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 4 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

Named alongside it

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

ConstructionCircleCompassConstructible numberMidpointOperation setReductionComplexityCross-ratioHarmonic conjugateIncidenceInversion

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