Matpic
mathematics, in pictures
Concept

Straightedge

An instrument that draws the line through two given points and nothing else. Its operations are preserved by every projective transformation, which is exactly why it cannot find a midpoint on its own.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

The midpoint of a segment, drawn with a compass and no straightedge. A segment with the arcs that step its length three times round one end to reach the point twice as far away, and the further arcs that send that point back to the midpoint, every one of them a circle.

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.

computation · Compass-only The circle is used once, and its centre is the point. A circle with its centre and one diameter, a point above it, and the straightedge-only construction of the parallel to that diameter through the point.

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 · Compass-only A compass that will not change its opening. A segment longer than twice the compass's fixed opening, with the opening stepped along it 2 times and the remaining piece bisected by two arcs of that same opening.

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 · Compass-only The midpoint of AB: simplicity 11. A construction drawn step by step — the midpoint of AB, using ruler and compass — with each circle and line numbered in order and Lemoine's count of its operations beneath.

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 · Compass-only A straightedge construction on a circle, and the same construction moved by a map that keeps the circle. Two copies of one straightedge construction on a circle — six points, five chords and their crossings — the second the image of the first under a projective map fixing the circle. Chords and crossings correspond exactly, but the centre (orange) is carried to (0.551, 0.000).

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 · Compass-only The spiral of Theodorus, √2 to √17, built from square corners and a unit length. 16 right triangles with legs √k and 1 arranged in a spiral around a common corner; their long sides have lengths √2 to √17, and together they turn through 351.2 degrees.

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 · Compass-only

Named alongside it

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

ConstructionCircleCompassConstructible numberOperation setMidpointReductionComplexityConjugateCross-ratioField extensionHarmonic conjugate

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