Matpic
mathematics, in pictures
Concept

Approximation

A value put in place of another, together with a stated bound on how far apart the two can be.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

0123456721 steps of a rotation by φ − 1 of a turnthe gaps between neighbouring points take 2 distinct values — never more than three, at any number of steps

Three gaps and no more

Turn a circle by the same irrational angle over and over. The points never repeat and never settle, and yet at every single stage the gaps they leave take at most three different lengths — never four, at any number of steps, for any angle.

dynamics · golden ratio 9 of the 24 arrangements put nothing back where it startedthe shaded diagonal is where an object stays put; a marked grid is one with no mark on it

Nobody gets their own hat

Hand back a pile of hats at random and ask for the chance that not one person gets their own. The answer barely moves as the crowd grows — it is a third and a bit at four people, and a third and a bit at four thousand.

probability · inclusion exclusion 10 → 11.201 → 21.502 → 323 → 434 → 565 → 614.70 draws expected in all6 kinds, and the last is the expensive onethe waits are 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.70 drawsthe last one alone costs 6 draws on average, which is why the total grows faster than the number of kinds

How long until every one turns up

Draw at random from six equally likely kinds until all six have appeared. The wait is not six draws, and it is not sixty; it is fourteen point seven, and the number is a harmonic sum wearing a hat.

probability · expectation 1 steplength 2.00gap 0.7072 stepslength 2.00gap 0.2584 stepslength 2.00gap 0.14316 stepslength 2.00gap 0.035every staircase is exactly 2 long; a quarter circle is 1.5708 longthe finest one here is never more than 0.035 away from the curve, and it is still 2 long

The staircase that is not the diagonal

A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.

analysis · arc length

Named alongside it

The objects these essays reach for when they reach for this one.

RecurrenceSample spaceAlternating seriesArc lengthCollisionComplementary countingContinued fractionsContinuityConvergenceConvergence rateCounterexampleCounting argument

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