The same thing twice — page 20
Dynamics
The silver mean at the flat point
At the golden-mean rotation number, the circle map with a flat point has two universal constants, 2.8336 and 1.2886. Repeat the measurement at the silver mean, √2 − 1, and two different numbers appear — 6.7992 and 1.5868 — shared by every map with the same kind of flat point. Below the critical line the silver gaps shrink by (1 + √2)², the square of the silver ratio, exactly as a rigid rotation predicts; on it they do not. Every rotation number with a repeating continued fraction carries its own pair, and a number whose expansion ends in ones inherits the golden pair, whatever came first.
Geometry
The price of curvature in Morley's triangle
On a sphere, Morley's trisectors meet in a triangle that is not quite equilateral, and its failure grows with the curvature the triangle encloses. For a small triangle the failure is proportional to the angle excess, with a coefficient set by the triangle's shape alone. Computed over every shape, the coefficient is nought for the equilateral triangle, at most 0.00303 for any shape with no angle below 11°, largest for obtuse triangles near 139°, 23° and 19° — and exactly the same on the hyperbolic plane, where the excess is negative.
Geometry
The longest fence of diameter one
Among polygons with n sides and diameter one, the one with the most area is regular only when n is odd. Ask for the most perimeter instead and the answer is kinder: Karl Reinhardt proved in 1922 that the longest perimeter is 2n sin(π/2n) whenever n has any odd factor at all, reached by polygons cut from curves of constant width, many of them not regular. Only when n is a power of two does every polygon fall short — and the best quadrilateral, found by searching every one, is an equilateral triangle with one corner bent out.
Probability
The best middle when the shape is unknown
The mean is the best estimate of a centre when the data are normal and worthless when they are Cauchy. The median is best for Laplace data and wastes a third of normal data. Between them lie the trimmed means — drop a share from each end, average the rest — and across seven symmetric shapes the best guarantee comes from trimming about 28 per cent from each end, which never does worse than 81 per cent of the best possible. Peter Bickel proved in 1965 that trimming can never cost more than a known factor against the mean, whatever the shape.