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mathematics, in pictures
Concept

Triangulation

A division of a shape into triangles meeting edge to edge, with no gap and no overlap. Refining one is the standard way to turn a continuous question into a finite one, which is how Sperner's lemma reaches Brouwer's theorem.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

Every triangulation of a 6-gon. All 14 ways of cutting a convex 6-gon into triangles with non-crossing diagonals — the 4th Catalan number, counted by drawing them.

One sequence, counting everything

The number of ways to cut a polygon into triangles is 1, 2, 5, 14, 42. So is the number of ways to bracket a product, the number of binary trees, and the number of paths that never cross a diagonal. They are the same count, and the reason is one picture.

discrete · Catalan numbers The plane divided by nearest neighbour. 10 sites, and every point of the rectangle shaded by which site is closest to it. The boundaries are the places where two sites tie.

The plane, divided by whoever is nearest

Scatter some points and colour every other point of the plane by which one is closest. The result is a tiling nobody designed, and its dual triangulation has a property that no part of the construction mentions.

geometry · Voronoi A lattice polygon of area 22.5. A polygon with all its corners on the integer grid, with the 20 grid points strictly inside and the 7 on its boundary marked; its area is the first count plus half the second, less one.

Area by counting dots

Draw a polygon with every corner on a grid of dots. Count the dots strictly inside, add half the dots on the edge, subtract one — and the answer is the area, exactly, with no measuring anywhere.

discrete · Pick theorem A three-coloured triangulation, and the walk that finds a rainbow triangle. A triangle cut into 36 smaller ones, its corners coloured under Sperner's rule. The 9 small triangles carrying all three colours are shaded, and a path enters through a door on one edge and ends inside one of them.

Three colours force a triangle

Cut a triangle into small ones and colour the corners under one restriction. However the cutting and the colouring are done, some small triangle ends up with all three colours — and the number of them is always odd.

discrete · Fixed points Every ear carried to a slice of a disc. The corridor's triangulation on the left and the same triangulation of a regular 20-gon on the right, with four points and their images marked; the map is affine on each triangle and agrees on every shared edge.

Every loop is a circle in disguise

Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.

topology · Jordan curve Nineteen circles, one per vertex. A circle packing of a triangulation with seven interior vertices and twelve on the boundary: two circles touch exactly when their vertices are joined, and the radii were solved for rather than chosen.

Every flat graph is a pile of circles

A graph that can be drawn without crossings can be drawn in one particular way: as circles, one per vertex, touching exactly when their vertices are joined. The picture is not a choice — it is determined, up to the group two inversions generate.

geometry · Inversion One word, four objects. The balanced word (()())() drawn as a lattice path, as nested brackets, as a triangulation of a 6-gon and as a binary tree. The four are the same object in four notations, and each is built here from the word itself.

One word, and four objects

A balanced string of brackets, a lattice path, a triangulated polygon and a binary tree are four different-looking things counted by the same numbers. They are not four things that happen to agree — each is a way of writing the others down, and the translation is mechanical.

discrete · Catalan numbers The 14 triangulations, joined by single flips. The flip graph of a 6-gon: 14 triangulations drawn as small polygons and joined by 21 edges, one for each pair differing in a single diagonal.

The solid whose corners are triangulations

Take the triangulations of a hexagon as points and join two of them when a single diagonal can be swapped for another. The result is not merely a graph — it is the edge skeleton of a genuine convex polyhedron, with fourteen corners, three square faces and six pentagonal ones.

discrete · Catalan numbers A labelled square and the edges that join opposite labels. A square grid of 121 vertices, each coloured by one of four labels, with opposite boundary vertices carrying opposite labels, and 3 edges drawn thick where a label meets its negative.

Opposite labels that have to meet

Cut a square into triangles, label every corner +1, −1, +2 or −2, and insist only that opposite points of the edge get opposite labels. Somewhere inside, an edge must join a label to its negative. The proof counts quarter-turns round a diamond — an odd number on the boundary, zero in any triangle that avoids opposites — and making the triangles smaller turns the count back into the theorem about opposite points on the Earth.

topology · Borsuk ulam Dividing a rent of 90 three ways, on a 9-step grid. A triangle of possible rent splits, triangulated into 81 small triangles, with each grid point coloured by the room its housemate would pick; 3 small triangles have all three rooms.

A rent nobody envies

Three housemates, three rooms that are not alike, one rent. Every way of splitting the rent is a point of a triangle; ask, at each point of a fine grid, which room one housemate would take at those prices, taking turns so that each small triangle has one corner for each of them. Sperner's lemma then promises a small triangle where all three would choose different rooms — and as the grid is refined, the envy at that triangle shrinks to nothing.

applied · Fair division

Named alongside it

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

Catalan numbersFixed pointParityBijectionBinary treesContinuityConvexityCounterexampleCounting argumentEuler characteristicExistence proofFair division

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