Matpic
mathematics, in pictures
Concept

Orientability

The property of a surface that a consistent sense of turning can be chosen at every point at once. It fails exactly when carrying a frame round some loop returns it mirrored, and it is what a one-sided surface lacks, stated without reference to any surrounding space.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The star polygon {5/2}. 5 equally spaced points joined every 2th, forming a closed path that winds 2 times about the centre with an interior angle of 36.0 degrees at each point.

The four that are allowed to cross themselves

Drop convexity from the definition of a regular solid and four more appear. Their faces are pentagrams, they pass through one another, and the alternating sum that gives two for every ordinary solid gives minus six for two of them.

geometry · regular polyhedra 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 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 A frame carried round a Möbius band. A flat rectangle whose ends are about to be joined, with a pair of arrows carried along it — one along the band and one across it — and the sign of the frame at each station.

Orientation is a sign

Carry a pair of arrows once round a loop and compare them with the pair they started as. The comparison is a determinant, its sign is the whole of the answer, and no room, no side and no normal vector appears anywhere in the statement.

topology · orientability

Named alongside it

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

Euler characteristicAntipodal mapClosed surfaceGluing diagramImmersionProjective planeBoundaryClassificationCodimensionConnected sumCovering spaceCross cap

All concepts