Summary. This is the first of two papers in which we classify the regular projective polyhedra in P 3 with planar faces. Here, we develop the basic notions; we introduce a new diophantine trigonometric equation, which plays a key role in the classification theorem, relating the combinatorial and geometric parameters of such polyhedra, and conclude with the case in which the polyhedron is an embedded surface.Mathematics Subject Classification (1991). 52B, 52C, 51M20, 51F15.

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger's theorem gives a generalization of the theorem of Tverberg on non-separated partitions.

Foundations for the topic of crystallizations are proposed through the more general concept of colored triangulations. Classic results and techniques of crystallizations are reviewed from this point of view. A new set of combinatorial invariants of manifolds is defined, and related to the fundamental group and other known invariants. A universal group theoretic approach for this theory is introduced.

Coexistence holes characterize the assembly and disassembly of multispecies systemsThe MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.

Given a finite family F of convex sets in R d , we say that F has the (p, q) r property if for any p convex sets in F there are at least r q-tuples that have nonempty intersection. The piercing number of F is the minimum number of points we need to intersect all the sets in F . In this paper we will find some bounds for the piercing number of families of convex sets with (p, q) r properties.

A carousel is a dynamical system that describes the movement of an equilateral linkage in which the midpoint of each rod travels parallel to it. They are closely related to the floating body problem. We prove, using the work of Auerbach, that any figure that floats in equilibrium in every position is drawn by a carousel. Of special interest are such figures with rational perimetral density of the floating chords, which are then drawn by carousels. In particular, we prove that for some perimetral densities the only such figure is the circle, as the problem suggests.

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