Circle
Named by 17 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.
Half a circle against a wall
Lay a fence of fixed length with both ends against a straight wall and the best shape is a half-circle, holding exactly twice what a full circle of the same fence holds. The proof is a mirror: doubled in the wall, any fence becomes a closed curve with twice the length and twice the area, and the closed-curve answer carries over. In a corner the same mirrors give a slice of a circle — until the corner's angle stops dividing a half-turn.
Nearly the most means nearly round
A shape that holds almost as much as a circle of the same perimeter must almost be a circle. Bonnesen made that exact: the ring between a convex shape's largest inscribed circle and smallest enclosing circle is never wider than √(L² − 4πA)/π. Three quite different shapes holding 99% of the circle's area all have rings under 9.55 wide, and not one of 200 random convex shapes breaks the bound.
One circle touching four
The nine-point circle touches the inscribed circle and each of the three escribed ones. Nothing in its construction mentions them, the two families of centres are built from different kinds of number, and the tangency is four exact equalities between distances and radii.
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.
One number for every chord through a point
Draw any line through a point and let it cut a circle twice. The two distances from the point to the circle multiply to the same number whichever line is drawn — inside, outside, or grazing as a tangent. The number belongs to the point, and the reason it does not depend on the line is the inscribed angle: two chords through a point cut out two triangles with the same angles.
Named alongside it
The objects these essays reach for when they reach for this one.
ConstructionConvexitySimilar trianglesInscribed angleInversionTangencyAreaConstructible numberIncidenceLocusOptimalityStraightedge