Paper Theorem

A paper theorem, on Drawn Mathematics, is a mathematical fact whose construction you can fold from a single sheet of paper, such as a length of √2 or the trisection of an angle. "Paper theorem" is the name of a section of this site, not a standard term in mathematics. You fold two A4 sheets before reading the proof in the section's first article, A4 Paper Size: Built So Each Half Keeps the Same Shape.

Two halves, always

An article in this section always has two halves, in this order:

1. The demonstration. Folds you can make with the sheet in front of you, to build the length, angle or point in question.
2. The theorem. The statement, the assumptions it depends on, and a proof, or a pointer to where the proof was published.

The order is fixed. The demonstration comes first, so the statement has something concrete to point at; the theorem comes second, so the article does not stop at a single folded sheet.

An article with a demonstration but no proof, and no pointer to one, does not carry the paper theorem name.

Why the demonstration is not the proof

A folded sheet shows one case. Real paper has thickness, and every crease lands a little off the ideal line, so a fold can agree with the statement only as closely as your eye can judge. The statement in the second half is about exact lines and points, and only the proof covers every case its assumptions allow.

The demonstration is there so you can see and hold the construction before reading the proof. It never replaces the proof.

What the folds are, and are not

The folds in a paper theorem build a point, a length or an angle. They do not make an origami model, so they are neither folding instructions for one nor a crease pattern for one.

Paper theorems sit in the Crease Patterns category, which also covers how to read crease patterns and the theorems of flat folding. How crease patterns are checked is described on Checking the construction.