Matpic
mathematics, in pictures
Concept

Immersion

A smooth map whose derivative never collapses, so that every small patch of the source is a faithful patch of the image. It is weaker than an embedding, which is also one to one, and it is what allows a Klein bottle to be drawn in three dimensions at the cost of a crossing.

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

The Klein bottle, drawn where it does not fit. A closed one-sided surface in three dimensions, drawn as a tube with a figure-eight cross-section that turns over once on the way round, with the circle where the drawing passes through itself marked.

The bottle that needs a fourth dimension

Take the Möbius band's rectangle and glue the second pair of edges too. The result is closed, one-sided, and cannot be built in three dimensions without passing through itself — which is a fact about the room rather than about the surface.

topology · Orientability The gluing abab makes a projective plane. A polygon whose edges carry the word abab, with arrows for the direction each edge is glued and the corners coloured by which vertex they become.

A disc sewn to a Möbius band

The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.

topology · Orientability Pascal's triangle mod 2, and the room for the 2-dimensional projective space. Pascal's triangle with odd entries filled, rows 0 to 15, with row 3 marked as the tangent ledger 1 + a + a² and row 1 as the normal ledger 1 + a.

The room a projective space needs, read off Pascal's triangle

The projective plane cannot sit in three-dimensional space without crossing itself, and the reason can be written as arithmetic: a polynomial that records how a shape twists, which a room must cancel. For the n-dimensional projective space that polynomial is a row of Pascal's triangle read mod 2, its inverse is another row, and the inverse's last term says how many extra dimensions the room must have — exactly enough, at every power of two.

topology · Orientability

Named alongside it

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

Closed surfaceEmbeddingEuler characteristicGluing diagramOrientabilityProjective planeAntipodal mapBinomial coefficientBoundaryClassificationCodimensionConnected sum

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