I describe below an elementary problem in Euclidean (or Hyperbolic) geometry which remains unsolved more than 10 years after it was first formulated. There is a proof for n = 3 and (when the ball is the whole of 3-space) when n = 4. There is strong numerical evidence for n 6 30. Let (x1, x2, ...xn) be n distinct points inside the ball of radius R in Euclidean 3-space. Let the oriented line xixj...

, for t ∈ R. (1) The line lab is the set of parameterizations of the rotations of R 2 that take a to b. We consider the set of n lines L = {lab : a, b ∈ P}. (2) To prove that P determines Ω(n/ log n) distinct distances, it suffices to prove that the number of pairs of intersecting lines in L is O(n logn). Let Nk denote the number of points in R that are incident to exactly k lines of L, and let...

If φ : L → L′ is a bijection from the set of lines of a linear space (P,L) onto the set of lines of a linear space (P ′,L′) (dim (P,L), dim (P ′,L′) ≥ 3), such that intersecting lines go over to intersecting lines in both directions, then φ is arising from a collineation of (P,L) onto (P ′,L′) or a collineation of (P,L) onto the dual linear space of (P ′,L′). However, the second possibility can...

Quantifying the dissimilarity (or distance) between two sequences is essential to the study of action potential (spike) trains in neuroscience and genetic sequences in molecular biology. In neuroscience, traditional methods for sequence comparisons rely on techniques appropriate for multivariate data, which typically assume that the space of sequences is intrinsically Euclidean. More recently, ...

The theory of locally homogeneous geometric structures on manifolds is a rich playground of examples on the border of topology and geometry. While geometry concerns quantitative relationships between collections of points, topology concerns the loose qualitative organization of points. Given a geometry (such as Euclidean geometry) and a manifold with some topology (such as the round 2-sphere), ...

In this paper we show that the θ-graph with 4 cones has constant stretch factor, i.e., there is a path between any pair of vertices in this graph whose length is at most a constant times the Euclidean distance between that pair of vertices. This is the last θ-graph for which it was not known whether its stretch factor was bounded.

We are studying the dynamics of a one-dimensional field in a noncommutative Euclidean space. The non-commutative space we consider is the one that emerges in the context of three dimensional Euclidean quantum gravity: it is a deformation of the classical Euclidean space E3 and the Planck length lP plays the role of the deformation parameter. The field is interpreted as a particle which evolves ...

Euclidean geometry has been a de facto model for representing and processing data spaces. We have, however, started to see non-Euclidean geometries being used in many applications, including visualization, network measurement, and geometric routing. For these applications, Euclidean-based techniques may not be effective. As such, we need new mechanisms for managing and exploring non-Euclidean d...