Circle
Named by 26 essays across 4 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.
The least wall for equal rooms
Divide the plane into rooms of equal area using as little wall as possible, and every wall does double duty. Bees settled on hexagons long ago, and the proof that nothing does better — not even rooms with curved walls — came in 1999. The straight-walled half of it is two facts: the rooms of any division average six sides, and more sides never cost more wall.
Every chord slid to the middle
Take a shape, pick a line, and slide every chord that crosses the line at right angles until the line cuts it in half. The area cannot change, the boundary can only get shorter, and the result is symmetric. Do it again about another line, and another, and the shape is squeezed towards a disc — unless the lines are badly chosen, in which case it stops short.
The skeleton inside a shape
Take the Voronoi diagram not of a handful of points but of a shape's whole outline: the points inside the shape that have two nearest points on its edge. What comes out is a skeleton — a thin branching tree that describes the shape and, with one number attached to each point, rebuilds it exactly. It is also where fires lit all round the edge would meet, and it is violently unstable: a tooth too small to notice grows a branch several times its own size.
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.
Random roots crowd onto the circle
The roots of a random polynomial are not scattered across the plane. They gather within about 1/n of the unit circle, and their directions spread evenly round it. Paul Erdős and Pál Turán proved in 1950 that this is not a fact about randomness at all: any polynomial whose coefficients are all of roughly one size has roots whose directions are nearly even, and the inequality says exactly how nearly, in terms of nothing but the coefficients.
How many pairs can be one apart
Place n points in the plane and count the pairs exactly one unit apart. A square grid with its spacing as the unit gives two pairs per point; the same grid with the unit chosen as a length it realises in many directions — the square root of 5, of 25, of 65, of 325 — gives three, five, eight. Counting points on circles caps the total at a constant times n to the four-thirds. Between that cap and the grids the truth has not moved since 1984, and at every size anyone can draw, the two look alike.
Wings that cut a chord equally
Take a chord of a circle and its midpoint. Draw any two more chords through the midpoint, join their ends crosswise, and the two crossing lines — the butterfly's wings — cut the first chord at equal distances from the middle. The proof is the inscribed angle and the power of a point working together; move the point off the middle and what survives is a law about reciprocals; replace the circle by any conic and the theorem does not notice.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
ConstructionConvexitySimilar trianglesAreaInscribed angleInversionOptimalityCompassIncidenceInvariantStraightedgeSymmetry