Polyhedral Theory Research Articles

A Non-Shellable 3-Sphere

Shellability of simplicial complexes has been a useful concept in polyhedral theory, in piecewise linear topology and recently in connection with Cohen Macauley rings. In this paper we initiate a study of shellability in a graph theoretic context. For certain graphs, necessary and sufficient conditions for shellability are given. A correspondence is established between n -colored graphs and ( n - 1)-dimensional simplicial complexes. Via this correspondence, results on shellability of graphs have implications for problems in P.L. topology. For example, it is known that all 2-spheres and convex d-spheres are shellable. It has been conjectured that all 3-spheres are shellable. A counter-example is provided in the following sense. We construct a finite cell complex whose cells are simplexes and such that the underlying space is a 3-sphere whose simplexes cannot be shelled.

Regular polyhedra—old and new

Although it is customary to define polygons as certain families of edges, when considering polyhedra it is usual to view polygons as 2-dimensional pieces of the plane. If this rather illogical point of view is replaced by consistently understanding polygons as 1-dimensional complexes, the theory of polyhedra becomes richer and more satisfactory. Even with the strictest definition of regularity this approach leads to 17 individual regular polyhedra in the Euclidean 3-space and 12 infinite families of such polyhedra, besides the traditional ones (which consist of 5 Platonic polyhedra, 4 Kepler—Poinsot polyhedra, 3 planar tessellations and 3 Petrie—Coxeter polyhedra). Among the many still open problems that naturally arise from the new point of view, the most obvious one is the question whether the regular polyhedra found in the paper are the only ones possible in the Euclidean 3-space.

Manifold Phenomena in the Theory of Polyhedra

Manifold Phenomena in the Theory of Polyhedra

Manifold phenomena in the theory of polyhedra

Introduction. When can an isotopy be covered by an ambient isotopy? Let us restrict attention to the compact p.l. category. Hudson and Zeeman have shown that a locally unknotted isotopy of a manifold in a manifold can be covered by an ambient isotopy of the big manifold. By Zeeman's codimension > 3 unknotting theorem, an isotopy of manifolds is locally unknotted if the codimension is greater than or equal to 3. Hence, any isotopy of a manifold in a manifold of dimension at least 3 higher can be covered. Lickorish has generalized Zeeman's unknotting theorem to the case of a proper embedding of a cone in a ball of dimension at least three higher. From this, Hudson has shown any isotopy of a polyhedron in a manifold can be covered if the polyhedron has codimension at least 3. As a modest aim we would like a criterion of local unknottedness of a polyhedron in a manifold so that the original Hudson-Zeeman theorem would generalize. What we actually obtain is more general. We present a characterization of those isotopies of a polyhedron in a polyhedron which can be covered by ambient isotopies. Perhaps surprisingly, this question admits a rather elegant general solution. Intrinsic Dimension. Our major tool is the theory of intrinsic dimension developed by Armstrong. Given a polyhedral pair (X, X0) and a point x E X0, we define the intrinsic dimension of x in (X, X0), denoted d(x; X, X0), to be

Critical points and curvature for embedded polyhedra

Recently a new insight into the Gauss-Bonnet Theorem and other problems in global differential geometry has come about through the connection between total curvature of embedded smooth manifolds and critical point theory for non-degenerate height functions. This paper presents an analogous program for embedded polyhedra. The methods are completely elementary, using the techniques neither of differential geometry nor of algebraic topology. As such the paper has a twofold purpose —to study global geometry of polyhedra for its own sake, and to give a deeper understanding of the theorems of global differential geometry through an elementary presentation of their finite or combinatorial content. Moreover the polyhedral theory applies to a wider class of objects, and gives a new interpretation of the relation between intrinsic and extrinsic curvature. Although the polyhedral part of the paper is relatively self-contained, the remarks which show the connection with the differentiable theory presuppose a familiarity with the classical differentiable results. For a bibliography on these and related problems, see Kuiper [4]. This paper will contain no discussion of the possible convergence theorems relating the polyhedral and differentiable theories—for a presentation of this topic, expecially in the 2dimensional and convex cases we refer to A. D. Alexandrow [1]. For related total curvature concepts see also Chern-Lashof [3]. A subsequent paper of the author will deal with critical points and curvature for mappings of complexes into Eι for / > 1, and into /-dimensional manifolds. The author wishes to thank Professor Kuiper for his interest and advice throughout the development of this research. 1. The critical point theorem Definition. A convex cell complex Mk embedded in E n is a finite collection of cells {Cr}, where each C° is a point, and each Cr is a bounded convex set with interior in some afϊine Eτ c En, such that the boundary dCτ of C r is a union of Cs with s < r, and such that if s < r and Cs Γ\Cr Φ 0, then O C dC Mk is called k-dimensional if there is a Ck in Mk but no C*+1.

Embryology and the Theory of Polyhedra

A NEW volume of Proceedings of the Bologna Mathematical Congress begins with a short paper by Max Bruckner, on the old problem of how many different polyhedra are possible of n sides—with the limiting condition that all the corners shall be trihedral. In my book on “Growth and Form” I dealt with the kindred problem of the possible number of arrangements of a plane assemblage of cells, their partition-walls all meeting three-by-three, as is actually the case in a system of soap-bubbles or of living cells. The problem of the polyhedra is just as interesting to the biologist; for any natural clump of cells, such as a totally segmented egg or ‘morula’, may be looked on as a polyhedron and may be studied accordingly.