Concept

Unit distance graph

A graph whose vertices are points of the plane, with two joined exactly when they are one unit apart. How many edges it can have, and how many colours it needs, are two of the oldest open questions in discrete geometry.

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

The Moser spindle: 7 points, every edge one unit, and no three-colouring. the Moser spindle drawn to scale with 7 vertices and 11 unit-length edges, coloured with four colours; none of its 2187 three-colourings is proper.

How many colours the plane needs

Colour every point of the plane so that no two points exactly one unit apart share a colour. Seven colours are enough, by a tiling with hexagons, and four are necessary, by a graph of seven points. For sixty-eight years nothing better was known on either side; then in 2018 a graph of 1,581 points showed four is not enough — and the answer is still somewhere from five to seven.

discrete · Graph colouring

456 pairs at one distance among 144 grid points. A 12-by-12 grid of points with 456 segments joining every pair at distance √25, in 6 directions.

How many pairs can be one apart

Place n points in the plane and count the pairs exactly one unit apart. A square grid with its spacing as the unit gives two pairs per point; the same grid with the unit chosen as a length it realises in many directions — the square root of 5, of 25, of 65, of 325 — gives three, five, eight. Counting points on circles caps the total at a constant times n to the four-thirds. Between that cap and the grids the truth has not moved since 1984, and at every size anyone can draw, the two look alike.

geometry · Inscribed angle

Named alongside it

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

Axiom of choice Chromatic number Circle Compactness Exhaustive search Graph colouring Incidence Lattice Prime Sums of two squares Tiling

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