Concept

Recurrence

The return of a system arbitrarily close to a state it has already been in. For a fair walk on the line it happens with certainty, and in three dimensions it does not, which is a fact about dimension rather than about walks.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

Nine walks, and the square root. 9 independent walks of 400 steps, each step one place left or right. The dashed curves are ±√n: the walks stay near them, spill past them, and come back — which is what a typical distance means as opposed to a limit.

A walk that always comes home, until it does not

Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.

probability · Random walk

Rotating by √2 − 1 of a turn, 40 times. Points on a circle produced by repeatedly turning through the same angle.

The orbit that must come back

A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

dynamics · Pigeonhole

All 24 arrangements of 4 objects, and the 9 that move every one. Every permutation of 4 objects drawn as a grid of cells, with the diagonal — where an object stays where it began — shaded, and the arrangements that avoid it entirely marked.

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

Waiting for all 6 kinds. One bar per new kind: the expected number of draws needed to see a kind not yet seen, rising as fewer of them are left, and adding to 14.70 draws in total.

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

The tree of Pythagorean triples. A tree rooted at 3-4-5. Each triple has three children, obtained by three fixed integer matrices, and every primitive triple appears exactly once somewhere in it.

A tree that holds every triple

Three fixed matrices, applied to 3-4-5 over and over, produce every primitive Pythagorean triple there is — each of them once, none of them twice, and with no test for common factors anywhere in the procedure.

number · Pythagoras

256 consecutive pairs from xₙ₊₁ = 137xₙ + 187 mod 256. Consecutive outputs of a linear congruential generator plotted as points of a square, falling on a small family of evenly spaced parallel lines.

The planes a recurrence cannot leave

One multiplication and one addition, taken modulo a fixed number, produce a sequence that passes for random one value at a time. Taken two or three at a time it does not, and the reason is a whole-number relation that pins every point onto one of a small family of parallel lines.

computation · Pseudorandomness

A 4-bit register that visits all 15 nonzero states. The first 15 states of a 4-bit linear feedback shift register with taps at 4 and 1, with the bit that leaves the register at each step; the output shows every nonzero window of 4 bits exactly once.

A memory of four bits

A register holding four bits, shifting them along and adding two of them back, runs through all fifteen nonzero states before it repeats. Which two are added back is a question about a polynomial, and getting it wrong costs fourteen of the fifteen.

computation · De bruijn

One walk on the whole numbers, three chances, three different fates. The relative weight of each state for three step-up chances, drawn as bars, with the running total of those weights and what each case means beneath.

Where the shares have nowhere to go

On finitely many states, a chain that can reach everywhere and is not forced into a rhythm settles down. Give it infinitely many and both conditions can hold while the walk leaves and never returns — or returns with certainty and takes an unbounded average time about it.

probability · Markov chains

Named alongside it

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

Approximation Counting argument Expectation Limit Random walk Sample space Alternating series Bijection Binomial distribution Collision Complementary counting Convergence

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