Concept

Polytope

The generalisation of a polygon and a polyhedron to any number of dimensions, bounded by flat pieces of one dimension lower. Regular ones are infinite in number in two dimensions, five in three and six in four, after which exactly three exist in every dimension.

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

The splits no group can beat, for three partners. The triangle of ways to split a fixed total between three players, with each coalition's demand drawn as a straight cut across it, and the region surviving every cut shaded.

A split nobody can walk away from

Every way of dividing what a group earns is a point of a triangle, and every coalition's threat to leave cuts a straight line across it. What survives all the cuts is the set of stable divisions — and for one three-player game there is nothing left.

applied · the core

The regular solids of four dimensions. 5-cell, tesseract, 16-cell, 24-cell, each turned in four dimensions and projected to the page; the edges are the pairs of vertices at the shortest distance apart.

Six in four dimensions, and three forever after

The count of regular solids goes five in three dimensions, six in four, and then three in every dimension above — for good. Four dimensions is the last place anything unusual happens, and it happens twice.

geometry · regular polyhedra

Named alongside it

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

Simplex Allocation Classification Coalition Constraint Cooperative game Core Dimension Duality Exhaustive search Fairness Platonic solids

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