Generator

polyhedron

A generator in the geometry library, called 11 times across 2 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.

polyhedron 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

`show: "all"`

`show: "dual"`

`show: "budget"`

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.

3 triangles at a vertex leave a gap ×3
3 hexagons at a vertex cannot leave a gap ×1
3 pentagons at a vertex leave a gap ×1
3 squares at a vertex leave a gap ×1
4 pentagons at a vertex cannot leave a gap ×1
4 squares at a vertex cannot leave a gap ×1
6 triangles at a vertex cannot leave a gap ×1
exactly five vertex figures close up ×1

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.

Topology

Every corner pays for itself

Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.

Geometry

Why the list of perfect solids stops at five

There are infinitely many regular polygons and exactly five regular solids. The reason is not deep, but it is very sharp, and it can be checked on a single row of corners.

The whole library · What the figures prove