Matpic
mathematics, in pictures
Generator

triangulation

A generator in the discrete library, called 5 times across 1 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

triangulation is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

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

show: "catalan"

The Catalan numbersC(0) to C(8), each computed twice — by the convolution that defines them and by the closed form — and required to agree.nC(n)011122354145426132742981,430ways to cut a polygon · ways to bracket a productpaths that never cross the diagonal · binary treesone sequence, counting all of them

What it checks while it draws

Collected by running the family and listening to lib/verify.js, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Discrete

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 whole library · What the figures prove