Matpic
mathematics, in pictures
Generator

loops

A generator in the topology library, called 6 times across 2 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

loops is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

3 loops in one ring, and the number that separates themLoops drawn in an annulus, each labelled with how many times it goes round the hole. Loops with different counts cannot be deformed into one another without leaving the ring.0 times round1 time round2 times roundthe number under each loop is counted by walking it and adding the angle turned through, as seen from the holea loop can be slid and stretched at will inside the ring without changing that number, and there is no way tochange it without leaving the ring

show: "shrink"

One shrink, allowed in a disc and not in a ringA loop pulled straight in toward the centre, drawn at six stages. In a disc every stage lies inside the region; in an annulus the later stages cross the hole, which is not a deformation inside the ring at all.in the disc: down to a pointin the ring: through the holethe same six stages in both: the disc allows every one of them, the ring stops allowingthem at stage 3 of 6this settles one attempt, not all of them — that the count cannot change under anydeformation is what rules the rest out

show: "compose"

1 round, then 2 round, is 3 roundTwo loops in a ring, and the single loop got by running the first and then the second. Its count is the sum of theirs.1 round2 round3 roundthenis1 then 2 gives 3: the count of the concatenation is computed from the joined samples, not from the sumwhich makes the classes a copy of the whole numbers under addition, with the constant loop as zero

show: "cover"

A loop that ends 3 turns above where it startedThe loop in the ring on the left, and the angle it has turned through followed continuously on the right. The path downstairs closes; the one upstairs finishes a whole number of turns higher.the loop, in the ring00.20.40.60.81123along the loopturns accumulatedthe loop closes up downstairs and does not close up upstairs: it ends 3 turns above where it beganthe count is exactly that gap, which is why it cannot change a little — the two ends are either level or a wholeturn apart

show: "torus"

Loops on a torus, and the two counts that classify themThe torus drawn as a square with opposite edges identified, and three loops on it. Each is labelled with how many times it crosses each pair of edges.1 across, 0 up0 across, 1 up2 across, 1 upopposite edges of the square are the same edge, so a path leaving the right side comes back in onthe leftthe two counts cannot be traded against each other: no amount of sliding turns a path that crossesone edge into one that crosses the other

show: "odd"

An odd map turns an odd number of timesTwo maps of the circle to itself, drawn as spirals so that repeated passes can be counted. The first sends opposite points to opposite points and turns an odd number of times; the second does not and turns an even number.opposite points to opposite pointsturns 3 timesnot that kind of mapturns 2 timesthe dashed chord joins the images of one pair of opposite points: on the left it runsthrough the centre, on the right it does notthe left map turns 3 times and the right one 2, and it is the first that cannot be an evennumber

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Topology

A loop that cannot be pulled tight

A hole is a strange thing to point at, because it is precisely where the surface is not. What can be pointed at is a loop of string lying on the surface — and the hole announces itself by refusing to let that loop be pulled in to a point.

 Topology

One line that halves them both

Two shapes lying anywhere on a page, of any sizes and any shapes at all. There is always a single straight line that cuts both of them into two equal halves at once — and finding it needs no cleverness, only the observation that a quantity which reverses sign has to pass through zero.

The whole library · What the figures prove