The collection

Every essay — page 22

Page 22 of 47, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Topology

What survives bending, and what does not.

A Möbius band from a strip of length √3. length 1.732050807569; closure error 5.55e-16; folded triangle 1.154701 by 1.000000.

How short a paper Möbius band can be

Two random loops and the crossings between them. 56 crossings, 25 positive, 31 negative, linking number -3.

Many crossings and a small linking number

How many subgroups of each index a surface group has. genus 1: 1,3,4,7,6,12,8,15; genus 2: 1,15,220,5275,151086,6605004,362069912,26370058035; genus 3: 1,63,7924,2757307,2081946006,3233368740252,9037199115520808,42295362888015649395; free F2: 1,3,13,71,461,3447,29093,273343.

The subgroups of a surface group

A curve that crosses a line twelve times. n=6: upper 9,8,7,6,5,4,3,2,1,0,11,10; lower 3,2,1,0,7,6,5,4,11,10,9,8; meanders 1828 of 17424.

Curves that cross a line and close

Random discs covering 63% of the plane. Boolean model of unit discs at eta=1 (seed 5) in an 18×18 square: 138 discs, 16 pieces, 3 holes.

Where random discs change sign

7 figures · new

A disc needs three pieces to shrink. Unit disc: two half-discs keep diameter 2; three 120-degree sectors have diameter 1.7321.

How many pieces to shrink a shape

7 figures · new

Probability

Randomness with a shape.

A Galton board after 600 balls. 600 balls fall through 12 rows of pegs, each bouncing left or right at random, and pile up in a bell-shaped heap.

A bell curve assembled out of coin flips

120 needles on a lined floor. 120 needles dropped at random across evenly spaced lines; 83 of them cross a line.

Getting pi by dropping needles on the floor

When a shared birthday becomes likely. The chance that some pair in a group shares a birthday, against group size. It passes a half at 23 people, where the probability is 50.7%.

Twenty-three people

7 figures $Bayes' theorem as two rectangles. A unit square split by how common the condition is (1.0%) and then by how the test behaves. Of everyone who tests positive, the fraction who have it is 16.7%.$

Bayes' theorem is a picture of a square

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

Three doors, as areas. Staying wins 33.3% of the time and switching wins 66.7%, because the host's choice is constrained by what the host can see, so opening a door rules a region out without moving any boundary.

The door that was not opened

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

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

The rule that forgets where it came from

five values, unevenly weighted, and the mass outside 3 standard deviations. A distribution drawn as bars, with the windows one and a half, two and three standard deviations wide marked. The probability outside each window is summed and compared with the bound that knows only the variance.

How far from the average a thing can be

One set of sums, two scalings, two different limits. The exact distribution of a sum of n independent copies, scaled two ways. Divided by n it collapses onto the mean; divided by the square root of n it holds a fixed width and settles into a shape.

The average settles and the wobble does not

When to stop looking

A path folded about the first time it touches. A walk from 2 to 4 that touches the axis, with the part before its first touch reflected. The reflection is a path from the mirrored start to the same endpoint, and the correspondence is exact.

The path folded at its first touch

Time spent on one side of the axis. The exact distribution of the number of steps a 40-step fair walk spends above the axis. It is U-shaped: the extremes are the likeliest outcomes and an even split is the rarest.

Half the time is the rarest answer

A walk with a barrier at each end. Three games played to absorption on a table of 12, beside the chance of ruin from each starting stake — a straight line, because the walk is fair.

Two barriers and a fair game

One walk at three magnifications, and the shape it is heading for. The same random walk over three windows, each ten times longer than the last and scaled vertically by the square root of ten, so all three look alike. Beside them, the exact distribution of the position after a few step counts, standardised, closing on the bell curve.

The walk that becomes a curve

A lopsided distribution added to itself, and the shape that returns. On the left, the exact distribution of a sum of copies of one lopsided distribution, standardised, for several counts: the shapes converge. On the right, the bell curve convolved with itself, which is the bell curve again.

The shape that averaging leaves alone

Averages of a heavy-tailed quantity, which never settle. Running averages of draws from a Cauchy distribution, which jump rather than converge, beside the cumulative distributions of averages of 1, 4 and 16 draws, which lie on top of one another.

An average that never settles