Concept

Eigenvalue — where it appears

A number a matrix multiplies one of its own directions by. Here they are used as a verdict rather than as a calculation: the signs of a distance matrix's eigenvalues say whether a set of places has a flat picture, and their sizes say how badly it fails to.

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

Also named here as gram matrix — the same set of essays touches all of them, so they are one junction rather than several.

The four places, and every distance between them. London, New York, Tokyo, Sydney on a Mollweide projection, with every great circle between a pair drawn. The six ground distances run from 5,570 km to 16,994 km, and they are the whole of the input to every figure in this ladder — no coordinate, no projection and no coastline enters any of them. The arcs are drawn only to say which pair each number belongs to; on this page they are curves, and on the ground they are the shortest routes.

Four cities that cannot be drawn to scale

Every impossibility in this field so far has been about a surface. This one is about four numbers: London, New York, Tokyo and Sydney have six distances between them, and no four dots on any sheet of paper have those six separations. The test is a determinant Cayley wrote down in 1841, and it comes out −3.34 × 10²³ where zero is required.

impossibility · Embedding

The page's curvature is not a free parameter. The same trilateration, carried out on a sphere of stated radius instead of on a plane, with the error of the distances the construction decides rather than holds. At the radius the distances were measured on it is 5.1e-15 — exact, by construction, which is the refusal this figure exists to make. Two per cent either side of it the worst decided distance is already out by around 10%. Below 90 per cent of the Earth's radius the construction cannot be completed at all: two circles that must cross do not, and the page is simply too small to hold the places. The flat sheet is the right-hand limit, at 152%.

The escape is not a dimension

Four places that will not lie in a plane surely lie in a space — and they do not. The double-centred matrix of their great-circle distances has a negative eigenvalue, so no Euclidean space of any dimension holds them, and buying a third dimension improves the picture by nothing whatever, to fifteen decimal places. What does work is a page with curvature, and the curvature is pinned to within two per cent.

impossibility · Embedding

Named alongside it

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

Cayley menger Closed form Distance matrix Embedding Gaussian curvature Gram matrix Isometry Verification Chord Constraint Dimension Theorema Egregium

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