f-Vectors of Polyhedra

The f-vector of a triangulation of a polyhedron X is the numbers of simplices at various dimensions. We prove that the affine span of f-vectors of X has dimension (n+s+1)/2, where n is the dimension of X, and s is the dimension of the part of X that is singular with respect to the local Euler characteristic.     

On the Capacity of Regular Polyhedra

Upper bounds for the capacity of regular solids are given throughan application of Dirichlet's principle, using simple trialfunctions suggested by the symmetry of the problem.     

A Rigidity Criterion for Non-Convex Polyhedra

Let P be a (non-necessarily convex) embedded polyhedron in R³, with its vertices on the boundary of 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 C₁, . . ., 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 properties of hyperideal hyperbolic polyhedra.     

On the Notion of the Combinatorial p-Parameter Property for Polyhedra

We introduce some versions of the notion of the combinatorial p-parameter property for polyhedra whose general meaning reduces to search for the number of parameters ensuring unique determination of a polyhedron locally on assuming given edge lengths and combinatorial structure.     

Resolvable Representation of Polyhedra

The paper proposes a new method for the boundary representation of three-dimensional (not necessarily convex) polyhedra, called a resolvable representation , in which small numerical errors do not violate the symbolic part of the representation. In this representation, numerical data are represented partly by the coordinates of vertices and partly by the coefficients of face equations in such a way that the polyhedron can be reconstructed from the representation in a step-by-step manner. It is proved that any polyhedron homeomorphic to a sphere has a resolvable representation, and an algorithm for finding such a representation is constructed. Received January 21, 1997, and in revised form April 29, 1998.     

A Trick for Calculating Volumes of Non-Euclidean Polyhedra

Mathematical Notes -     

Closed-Form Expressions for Uniform Polyhedra and Their Duals

A uniform polyhedron has equivalent vertices and regular polygonal faces. An established set of 77 Wythoff symbols effectively describes the dynamic kaleidoscopic constructions of uniform polyhedra. The main combinatorial and metrical quantities of uniform polyhedra and their duals are presented as closed-form expressions derived from the Wythoff symbols. Received September 20, 2000, and in revised form September 20, 2001. Online publication February 19, 2002.     

On Sub-determinants and the Diameter of Polyhedra

We derive a new upper bound on the diameter of a polyhedron \(P = \{x {\in } {\mathbb {R}}^n :Ax\le b\}\) , where \(A \in {\mathbb {Z}}^{m\times n}\) . The bound is polynomial in \(n\) and the largest absolute value of a sub-determinant of \(A\) , denoted by \(\Delta \) . More precisely, we show that the diameter of \(P\) is bounded by \(O(\Delta ^2 n^4\log n\Delta )\) . If \(P\) is bounded, then we show that the diameter of \(P\) is at most \(O(\Delta ^2 n^{3.5}\log n\Delta )\) . For the special case in which \(A\) is a totally unimodular matrix, the bounds are \(O(n^4\log n)\) and \(O(n^{3.5}\log n)\) respectively. This improves over the previous best bound of \(O(m^{16}n^3(\log mn)^3)\) due to Dyer and Frieze (Math Program 64:1–16, 1994).     

Randomized Complexity Lower Bound for Arrangements and Polyhedra

The complexity lower bound Ω (log N ) for randomized computation trees is proved for recognizing an arrangement or a polyhedron with N faces. This provides, in particular, the randomized lower bound Ω (n log n ) for the DISTINCTNESS problem and generalizes [11] where the randomized lower bound Ω (n ² ) was ascertained for the KNAPSACK problem. The core of the method is an extension of the lower bound from [8] on the multiplicative complexity of a polynomial. Received May 14, 1997, and in revised form October 27, 1997, and February 16, 1998.     

Dihedral Angles of Convex Polyhedra

Convex polyhedra in H ³ are not determined by (their combinatorics and) their edge lengths. Convex space-like polyhedra in the de Sitter space S ³ ₁ are determined neither by their dihedral angles nor by their edge lengths. The same holds of convex polyhedra in S ³ . Received November 16, 1998, and in revised form March 8, 1999.     

Continuous Blooming of Convex Polyhedra

We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming.     

Children's Evolving Understanding of Polyhedra in the Classroom

To better understand the development of children's thinking in three-dimensional geometry, we conducted a teaching experiment with 8- and 9-year olds in which children built and described polyhedra during several lessons. Analysis of pre-/post-assessments showed that children advanced in their geometric reasoning and began to identify, enumerate, and notice relationships between component parts of polyhedra. Our consideration of a class activity showed how examining a range of examples and non-examples enculturated students into the practice of attending to component parts. Promoting precise, formal definitions for components proved to be a significant challenge for the teacher in establishing norms for class discussions.     

Branching Numbers for Euclidean Projections onto Convex Polyhedra

The purpose of this note is to give a short proof of the theorem of Ralph in Linear Algebra Appl. 178 (1993), 249–260, on the geometry of projection maps. This theorem implies that the so-called normal maps derived from projections satisfy a general geometric criterion for bijectivity of piecewise affine maps given by Kuhn and Löwen in Linear Algebra Appl. 96 (Section 5.3), (1987), 109–129. As a consequence, one obtains Robinson's characterization of bijectivity of normal maps.