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Concept

Compass

An instrument that draws a circle of a given radius about a given point, and thereby carries a length from place to place. Removing the straightedge entirely, or fixing its opening, changes neither what it reaches nor what its partner reaches.

Named by 2 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 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

Named alongside it

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

Constructible numberConstructionMidpointStraightedgeCircleCross-ratioIncidenceInversionOperation setProjective mapReductionSimilar triangles

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