Concept

Antipodal map

The map sending each point of a sphere to the point diametrically opposite it. It moves every point, so the quotient by it is again a surface, and that quotient is the projective plane — the standard way a one-sided surface is built out of a two-sided one.

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

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

A sphere over a projective plane, cell by cell. An icosahedron with opposite faces drawn in matching colours, beside the count of cells it has and the count the quotient by the antipodal map has — every number halved, including the Euler characteristic.

Two sheets over a one-sided surface

Above every one-sided surface sits a two-sided one, exactly twice as large, and the map between them forgets which of the two senses of turning a point was carrying. Building it turns a question about sides into a question about covers.

topology · orientability

Named alongside it

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

Euler characteristic Orientability Projective plane Classification Closed surface Connected sum Covering space Cross cap Deck transformation Free action Fundamental group Gluing diagram

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