Matpic
mathematics, in pictures
Concept

Torus — where it appears

The surface of a doughnut — a sphere with one handle, on which a closed curve need not separate the surface in two.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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

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 · homotopy a slab with one hole through it: 32 vertices,64 edges, 32 faces — and 32 − 64 + 32 = 0every face is flat and every edge is straight,and the answer is 2 − 2g with g = 1 ratherthan 2

The solid where the answer is not two

A slab with a hole through it has flat faces, straight edges and sixteen corners, and its alternating sum is zero. It is not a trick and not a degenerate case — it is the object that shows the theorem had a hypothesis nobody had written down.

topology · euler characteristic 012345623456012601234512345601→ the right edge is the left edge →↑ the top edge is the bottom edge ↑each row is offset by half a brick, so every brick touches two above and two below as wellas one on each side — six neighbours, and the labels differ by one, two and threethe left and right edges are the same edge and so are the top and bottom, which is whatmakes this a doughnut rather than a rectangle; all seven regions meet all six others, soseven colours are needed

Seven regions on a doughnut

A map on a torus can need seven colours, and the proof is a picture — seven regions, each sharing a border with all six others. The plane needed a computer and eighty-six years; the harder surface was settled in 1890 by drawing something.

discrete · graph colouring

Named alongside it

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

GenusEuler characteristicComplete graphConnectednessContinuityCounterexampleCovering spaceDeformationFundamental groupGraph colouringHeawood numberHomotopy

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