Matpic
mathematics, in pictures
Concept

Method of exhaustion

Pinning a quantity down by bounding it above and below with pieces that can be made as small as needed. It is what Archimedes used for the circle and it is the ancestor of the limit, arriving two thousand years before one was defined.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

A disc unrolled into a triangle. A disc cut into 12 concentric rings, and the same rings straightened and stacked. The longest is the outer circumference; the shortest is nearly a point; the stack is a triangle.

A circle unrolled into a triangle

Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.

geometry · Circle area A square cut into 7 pieces and a remainder. A square divided by cutting off a fixed fraction of what is left, over and over, so that the pieces are the terms of a geometric series and the uncut corner is the tail.

The sum that fits in one square

Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.

analysis · Geometric series A circle trapped between two 12-sided polygons. A circle with a regular polygon of 12 sides inscribed in it and another circumscribed about it, beside a table of the bounds on pi obtained by doubling the side count.

Pinned between two sequences

The ring dissection makes the answer obvious and proves nothing. Archimedes' method proves it and makes nothing obvious — it never exhibits the area at all, it rules out every other value — and the recursion that drives it computes π by hand with one square root a step.

geometry · Circle area A hemisphere and a cylinder with a cone taken out, sliced at one height. Two solids drawn in profile — a hemisphere, and a cylinder with a cone removed — each cut at the same height, with the disc and the annulus the cut produces marked and their equal areas given.

The slice that has to match

The same slicing one dimension up gives the sphere's volume in a line, once one comparison is noticed: at every height a hemisphere's disc has exactly the area of a cylinder's slice with a cone's taken out of it. The principle that licenses that comparison also returns a false answer the moment the slices are not parallel.

geometry · Circle area Bands of a sphere and of its cylinder, cut by the same planes. A unit sphere inside its cylinder with 3 horizontal bands; each sphere band has area 1.885, 1.885, 1.885, equal to the cylinder band of the same height.

A band of a sphere is a band of its cylinder

Cut a sphere with two parallel planes and the band between them has exactly the area of the band the same planes cut from the cylinder wrapped round the sphere — whatever the band's latitude. Near the equator the sphere's band is wide and nearly upright; near a pole it is narrow and nearly flat; and the two effects cancel exactly. Archimedes proved it, wanted it on his tomb, and it gives the area of the sphere, the only honest way to pick a random point on it, and a map on which no country is the wrong size.

geometry · Circle area 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

Named alongside it

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

LimitPiArchimedesAreaCircle areaDissectionConvergence rateAngle trisectionApproximationArchimedean spiralConstructible numberConvergence

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