Circle
Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.
An angle that does not care where it stands
Fix two points on a circle and look at them from anywhere else on the far arc. The angle is the same from every one of those places, and it is exactly half the angle at the centre.
The plane, divided by whoever is nearest
Scatter some points and colour every other point of the plane by which one is closest. The result is a tiling nobody designed, and its dual triangulation has a property that no part of the construction mentions.
What two points can build
A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.
The most area a fence can hold
One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.
Nine points on one circle
Three midpoints, three feet of altitudes and three more midpoints. Nine points defined in three unrelated ways, on an arbitrary triangle, and all nine sit on one circle — checked here on two hundred and forty triangles as well as on the drawn one.
The map that trades circles for lines
Send every point to the one on the same ray whose distance multiplies with it to a fixed number, and circles become lines, lines become circles, angles survive untouched, and a ring of tangent circles falls out of a ring of equal ones.
Eight circles touching three
Draw three circles. How many circles touch all three? The answer is eight, the count is a fact about signs rather than about geometry, and the classical way to find them is to move the problem somewhere it becomes easy.
Curvatures that stay whole
Four circles touching one another satisfy an equation in their curvatures. Read it as a quadratic and the second solution is the first subtracted from something — so a packing that starts with whole numbers stays whole forever.
When the sites are not the same size
Give every site a weight and the boundaries slide. The cells stay convex and the edges stay straight, which is surprising, and one thing happens that the unweighted diagram never allows — a site can end up owning nothing at all.
Angles survive and areas do not
Stereographic projection takes every circle on the sphere to a circle or a line, and every crossing angle to itself. It does both exactly, with no approximation anywhere, and it destroys area so thoroughly that a patch near the pole can be a thousand times its neighbour's size.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
ConstructionInversionConstructible numberConvexitySimilar trianglesTangencyAngleAreaConformal mapDualityIncidenceInscribed angle