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mathematics, in pictures
Concept

Binary trees

Trees in which every node has a left child and a right child, either of which may be absent. They are counted by the Catalan numbers, and the count is where several apparently unrelated families of objects turn out to be the same family.

Named by 5 essays across 2 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 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 Seven flips between the octagon's farthest triangulations. A shortest flip path of length 7 between two of the 132 triangulations of the octagon.

How far apart two triangulations can be

Any triangulation of a polygon can be turned into any other by flipping one diagonal at a time. On an octagon seven flips always suffice; on a polygon with m vertices the worst case is 2m − 10 flips once m reaches thirteen, a value Sleator, Tarjan and Thurston proved with hyperbolic geometry in 1988. Breadth-first search over all 58,786 triangulations of the 13-gon finds the sixteen.

discrete · Catalan numbers The flip graph of the hexagon, given a direction. Tamari lattice T4: 14 elements 3210, 3200, 3010, 3100, 3000, 0210, 0200, 1010, 0010, 2100, 2000, 0100, 1000, 0000; 21 covering relations; 68 intervals.

Pairs of trees that count planar maps

Give every flip of a triangulation a direction and the triangulations become an order, the Tamari lattice. Counting the pairs that are in order — 1, 3, 13, 68, 399, 2,530, 16,965, 118,668, 857,956 — gives 2(4n + 1)!/((n + 1)!(3n + 2)!), which is exactly Tutte's 1962 count of planar triangulations. Chapoton noticed the match in 2006; a pair of trees and a map on a sphere turn out to be the same thing.

discrete · Catalan numbers The tree of solutions of x² + y² + z² = 3xyz. Markov triples from (1, 2, 5) to depth 4; 231 Markov numbers below 10^15; first 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325.

The numbers that come after the golden ratio

The golden ratio is the number worst approximated by fractions: no constant larger than √5 works in Hurwitz's theorem. Set it aside and √2 is worst, with √8. Set that aside and the next is a number whose continued fraction repeats 2, 2, 1, 1, with √221/5. Markov proved in 1879 that the list continues for ever below 3, one entry for each solution of x² + y² + z² = 3xyz.

geometry · Golden ratio

Named alongside it

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

Catalan numbersTriangulationBijectionCounting two waysExhaustive searchLattice pathsRecursionConjectureContinued fractionsConvexityDiophantine approximationFlip

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