Matpic
mathematics, in pictures
Concept

Quotient

The object obtained by declaring some elements of a structure to be the same and keeping only what survives that identification. Its points are the classes of the relation, and its structure is whatever the original structure induces on them.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

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 chain of 18 worlds, and the 3 the formulas can tell apart. A row of 18 circles for the worlds of the model, shaded by which of the 3 classes each falls into, above the quotient model's 3 worlds with the arrows the collapse gives them.

How many worlds a formula can need

A modal formula can be true in a model with infinitely many worlds. It can also be true in a small one — and the small one is built from the large one by throwing away every distinction the formula was never able to make.

logic · Modal logic The three pairings of the roots of x⁴ + x + 1. Three panels each showing the same four roots of a quartic in the complex plane, joined in a different way into two pairs, with the value of the sum of the pair products beneath.

Three ways to pair four roots

Four roots can be split into two pairs in exactly three ways, and the three numbers r·r′ + r″·r‴ those pairings give are the roots of a cubic whose coefficients can be read straight off the quartic. That cubic is where Ferrari's formula gets its cube roots, and it is also a verdict: whether its roots are rational decides which of the twenty-four symmetries the quartic's roots actually have.

algebra · Galois correspondence

Named alongside it

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

Antipodal mapBisimulationCompletenessCovering spaceDecision procedureDeck transformationDiscriminantEquivalence relationEuler characteristicFinite model propertyFree actionFundamental group

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