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mathematics, in pictures
Concept

Dissection

A cutting of a shape into pieces that reassemble into another shape with no gap, no overlap and nothing left over. It is the site's flagship kind of proof, and the assertion behind one is that the pieces tile the target with no gap and no overlap.

Named by 19 essays across 5 fields — each of them below, with the objects they name alongside it.

The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right.

Two squares, four triangles, and no algebra

The Pythagorean theorem is usually met as a formula to be memorised. It is much better met as a rearrangement that can be checked by eye.

geometry · Pythagoras A disc unrolled into a triangle. A disc cut into 12 concentric rings, and the same rings straightened and stacked. The longest is the outer circumference; the shortest is nearly a point; the stack is a triangle.

A circle unrolled into a triangle

Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.

geometry · Circle area Euclid's proof, without moving anything. The square on a leg and its share of the square on the hypotenuse are each exactly twice the same triangle, so they are equal. Nothing in the figure is cut or rearranged; the triangle is only looked at from the other side.

Euclid proves it without moving anything

The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.

geometry · Pythagoras Completing the square, as a square. An x by x square with the strip split in half and laid along two sides, leaving a square hole of side 1.5. Filling the hole costs 2.25 and buys a perfect square.

Completing the square, by completing a square

The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.

algebra · Completing the square A lattice polygon of area 22.5. A polygon with all its corners on the integer grid, with the 20 grid points strictly inside and the 7 on its boundary marked; its area is the first count plus half the second, less one.

Area by counting dots

Draw a polygon with every corner on a grid of dots. Count the dots strictly inside, add half the dots on the edge, subtract one — and the answer is the area, exactly, with no measuring anywhere.

discrete · Pick theorem A square cut into 7 pieces and a remainder. A square divided by cutting off a fixed fraction of what is left, over and over, so that the pieces are the terms of a geometric series and the uncut corner is the tail.

The sum that fits in one square

Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.

analysis · Geometric series A rectangle grown on two sides. A rectangle x by √x, with both sides grown by the change a step of h makes. The new area is the old one, two strips, and a small corner rectangle that has both increments in it.

A rectangle grown on two sides

A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.

analysis · The derivative 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.

computation · Scissors congruence 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 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 · Scissors congruence 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.

computation · Scissors congruence 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.

computation · Scissors congruence 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.

computation · Scissors congruence 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.

computation · Scissors congruence A hemisphere and a cylinder with a cone taken out, sliced at one height. Two solids drawn in profile — a hemisphere, and a cylinder with a cone removed — each cut at the same height, with the disc and the annulus the cut produces marked and their equal areas given.

The slice that has to match

The same slicing one dimension up gives the sphere's volume in a line, once one comparison is noticed: at every height a hemisphere's disc has exactly the area of a cylinder's slice with a cone's taken out of it. The principle that licenses that comparison also returns a false answer the moment the slices are not parallel.

geometry · Circle area 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.

computation · Scissors congruence 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.

computation · Scissors congruence 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.

computation · Scissors congruence A square cut into rectangles of ratio φ squared. 1/r@(0.0000,0.0000) 0.1273×0.3333; r@(0.1273,0.0000) 0.8727×0.3333; 1/r@(0.0000,0.3333) 0.1273×0.3333; r@(0.1273,0.3333) 0.8727×0.3333; 1/r@(0.0000,0.6667) 0.1273×0.3333; r@(0.1273,0.6667) 0.8727×0.3333

The rectangles a square can be cut into

Cut a square into rectangles that all have the same shape — the same ratio of long side to short, in any sizes, lying either way. Two to one is easy. The golden ratio is impossible, however many pieces are used and however small. The ratio φ squared works with six. What decides it is not the ratio but its partner: the number obtained by changing the sign of the square root, which must be positive. Changing that sign is a map that keeps every sum and product and scrambles every size, and areas that must add up to one cannot all turn negative.

computation · Scissors congruence Dudeney's four pieces, as a triangle and as a square. Dudeney dissection of the equilateral triangle of side 2 (area 1.732051) into a square of side 1.316074: J at (0.25451, 0.44082), K at (0.75451, 1.30685); piece areas 0.40568, 0.68076, 0.39787, 0.24775.

Dudeney's four pieces are the fewest

In 1902 Henry Dudeney cut an equilateral triangle into four pieces that rearrange into a square, and hinged them so the triangle folds into the square in one motion. For a hundred and twenty-two years nobody could say whether three pieces would do. In December 2024 Erik Demaine, Tonan Kamata and Ryuhei Uehara proved they would not. The construction, built exactly here, shows why four is natural; the proof shows why it is necessary.

computation · Scissors congruence

Named alongside it

The objects these essays reach for when they reach for this one.

AreaInvariantOperation setCongruencePolygonConstructionImpossibilityLimitMethod of exhaustionRigid motionTriangulationArchimedes

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