Tessellations of R 3 that use convex polyhedral cells to fill the space can be extremely complicated. This is especially so for tessellations which are not ‘facet-to-facet’, that is, for those where the facets of a cell do not necessarily coincide with the facets of that cell's neighbours. Adjacency concepts between neighbouring cells (or between neighbouring cell elements) are not easily formulated when facets do not coincide. In this paper we make the first systematic study of these topological relationships when a tessellation of R 3 is not facet-to-facet. The results derived can also be applied to the simpler facet-to-facet case. Our study deals with both random tessellations and deterministic ‘tilings’. Some new theory for planar tessellations is also given.

We give a new type of geometric construction that allows for the construction of families of quintic irrationalities, and is quite rich in algebraic properties. This construction may be considered as our first attempt at characterizing points constructible with compass and twice-notched ruler, a problem that seems to have been known in some form for more than two millennia.

Given a set D of positive real numbers, let Xn(D) denote the graph with ℝn as the vertex set such that two points are joined if their distance is in D. Bukh conjectured in [Measurable sets with excluded distances. Geom. Funct. Anal.18 (2008), 668–697] that if D is algebraically independent, then Chr(Xn(D)), the chromatic number of Xn(D), is finite. Here we prove that Chr(Xn(D)) is countable and that, if n=2 , even the coloring number is countable. Furthermore, we prove that Chr (Y )is countable, where Y is the following graph on ℂn: let 𝔽 be a countable subfield of ℂ and let D⊆ℂ be algebraically independent over 𝔽; join a,b∈ℂn if there is some p(x,y)∈𝔽[x,y]such that p(x,x)is identically zero and p(a,b)≠0is algebraic over some d∈𝔽∪D.

Poincaré's Polyhedron Theorem is a widely known valuable tool in constructing manifolds endowed with a prescribed geometric structure. It is one of the few criteria providing discreteness of groups of isometries. This work contains a version of Poincaré's Polyhedron Theorem that is applicable to constructing fibre bundles over surfaces and also suits geometries of non-constant curvature. Most conditions of the theorem, being as local as possible, are easy to verify in practice.

We define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.

Two graphs, the edge crossing graph E and the triangle graph T are associated with a simple lattice polygon. The maximal independent sets of vertices of E and T are derived including a formula for the size of the fundamental triangles. Properties of E and T are derived including a formula for the size of the maximal independent sets in E and T. It is shown that T is a factor graph of edge-disjoint 4-cycles, which gives corresponding geometric information, and is a partition graph as recently defined by the authors and F. Harary.

In this paper, we give a sufficient condition (Theorem) in order that one domain D1 bounded by a C2-smooth boundary can be enclosed in, or enclose, another domain D0 bounded by the same kind of boundary. A same kind of sufficient condition for convex bodies (Corollary) is also obtained.

We give a rigorous definition of tropical fans (the ‘local building blocks for tropical varieties’) and their morphisms. For a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point; a statement that can be viewed as one of the important first steps of tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some ℝr) together with the evaluation and forgetful morphisms. Using our results this gives new, easy and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any ℝr) through given points are independent of the points.

A star is a planar set of three lines through a common point in which the angle between each pair is 60∘.A set of lines through a point in which the angle between each pair of lines is 60 or 90∘ is star-closed if for every pair of its lines at 60∘ the set contains the third line of the star. In 1976 Cameron, Goethals, Seidel and Shult showed that the indecomposable star-closed sets in Euclidean space are the root systems of types An, Dn, E6, E7 and E8. This result was a key part of their determination of all graphs with least eigenvalue −2. Subsequently, Cvetković, Rowlinson and Simić determined all star-closed extensions of these line systems. We generalize this result on extensions of line systems to complex n-space equipped with a hermitian inner product. There is one further infinite family, and two exceptional types arising from Burkhardt and Mitchell’s complex reflection groups in dimensions five and six. The proof is a geometric version of Mitchell’s classification of complex reflection groups in dimensions greater than four.

A planar graph contains faces which can be classified into types depending on the number of edges on the face boundaries. Under various natural rules for randomly dividing faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

Abstract. We study the envelope of the family of planes which bisect the volume of a tetrahedron.

For each odd prime p there is a finite regular abstract 4-dimensional polytope of type {3, 3, p}. Its cells are simplices, and its vertex figures belong to an infinite family of regular polyhedra. We also give a geometric realization for these polytopes.

This paper is concerned with the solution of the following interesting geometrical problem. For what set of n points on the sphere is the sum of all Euclidean distances between points maximal, and what is the maximum?

Our starting point is the following surprising “invariance principle” due to K. B. Stolarsky: The sum of the distances between points plus the quadratic average of a discrepancy type quantity is constant. Thus the sum of distances is maximized by a well distributed set of points. We now introduce some notation to make the statement more precise.

The classification of conies in the real or complex projective plane under the action of the appropriate group is simple and well known. We consider here the more complicated question of classifying conies in the complex projective plane ℂℙ2 under the action of the real projective group PGL (3, ℝ).