Two at once

Metrics for more than two points at once

The conventional definition of a topological metric over a space specifies properties that must be obeyed by any measure of “how separated” two points in that space are. Here it is shown how to extend that definition, and in particular the triangle inequality, to concern arbitrary numbers of points. Such a measure of how separated the points within a collection are can be bootstrapped, to measu...

Seeing two doctors at once in general practice.

We report the reactions of 250 patients who saw two doctors together, a general-practitioner trainer and a vocational trainee, when they came to a general practice for consultation.Over 80 per cent were neutral and the remainder were almost equally divided between those who preferred to see two doctors and those who preferred to see their own doctor alone.Selected favourable statements outnumbe...

How can you be two places at once when you're not anywhere at all?

Looking at two paintings at once: Luminance edges can gate colors

UNLABELLED Two paintings, O1 and O2, were split into their luminance (grayscale) components L1, L2 and their color components C1, C2. The two color components, C1, C2, were transparently superimposed. Adding the grayscale of the first painting (= C1 + C2 + L1) looked like the original O1, while adding the grayscale of the second painting (= C1 + C2 + L2) looked like the original O2. CONCLUSIO...

Arcs Intersecting at Most Once

We prove that on a punctured oriented surface with Euler characteristic χ < 0, the maximal cardinality of a set of essential simple arcs that are pairwise nonhomotopic and intersecting at most once is 2|χ|(|χ|+1). This gives a cubic estimate in |χ| for a set of curves pairwise intersecting at most once on a closed surface. We also give polynomial estimates in |χ| for sets of arcs and curves pai...