Matpic
mathematics, in pictures
Generator

A triangle cut into three pieces that make a rectangle

A generator in the computation library, called 49 times across 9 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

dissect 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.

With nothing chosen

A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

A dissection that never comes apart

A dissection that never comes apart. The three pieces of the triangle-to-rectangle dissection drawn at 4 moments of the swing. Each top piece turns about a pin at the end of the slice it stands on, and the pieces stay joined throughout.

A 9 by 4 rectangle and a 6 by 6 square, from one staircase cut

A 9 by 4 rectangle and a 6 by 6 square, from one staircase cut. A rectangle cut by a staircase into two pieces, beside the square the same two pieces make when one of them is slid by a single step.

A 33 × 32 rectangle cut into 9 unequal squares

A 33 × 32 rectangle cut into 9 unequal squares. A squared rectangle of 9 squares with sides 18, 15, 14, 10, 9, 8, 7, 4, 1, each labelled with its size.

Squaring a 13 × 8 rectangle greedily, which is Euclid's algorithm

Squaring a 13 × 8 rectangle greedily, which is Euclid's algorithm. A 13 by 8 rectangle divided into squares of 8, 5, 3, 2, 1 and 1 by repeatedly cutting off the largest square.

A squared rectangle and the electrical network hidden in it

A squared rectangle and the electrical network hidden in it. The 33 × 32 squared rectangle with its 6 horizontal segments highlighted, beside a network in which those segments are nodes and each of the 9 squares is a wire joining two of them.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Computation

A dissection that never comes apart

The plane theorem lets the pieces be picked up and put down anywhere. Require instead that they stay joined at their corners and swing, and the theorem survives — which was open for a century and is a much stronger statement about the same cuts.

 Computation

A rectangle made only of squares

A rectangle can be cut into finitely many squares — of any sizes, as many as wanted — exactly when its two sides are in whole-number proportion. Max Dehn proved it in 1903, and the proof that stuck, found by four Cambridge undergraduates in 1940, reads the squares as currents in an electrical circuit.

 Computation

Equal area is enough, and equal volume is not

Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.

 Computation

Equal area on a sphere, without a rectangle

On a sphere, two polygons of the same area can still be cut into each other, exactly as in the plane. Almost nothing in the plane proof survives the move: a sphere has no rectangles, no parallel strips and no similar triangles of different sizes. What carries the theorem instead is a quadrilateral with two right angles, built from a triangle's midline.

 Computation

Finitely many, and nobody says how many

The theorem promises a dissection exists and the proof produces one. Running the proof on a hexagon produces thirty-nine pieces, ingenuity produces five, and there is no method for proving that five cannot be four.

 Computation

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.

 Computation

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.

 Computation

Slid, but never turned

The classical dissections all turn their pieces. Forbid the turn — allow the pieces to be slid and nothing else — and equal area stops being enough, for a reason that is a single number attached to each direction and that a cut cannot change.

 Computation

The obstruction that was the only one

Dehn showed in 1901 that a cube cannot be cut into a regular tetrahedron of the same volume, because a number built from edges and angles disagrees. For sixty-four years nobody knew whether that number was the whole story. Sydler proved in 1965 that it is: volume and Dehn's number together decide every case.

The whole library · What the figures prove