Triangulation
Named by 16 essays across 6 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
No odd number of equal triangles
A square can be cut into two triangles of equal area, or four, or any even number. It cannot be cut into three, or five, or any odd number — whatever shapes the triangles take. Paul Monsky's proof of 1970 has no geometry in its engine at all: it colours the points of the plane by how divisible their coordinates are by two.
The fewest corners a surface needs
Build a closed surface from triangles, any two meeting along a whole edge, at a single corner or not at all, and ask for the fewest corners. Two lines of counting give a floor for every surface, in terms of its Euler characteristic alone. The torus meets it with seven, the projective plane with six — and the Klein bottle, which the counting says could be built from seven, cannot, as a search of every possible arrangement shows.
A hole is a cycle that bounds nothing
A hollow triangle and a filled one have the same three edges and the same loop round them. In one the loop is the edge of something and in the other it is not, and that difference — a cycle that is not a boundary — is what a hole is. Counting holes is rank and nullity applied twice, and the Euler characteristic is what is left when the two applications cancel.
What it costs to find the point that stays put
Brouwer's theorem promises a point a map leaves where it is, and Sperner's corridor walks to it. On a gentle map the walk is short; twist the map about its fixed point and the corridor follows every turn, so its cost grows like the square of the grid. Halving the square by winding number finds the same point in a number of readings proportional to the grid's side whatever the map does — and no method that only reads the map can do fundamentally better.
How nearly equal an odd number of triangles can be
A square cannot be cut into an odd number of triangles of equal area. It can be cut into five triangles whose areas differ by about two hundredths, seven that differ by three thousandths, nine by a ten-thousandth and a half — the spread falling by a factor of seven or more with every two triangles added, closing on equality and never reaching it. A search for the closest finds that the best five it can make have the golden ratio in their areas.
A total hung in a temple
Put a polygon's corners on a circle, cut it into triangles, and add up the radii of the circles inscribed in the triangles. Cut it a different way and the total is the same — for all fourteen ways of cutting a hexagon, to every digit. The fact was painted on a wooden tablet in a Japanese temple around 1800, and the reason for it is a theorem about one triangle and the distances from its circumcentre to its sides.
Named alongside it
The objects these essays reach for when they reach for this one.
Catalan numbersEuler characteristicFixed pointInvariantParityAreaDissectionBijectionBinary treesBoundaryBrouwerContinuity