Concept

Frame — where it appears

An ordered list of independent directions at a point, one for each dimension. Two frames at the same point are compared by the determinant of the matrix carrying one to the other, and the sign of that determinant is the whole of what an orientation is.

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

Which axioms hold on which frames. A table of frames against modal axioms, each cell decided by checking the axiom under every valuation.

The axiom is the shape of the graph

Add one operator meaning necessarily and the choice of which axioms to accept stops being a matter of taste. Each candidate axiom is true of exactly those worlds-and-arrows diagrams whose arrows have a stated property, and a logic is a class of graphs.

logic · modal logic

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.

Accessibility Axiom Determinant Exhaustive search Handedness Intrinsic property Kripke model Modal logic Necessity Normal vector Orientability Orientation

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