We consider a stationary face-to-face tessellation X of Rd and introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probability p and white otherwise. We are interested in geometric properties of the union Z of black faces. Under natural integrability assumptions, we first express asymptotic mean values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we focus on cell percolation on normal tessellations and study asymptotic covariances of intrinsic volumes of Z ∩ W, where the observation window W is assumed to be a convex body. Special emphasis is given to the planar case where the formulae become more explicit, though we need to assume the existence of suitable asymptotic covariances of the face processes of X. We check these assumptions in the important special case of a Poisson-Voronoi tessellation.

Tessellations of R3 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 R3 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 present new ideas about the type of random tessellation which evolves through successive division of its cells. These ideas are developed in an intuitive way, with many pictures and only a modicum of mathematical formalism–so that the wide application of the ideas is clearly apparent to all readers. A vast number of new tessellation models, with known probability distribution for the volume of the typical cell, follow from the concepts in this paper. There are other interesting models for which results are not presented (or presented only through simulation methods), but these models have illustrative value. A large agenda of further research is opened up by the ideas in this paper.

Population-based studies of retinal neurons have helped to reveal their natural types in mammals and teleost fishes. In this, the first such study in a frog, labeled ganglion cells of the mesobatrachian Xenopus laevis were examined in flatmounts. Cells with large somata and thick dendrites could be divided into three mosaic-forming types, each with its own characteristic stratification pattern. These are named αa, αab, and αc, following a scheme recently used for teleosts. Cells of the αa mosaic (~0.4% of all ganglion cells) had very large somata and trees, arborizing diffusely within sublamina a (the most sclerad). Their distal dendrites were sparsely branched but achieved consistent coverage by intersecting those of their neighbors. Displaced and orthotopic cells belonged to the same mosaic, as did cells with symmetric and asymmetric trees. Cells of the αab mosaic (~1.2%) had large somata, somewhat smaller trees that appeared bistratified at low magnification, and dendrites that branched extensively. Their distal dendrites arborized throughout sublamina b and the vitread part of a, tessellating with their neighbors. All were orthotopic; most were symmetric. Cells of the αc mosaic (~0.5%) had large somata and very large, sparse, flat, overlapping trees, predominantly in sublamina c. All were orthotopic; some were asymmetric. Nearest-neighbor analyses and spatial correlograms confirmed that each mosaic was regular and independent, and that spacings were reduced in juvenile frogs. Densities, proportions, sizes, and mosaic statistics are tabulated for all three types, which are compared with types defined previously by size and symmetry in Xenopus and potentially homologous mosaic-forming types in teleosts. Our results reveal strong organizational similarities between the large ganglion cells of teleosts and frogs. They also demonstrate the value of introducing mosaic analysis at an early stage to help identify characters that are useful markers for natural types and that distinguish between within-type and between-type variation in neuronal populations.

Population-based studies of ganglion cells in retinal flatmounts have helped to reveal some of their natural types in mammals, teleost fish and, recently, the aquatic mesobatrachian frog Xenopus laevis. Here, ganglion cells of the semiterrestrial neobatrachian frogs Rana esculenta and Rana pipiens have been studied similarly. Ganglion cells with large somata and thick dendrites could again be divided into three mosaic-forming types with distinctive stratification patterns. Cell dimensions correlated inversely with density, being smallest in the visual streak. Cells of the αa mosaic (<0.2% of all ganglion cells) had the largest somata at each location (often displaced) and their trees were confined to one shallow plane within sublamina a of the inner plexiform layer. In regions of high regularity, many trees were symmetric. Elsewhere, asymmetric, irregular trees predominated and their dendrites, although sparsely branched, achieved consistent coverage by intersecting in complex ways. Cells of the αab mosaic were more numerous (≈0.7%) and had large somata, smaller (but still large) trees, and dendrites that branched extensively in two separate shallow planes in sublaminae a and b. The subtrees did not always match in symmetry, and each subtree tessellated independently with its neighbors. Cells of the αc mosaic (≈0.1%) had large, orthotopic somata and large, sparse trees (often asymmetric and irregular) close to the ganglion cell layer. Nearest-neighbor analyses and spatial correlograms confirmed that each mosaic was regular and independent. Densities, proportions, sizes, and mosaic statistics are tabulated for all three types, which are compared with types defined by size and symmetry in R. pipiens, by discriminant analysis in R. temporaria, by physiological response in both, and by mosaic analysis in Xenopus and several teleosts. The variable stratification of these otherwise similar types across species is consistent with other evidence that stratification may be determined, in part, by functional interactions.

We study the contribution made by three or four points to certain areas associated with a typical polygon in a Voronoi tessellation of a planar Poisson process. We obtain some new results about moments and distributions and give simple proofs of some known results. We also use Robbins' formula to obtain the first three moments of the area of a typical polygon and hence the variance of the area of the polygon covering the origin.

A network is a system of segments or edges in ℝd which intersect only in the segment endpoints, which are called vertices. An example is the system of edges of a tessellation. It is possible to give formulas for the specific connectivity number of a random network; in the stationary case, the intensity of the 0-curvature measure is equal to the difference of the intensities of the point processes of vertices and edge centres.

By using an adaptation of the radial generation method, we give an integral formula for the proportion of triangles in a Poisson-Voronoi tessellation, which gives a value of 0.0112354 to 7 decimal places. We also obtain the first four moments of some characteristics of triangles.

Population-based methods were used to study labeled retinal ganglion cells from the cane toad Bufo marinus and the treefrog Litoria moorei, two visually competent bufonoid neobatrachians with contrasting habitats. In both, cells with large somata and thick dendrites formed distinct types with independent mosaics. The αa, αab, and αc mosaics of Bufo in all major respects resembled those of ranids, studied previously, and could be provisionally matched to the same functional classes. As in other frogs, some αa cells were displaced and many α-cells of all types were asymmetric, but within each type all variants belonged to one mosaic. Nearest-neighbor analyses and spatial correlograms confirmed that all three mosaics were regular and independent. In Litoria, monostratified αa cells were not found. Instead, two bistratified types were present, distinguished individually by soma size and dendritic caliber and collectively by membership of independent mosaics: the larger (∼0.8% of all ganglion cells) was termed α1ab and the smaller (∼2.2%) α2ab. An αc cell type was also present, although too inconstantly labeled for mosaic analysis. Nearest-neighbor analyses and spatial correlograms confirmed that the two αab mosaics were regular and independent. Densities, proportions, soma sizes, and mosaic statistics are tabulated for each species. The emergence of a consensus pattern of α-cell types in fishes and frogs, from which this treefrog partly diverges, offers new possibilities for studying correlations between function, phylogeny, ecology, and neuronal form.

We consider a stationary germ-grain model Ξ with convex and compact grains and the distance r(x) from x ε ℝd to Ξ. For almost all points x ε ℝd there exists a unique point p(x) in the boundary of Ξ such that r(x) is the length of the vector x-p(x), which is called the spherical contact vector at x. In this paper we relate the distribution of the spherical contact vector to the times it takes a typical boundary point of Ξ to hit another grain if all grains start growing at the same time and at the same speed. The notion of a typical point is made precise by using the generalized curvature measures of Ξ. The result generalizes a well known formula for the Boolean model. Specific examples are discussed in detail.

Consider a forest of maple trees in autumn, with leaves falling on the ground. Those coming late cover the others below, so eventually the fallen leaves form a statistically homogeneous spatial pattern. In particular, the uncovered leaf boundaries form a mosaic. We formulate a mathematical model to describe this mosaic, firstly in the case where the leaves are polygonal and later for leaves with curved boundaries. Mean values of certain statistics of the mosaic are derived.

Many mean-value quantities of stationary random tessellations can be expressed in terms of three fundamental mean-value quantities. In this note we characterize the set of triples of mean values that can be realized, and show that every possible triple can arise from a suitable ergodic stationary isotropic tessellation.