Scissors congruence — the series

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.

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.

part 1 · 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.

part 2 · computation

The quantity a cut cannot change and a turn can. 4 polygons, each with the spikes of its translation invariant drawn round a dial: the length of the edges facing each direction, less the length of those facing the opposite way. It vanishes everywhere for 3 of them.

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.

part 3 · computation

What the chain costs on a 6-gon: 39 pieces. A regular 6-gon fanned into 4 triangles, each with the three cuts that turn it into a rectangle, beside the running count of the pieces the whole chain produces — 39 of them.

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.

part 4 · computation

Volume and one more number decide what a solid can be cut into. A table of a cube, a prism, a sixth of a cube, a regular tetrahedron, a regular octahedron and a collection of two tetrahedra with one octahedron, giving each one's volume, its Dehn invariant computed from its measured dihedral angles, and whether it can be cut into a box of equal volume.

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.

part 5 · computation

A spherical triangle becomes a quadrilateral with two right angles. A triangle on a sphere with the arc through the midpoints of two of its sides, the perpendiculars dropped from its three corners, and the quadrilateral with right angles at its base that the same area makes when the two corner pieces are moved.

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.

part 6 · computation

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.

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.

part 7 · computation

A triangulation of the square with 7 three-coloured triangles. A triangulation of the unit square into 34 triangles, vertices coloured by Monsky's rule, with the 7 triangles carrying all three colours shaded.

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.

part 8 · computation

The most nearly equal dissections of a square into 3, 5 and 7 triangles found. 3 triangles, areas 0.25000, 0.50000, 0.25000, spread 2.5000e-1; 5 triangles, areas 0.19098, 0.21353, 0.19098, 0.19098, 0.21353, spread 2.2543e-2; 7 triangles, areas 0.14151, 0.14458, 0.14151, 0.14458, 0.14151, 0.14174, 0.14458, spread 3.0711e-3.

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.

part 9 · computation