Matpic
mathematics, in pictures
Generator

Two points, and everything one round of compass and straightedge adds

A generator in the computation library, called 97 times across 22 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.

construct 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

Two points, and everything one round of compass and straightedge adds. Two starting points with the line and circles they permit, and the four points where those objects cross.

Hippias's quadratrix, traced by two uniform motions

Hippias's quadratrix, traced by two uniform motions. A unit square with a quarter circle, several positions of a turning radius and a falling horizontal line, their crossings, and the curve through them ending on the base at 2/π.

Cutting a 60° angle into 3 with the quadratrix

Cutting a 60° angle into 3 with the quadratrix. The quadratrix in a unit square, an angle of 60 degrees, the height where its arm meets the curve divided into 3 equal parts, and the 2 arms through the corresponding points of the curve.

Squaring the circle, once the quadratrix is drawn

Squaring the circle, once the quadratrix is drawn. Three steps: the quadratrix in a unit square with its foot at 2/π; similar triangles producing π/2; a rectangle of area π and a square and a disc of the same area.

The foot of the quadratrix, approached and never reached

The foot of the quadratrix, approached and never reached. The gap between the quadratrix and its foot at 2/π, against the height, on logarithmic axes: a straight line of slope two that never reaches zero.

Regular polygons with up to 24 sides, by three instruments

Regular polygons with up to 24 sides, by three instruments. A three-row grid for polygons with 3 to 24 sides, marking which can be drawn with compass and straightedge, with an angle trisector added, and with the quadratrix.

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 curve that divides any angle

Let a radius turn at a steady rate while a horizontal line falls at a steady rate, both finishing together, and mark where they cross. The curve they trace turns heights into angles, so dividing a height — which a ruler and compass can always do — divides the angle in the same ratio. The same curve meets its base at 2/π of the side, a length from which a square with the area of a circle follows. It reaches what no marked ruler or conic can, and the ancient objection to it is exact.

 Computation

A quintic a sliding mark reaches

The eleven-sided polygon needs a number of degree five, and five is not a product of twos and threes — so no conic and no angle trisector reaches it. A ruler with two scratches does, which places the marked ruler strictly above the conics and leaves its exact reach unknown.

 Computation

A third reached only in the limit

No compass-and-straightedge construction trisects every angle. But a quarter, plus a quarter of a quarter, plus a quarter of that, and so on, adds up to a third — and each of those pieces is two bisections away. So an angle can be trisected by bisecting forever, every stage exact and the shortfall shrinking to a quarter each time. The construction never ends, and that is precisely what Wantzel's proof forbids: a construction is a finite thing, and a third of a general angle is only reached in the limit.

 Algebra

A tower whose degrees multiply

Treat a field containing another as a vector space over it, and the size of an extension becomes a dimension — one that multiplies along a tower, so that three impossible constructions become arithmetic about which numbers divide which.

 Computation

Every step is a square root

A line meets a line by solving a linear equation and a circle by solving a quadratic one. There is no third case, so the numbers a construction reaches can only ever double in complexity — and a doubling is a thing that can be counted.

 Computation

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.

 Computation

The angle that will not divide by three

Halving an angle costs one circle. Cutting it in three means solving a cubic, and for sixty degrees that cubic has no rational root — but plenty of angles do trisect, and which ones is a question with a countable answer.

 Computation

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.

 Computation

The circle that will not square

The other three impossibilities are a number having the wrong degree. This one is a number having no degree at all — and that is a claim no finite search can establish, which makes it the one place in this field where the picture has to admit what it is not doing.

 Computation

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.

 Computation

The cube that will not double

Doubling a cube needs an edge in the ratio of the cube root of two. That number satisfies an equation of degree three, three does not divide any power of two, and the oldest open problem in geometry closes in a line.

 Computation

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.

 Computation

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.

 Computation

The mark that changes what is reachable

Two thousand years of failure to trisect an angle with compass and straightedge was failure at a stated set of operations. Scratch two marks on the straightedge and Archimedes trisects any angle in four steps — because the new operation solves a cubic, and the old ones could only ever solve quadratics.

 Algebra

The polygon an equation forces

The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.

 Computation

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.

 Computation

The spiral that measures its own circle

Let a ray turn steadily while a point moves steadily out along it, and the point draws Archimedes' spiral. Distance from the centre is then proportional to angle, so dividing a length divides an angle in any ratio. The spiral's second power is stranger: the tangent at the end of the first turn cuts off, on a line through the centre, a straight length exactly equal to the circumference of the circle through that end — a curved length laid out straight, and with it the circle squared. The catch is the tangent itself.

 Computation

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.

 Geometry

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.

 Computation

Two instruments with one reach

Allow every conic to be drawn at will, or allow an angle to be cut in three. The two permissions look nothing alike and reach exactly the same numbers — because what an operation buys is a degree, and both of these buy three.

 Computation

What two points can build

A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.

 Computation

Which polygons can be drawn

Three sides yes, seven no, seventeen yes. The list of constructible regular polygons is neither everything nor almost nothing, and the pattern in it is a fact about which numbers are one less than a power of two.

The whole library · What the figures prove