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mathematics, in pictures
Concept

Finite geometry

A geometry with finitely many points, whose lines and planes are finite sets obeying incidence rules. Built from a finite field or from a combinatorial design, it supplies exact structures for error-correcting codes and experimental plans.

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

The octonion multiplication table, drawn as seven lines. The Fano plane with its seven points labelled by the imaginary octonion units and its seven lines carrying an arrow each, giving the products of every pair.

What is lost at eight

Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.

algebra · Quaternions A code made of the flats of a space. All 32 words of the Reed–Muller code RM(1, 4), each drawn on the sixteen points of the affine space of four dimensions over two elements: one word of weight 0, one of weight 16 and thirty of weight 8.

Parallel planes take a vote

Draw a four-dimensional space with two values to a coordinate as a 4 × 4 grid, and mark its hyperplanes: thirty-two patterns any two of which differ on eight of sixteen squares. A polynomial's top coefficient can be read off any plane, and parallel planes do not overlap, so an error spoils only one vote. One of them flew to Mars, and in 2016 it turned out to reach Shannon's limit.

computation · Finite geometry

Named alongside it

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

AssociativityChannel capacityDimensionDivision algebraError-correcting codeExhaustive searchFinite fieldHamming distanceNormPolynomialProjective planeQuaternion

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