Matpic
mathematics, in pictures
Concept

Squaring the circle

The problem of constructing, with straightedge and compasses alone, a square whose area equals a given circle's. It is impossible, because it would require constructing the square root of π, and π is transcendental; curves drawn by other means, such as a spiral, can do it.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Looking for a polynomial with π as a root. A table of the closest an integer polynomial of each degree comes to vanishing at the number, over a bounded search.

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 · Constructible numbers The spiral's tangent lays the circumference out straight. Spiral r = aθ to θ = 6.2832; tangent at P meets the perpendicular through O at T with OT = 6.28319 = OP × θ = 6.28319.

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 · Neusis The five lunes that can be squared. Five crescents with arc ratios 2:1, 3:1, 3:2, 5:1, 5:3 and their equal-area kites; θ = 45.000°, 34.265°, 26.812°, 23.439°, 16.794°.

The five crescents that can be squared

The circle cannot be squared, but some regions bounded by circular arcs can: Hippocrates of Chios found a crescent with exactly the area of a triangle around 440 BC. The trick is to make two multiples of π cancel, and it works for a crescent whose two arcs turn through angles in the ratio m to n exactly when an angle defined by a polynomial can be drawn with ruler and compass. That happens for five ratios and no others.

geometry · Circle area

Named alongside it

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

Constructible numberPiAlgebraic numberAngle trisectionArchimedean spiralAreaDegree of an extensionField extensionGalois groupMethod of exhaustionMinimal polynomialPolynomial

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