Cobweb
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
The staircase that shows the whole orbit
Take a number, feed it to a rule, feed the answer back in. There is a way of drawing that on the rule's own graph which turns the entire future of a starting point into a shape — and the shape is legible.
A point that pulls, and a point that pushes
Every crossing of a curve with the diagonal is a value the rule leaves alone. Whether anything ever arrives there is decided by one number — the slope at the crossing — and the picture makes the reason obvious.
The sine of one degree
Every table of sines ever made by hand needed the sine of one degree, and no construction with ruler and compass can produce it: one degree is a third of three, and thirds are what the constructions cannot make. Ptolemy fenced it in with an inequality and got it right to the precision his table carried. Thirteen centuries later Jamshīd al-Kāshī wrote the triple-angle formula as a rule for improving a guess and let it run, gaining almost three decimal places at every turn — a cubic solved without Cardano, by doing the same thing again.
Named alongside it
The objects these essays reach for when they reach for this one.
IterationFixed pointLogistic mapOrbitAttractorChordConstructible numberContractionConvergenceCubicDegree of an extensionDerivative