Compass-only — the series
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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The centre a compass finds in six circles
A circle is drawn and its centre was never marked. A straightedge alone can never find it again. A compass alone can, from two points on the circle, in six circles — and a search through every construction of five circles or fewer shows that six is the least. The reason six works is an inversion that turns the circle into a straight line; the price of discarding the straightedge is exactly one move.
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The arcs that make the centre cheap
Finding a circle's centre with the compass alone, from two points on it, costs six circles when the points are placed with no special relation — and as few as two when the arc between them is 60°. Searching every arc in steps of five degrees maps the price exactly, and it is not a smooth function of the arc: it is a list of coincidences, each one a small number of compass steps that happens to land on a chord equal to the radius.