Critical point
Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.
The dark lines are one point's orbit
Past the end of the period-doubling cascade the bifurcation diagram turns into grey bands crossed by darker curves. Every one of those curves is the orbit of a single point — the top of the hump — and the places where they meet are exactly where the bands merge, in a second cascade that runs backwards at the same rate.
Pieces minus holes on a random surface
Cut a random landscape at some height and the region above it breaks into pieces with holes in them. How many pieces, and how many holes, has no formula. Their difference does: the average Euler characteristic of the region above height u is a constant times u multiplied by the bell curve at u, exactly, for every smooth Gaussian surface — because the Euler characteristic is a sum of peaks, pits and saddles, and the average number of those can be computed point by point.
Small Mandelbrot sets among the cubics
For most cubics Newton's method finds a root from almost every start. For some it does not: a whole region of starting points falls into a cycle and never finishes. Which cubics are the bad ones is decided by a single starting point — the one critical point that is not a root — and when the cubics are laid out in a plane and coloured by where that point goes, the bad ones form small, exact copies of the Mandelbrot set.
A knot must turn twice, and a little more
Follow a closed curve and add up how far its direction turns. Any closed curve turns through at least one full circle; a knotted one turns through more than two. The proof is a count of hilltops — the total turning equals 2π times the number of highest points the curve has, averaged over every direction to look from — and a curve with a single hilltop in any direction cannot be knotted.
One constant for each shape of top
Feigenbaum's 4.669 is the same for every smooth one-humped map — provided the hump is rounded like a parabola. Make the top flatter, like the graph of x to the fourth, and the doublings crowd together at 7.285; make it sharper, like x to the power 1.25, and they crowd at 3.260. Measured across twenty shapes of top, the constant is a smooth rising curve with 4.669 as one point on it, and the spatial constant falls as it rises.
The pieces no formula counts
A random vibration of a large membrane splits the surface into regions where it bulges up and regions where it dips down. Pieces minus holes has an exact average; the number of pieces has none. Nazarov and Sodin proved it is proportional to the area, Bogomolny and Schmit predicted the constant from percolation, 0.0624, and counting the pieces of 51 random waves gives 0.0599 — below the prediction by several standard errors. The sizes of the pieces follow percolation's law; their number does not.
Named alongside it
The objects these essays reach for when they reach for this one.
Period-doublingEuler characteristicFeigenbaum constantRandom fieldRenormalisationSaddle pointScalingAttractorBasin of attractionBifurcationChaosCurvature