Matpic
mathematics, in pictures
Generator

gluing

A generator in the topology library, called 5 times across 1 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.

gluing 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

The four ways to glue a square's edges in pairsSquares with their edges arrowed to show which is glued to which and which way round, each with the vertices, edges and faces the gluing leaves, and the surface those numbers name.abbaabb⁻¹a⁻¹V 3 − E 2 + F 1 = 2two-sideda sphereabababa⁻¹b⁻¹V 1 − E 2 + F 1 = 0two-sideda torusabababab⁻¹V 1 − E 2 + F 1 = 0one-sideda Klein bottleababababV 2 − E 2 + F 1 = 1one-sideda projective planethe corners are coloured by the vertex they become, so the count of colours is the V in V − E + Fthe characteristic and the orientability are computed from the word; the name underneath is looked up from that pair

show: "word"

The gluing aba⁻¹b⁻¹ makes a torusA polygon whose edges carry the word aba⁻¹b⁻¹, with arrows for the direction each edge is glued and the corners coloured by which vertex they become.ababcorners 1, 2, 3, 4 become one vertexthe 4 corners fall into 1 class, so V − E + F is 1 − 2 + 1 = 0every letter is used once each way round, so a consistent sense of turning survives the gluing

show: "handles"

A sphere, and a sphere with handlesTwo-sided closed surfaces in order: a sphere, then one with a handle, then two, then three, each with the Euler characteristic that counts them.a sphereχ = 2a torusχ = 0a surface with 2 handlesχ = −2a surface with 3 handlesχ = −4a sphere with g handles has characteristic 2 − 2g, so the number falls by two for every handleon the two-sided side of the classification, that number is the whole of the answer: two surfaces with the same characteristic are thesame surface

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

Every surface is a sphere with handles

Take a square and say which edges are to be glued to which, and which way round. Four such rules give four different surfaces — and two numbers computed from the rule, without ever building the surface, say which one.

The whole library · What the figures prove