature on a compact surface of genus greater than one. We prove that these metrics are actually realised by exactly one convex Fuchsian polyhedron (up to global isometries). This extends a famous theorem of A.D. Alexandro…
In this paper, isometric embedding results of Greene, Gromov and Rokhlin are extended to what are called "indefinite metric polyhedra". Two definitions for an indefinite metric polyhedron are given, an intuitive definition and a more useful definition. The more useful definition is used to show that every indefinite metric polyhedron (with the maximal degree of every vertex bounded above) admits a simplicial isometric embedding into Minkowski space of an appropriate signature. This result is then used to show that the two definitions coincide. A simple example is given to show that the dimensi…
A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2 if it is combinatorially regular and its geometric symmetry group has index 2 in its combinatorial automorphism group; thus its automorphism group is flag-transitive but its symmetry group has two flag orbits. The present paper completes the classification of finite regular polyhedra of index 2 in 3-space. In particular, this paper enumerates the regular polyhedra of index 2 with vertices on one orbit under the symmetry group. There are ten such polyhedra.
We consider triangle faced convex polyhedra inscribed in the unit sphere $S^2$ in ${\Bbb{R}}^3$. One way of measuring their deviation from regular polyhedra with triangular faces is to consider the quotient of the lengths of the longest and the shortest edges. If the number of faces tends to infinity, and the polyhedron with this number of faces varies, then the limit inferior of this quotient is $2 \sin 36^{\circ } = 1.1756 \ldots $.
We explore the perspective of a bug living on the two-dimensional surface of a polyhedron. Images of various kinds of effects like lensing and cloaking are shown via color pictures of three viewpoints: the first person perspective of the bug, a map of the bug's viewpoint, and a look at the bug on the embedded polyhedron from a three-dimensional exterior viewer. The pictures were constructed by computing the exponential map of a polyhedron by cutting and rotating faces into the tangent plane of the bug.
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 i…
The paper considers systems of contraction similarities in $\mathbb R^d$ sending a given polyhedron $P$ to polyhedra $P_i\subset P$, whose non-empty intersections are singletons and contain the common vertices of those polyhedra, while the intersection hypergraph of the system is acyclic. It is proved that the attractor $K$ of such system is a dendrite in $\mathbb R^d$. The ramification points of such dendrite fave finite order whose upper bound depends only on the polyhedron $P$, and the set of the cut points of the dendrite $K$ is equal to the dimension of the whole $K$ iff $K$ is a Jordan a…
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 $\mathbb{R}^d$ is the smallest integer $s$ such that $sP$ is an integral polyhedron. In this paper we show that, up to affine mappings preserving $\mathbb{Z}^d$, 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 integr…
Let $P$ be a (non necessarily convex) embedded polyhedron in $\R^3$, with its vertices on an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then $P$ is infinitesimally rigid. More generally, let $P$ be a polyhedron bounding a domain which is the union of polytopes $C_1, ..., C_n$ with disjoint interiors, whose vertices are the vertices of $P$. Suppose that there exists an ellipsoid which contains no vertex of $P$ but intersects all the edges of the $C_i$. Then $P$ is infinitesimally rigid. The proof is based on some geometric prop…
Let $P \subset \R^3$ be a polyhedron. It was conjectured that if $P$ is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. $P$ can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assumption of codecomposability. The proof relies on a result of independent interest concerning the Hilbert-Einstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In partic…
A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. The aim of this paper is to investigate three questions of Conway, regarding monostable polyhedra, which first appeared in a 1969 paper of Goldberg and Guy (M. Goldberg and R.K. Guy, Stability of polyhedra (J.H. Conway and R.K. Guy), SIAM Rev. 11 (1969), 78-82). In this note we answer two of these problems and make a conjecture about the third one. The main tool of our proof is a general theorem describing approximations of smooth convex bodies by convex polyhedra in terms of their static equil…
The topic of the paper are developments of $n$-dimensional Coxeter polyhedra. We show that the surface of such polyhedron admits a canonical cutting such that each piece can be covered by a Coxeter $(n-1)$-dimensional domain.
Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry gr…
Steinitz's theorem states that a graph $G$ is the edge-graph of a $3$-dimensional convex polyhedron if and only if, $G$ is simple, plane and $3$-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in $\mathbb{R}^3$.
A polyhedron $\textbf{P} \subset \mathbb{R}^3$ has Rupert's property if a hole can be cut into it, such that a copy of $\textbf{P}$ can pass through this hole. There are several works investigating this property for some specific polyhedra: for example, it is known that all 5 Platonic and 9 out of the 13 Archimedean solids admit Rupert's property. A commonly believed conjecture states that every convex polyhedron is Rupert. We prove that Rupert's problem is algorithmically decidable for polyhedra with algebraic coordinates. We also design a probabilistic algorithm which can efficiently prove t…
Let $Δ(d,n)$ denote the maximum diameter of a $d$-dimensional polyhedron with $n$ facets. In this paper, we propose a unified analysis of a recursive inequality about $Δ(d,n)$ established by Kalai and Kleitman in 1992. This yields much simpler proofs of a tail--polynomial and tail--almost--linear bounds on $Δ(d,n)$ which are recently discussed by Gallagher and Kim.
A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to Möbius transformations of the sphere. Motivated by this result, various papers investigate the problem of centering spherical configurations under Möbius transformations. In particular, Springborn proved that for any discrete point set on the sphere there is a Möbius transformation that maps it into a set whose barycenter is the origin, which implies that the combinatorial cla…
A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability obstruction for almost flat convex caps due to Tarasov. Our example, which has 340 vertices, significantly simplifies an earlier construction by Tarasov, and confirms that…
Let X be a space of constant curvature and P be a convex polyhedron in X. A Coxeter decomposition of the polyhedron P is a decomposition of P into finitely many Coxeter polyhedra, such that any two polyhedra having a common facet are symmetric with respect to this facet. In this paper we classify Coxeter decompositions of simplices in hyperbolic space of dimension greater than 3. The problem is close to the classification of the finite index subgroups in the discrete hyperbolic reflection groups.
The problem of counting the number of waves arriving at the vertex of a polyhedron is motivated by physics. In the article it was solved for the case of Platonic solid using three nontrivial results from number theory. This growth turns out to be subexponential. Also we prove a subexponential upper bound for all polyhedra with rational total angles at vertices.
