Construction
Named by 21 essays across 6 fields — each of them below, with the objects they name alongside it.
Two injections make a bijection
If each of two collections fits inside the other without collisions, they are the same size. That sounds obvious and is not, because neither injection needs to be onto — and the proof is a rule for deciding which of the two to follow, one chain at a time.
Three trisectors and a triangle nobody expected
Cut every angle of a triangle into three. The trisectors nearest each side meet in three points, and those three points are always the corners of an equilateral triangle — for every triangle there is, with no exceptions and no reason anybody finds obvious.
The rope that squares a corner
The theorem turns two sides into a third. Run it backwards and it turns three lengths into a right angle — which is a different statement, needs its own proof, and is the only one of the two that has ever been used to build anything.
The map that trades circles for lines
Send every point to the one on the same ray whose distance multiplies with it to a fixed number, and circles become lines, lines become circles, angles survive untouched, and a ring of tangent circles falls out of a ring of equal ones.
The straightedge buys nothing
Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.
Approached too fast to be algebraic
An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.
A field's worth of squares
Two orthogonal squares of order five are easy to stumble on. Four of them, every pair orthogonal, is not a stumble — it is one line of arithmetic over a field, and the field supplies as many as the order allows.
A page that knows where it is
A four-by-four array of bits, cyclic in both directions, in which every two-by-two block appears exactly once. Print it repeatedly across a sheet and any four marks on that sheet are an address.
Eight circles touching three
Draw three circles. How many circles touch all three? The answer is eight, the count is a fact about signs rather than about geometry, and the classical way to find them is to move the problem somewhere it becomes easy.
One circle, and a straightedge
A straightedge alone cannot bisect a segment, so it cannot draw a parallel, so it can construct almost nothing. Draw one circle anywhere and mark its centre and everything a compass could ever have done becomes available — the circle is never needed again.
The compass that will not open
Fix the compass at one opening and never change it. That looks like a serious loss — a circle of a given radius through a given point is the compass's whole job — and it turns out to cost nothing at all, for reasons that are arithmetic rather than geometric.
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.
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.
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.
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.
The price of a construction
Three theorems have shown that a compass alone, a straightedge with one circle, and a compass stuck at one opening all reach exactly the points a full set of instruments reaches. None of them said what the journey costs. Émile Lemoine priced every movement of the hand in 1888, and by his count a compass that collapses when lifted — Euclid's — pays twenty-one operations, by Euclid's own method, for what a compass that holds its opening does in four.
A walk that splices in its own detours
Euler proved that a walk crossing every bridge once needs every landmass to have an even number of bridges, and then stated, without proof, that this was enough. The missing half took 137 years, and it is not an argument but a procedure: walk until stuck, notice that stuck can only mean home, and splice in a detour from anywhere with edges left. The procedure never fails, and the reason fits in one sentence about arriving and leaving.
Every count a solid can have
Euler's formula ties the corners, edges and faces of a convex solid together, but it does not say which numbers are possible. Ernst Steinitz answered in 1906: V corners and F faces occur together exactly when neither is more than twice the other less four. The proof in one direction is two lines of counting; in the other it is two moves — cut off a corner, glue on a tetrahedron — that walk from the pyramids to every allowed pair.
The centre a straightedge cannot find
Give a straightedge one circle and its centre, and it can do everything a compass can. Take the centre away and it cannot even find it again — because to a straightedge a circle has no centre. The maps that keep a circle and its straight lines are the motions of the hyperbolic plane, and in that plane the centre is a point like any other.
The lengths dividers cannot reach
A pair of dividers carries a length from one place to another and draws nothing. With a straightedge it finds midpoints, parallels and right angles, and it draws the regular 17-gon. It cannot draw a segment of length √(1 + √2) — and the reason is not on the page at all, but in the other root of the equation that number solves.
The fewest moves to draw it
Every construction with ruler and compass is a sequence of moves — draw this line, draw that circle — and it is natural to ask for the shortest. Nobody can answer that by cleverness alone, but a machine can answer it by trying everything: the midpoint of a segment takes four moves and no fewer, the square on it five, a third of it five. Take the ruler away and the compass pays for it: six circles for the midpoint, seven for the square.
Named alongside it
The objects these essays reach for when they reach for this one.
Operation setCircleStraightedgeCompassConstructible numberDissectionInvariantAreaCongruenceCounting argumentExhaustive searchInversion