Concept

Embedding

A way of placing one shape inside another without the copy crossing itself, so that the shape keeps its own form in the larger space. How much room a shape needs to embed is a measure of how twisted it is.

Named by 2 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

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.

Immersion Binomial coefficient Boundary Closed surface Codimension Euler characteristic Gluing diagram Klein bottle Lucas' theorem Möbius band Orientability Pascals triangle

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