Sphere
Named by 13 essays across 4 fields — each of them below, with the objects they name alongside it.
Two pieces, in every dimension
A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.
A ball whose outside is not one
Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.
The number four points agree on
One inversion is a reflection and reverses orientation. Two of them compose to a motion, and what that motion leaves alone is a single number computed from any four points.
Why the second group commutes
Replace loops by spheres and the same construction gives a second homotopy group. It is always commutative, and the reason is not a fact about spheres or about any space — it is a two-line argument about any set carrying two compatible operations.
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.
Zero in four dimensions
Corners minus edges plus faces is two for every solid. One dimension up, corners minus edges plus faces minus cells is zero for every one of the six regular four-dimensional solids, from the five-cell to the six-hundred-cell, and for every other convex solid in four dimensions. The alternating sum does not break when the dimension rises: it alternates, two in odd dimensions and zero in even ones, because it is measuring a sphere and not a solid.
The highest point on the sphere is an eigenvalue
For a symmetric matrix, walk a unit arrow over every direction and record the value of xᵀAx. The highest value reached is the largest eigenvalue, the lowest is the smallest, and every eigenvalue in between is a saddle height, a minimum of maxima. From that one description comes a theorem no formula for the roots could give: delete a row and its column, and every eigenvalue of what is left sits between two of the original's.
A sphere that is nearly all equator
On an ordinary globe, the band within a tenth of the radius of the equator holds a tenth of the surface. On a sphere in a thousand dimensions the same band holds 99.85% of it, and the band of a fifth holds all but about two parts in ten billion. Almost every point of a high-dimensional sphere is near every equator at once — and so any function that cannot change quickly is, over almost all of the sphere, almost constant.
Three mirrors make every solid
Every symmetry of a regular solid, reflections included, is produced by just three mirrors meeting at its centre, reflected in one another over and over — a kaleidoscope. Put a single point between the three mirrors and its reflections are the corners of a solid: the regular solid itself if the point sits in a corner, and every one of its truncated and expanded relatives if it sits anywhere else.
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.
The ball that is largest in five dimensions
A disc of radius one has area π, a ball of radius one volume 4π/3, and in each further dimension the unit ball grows — until five, where its volume is 5.264, after which it shrinks towards nothing. The recursion that shows it is the ring dissection of the disc, done one dimension at a time, and the shrinking is not the ball getting small: it is almost all of a high-dimensional cube lying outside the ball, and almost all of the ball lying in a thin rind at its surface.
An instrument that computes with circles
An astrolabe is a stereographic projection of the sky cut in brass. Because the projection sends circles to circles, every line on it can be drawn with compasses, and turning the star map over the plate is a rotation that answers questions in spherical astronomy without a single calculation. Of the lines medieval makers engraved, one family is not a circle on the sphere — and the arcs they drew for it miss the truth by a third of a millimetre.
Valleys a search cannot count
Let equal charges on a sphere push each other apart from a random start, and they settle into an arrangement where nothing moves. Up to fifteen charges every start settles into the same one. Sixteen is the first number with two, a hundredth of a unit apart in energy; by fifty-two there are at least four, and for fifty-six charges only one start in ten finds the lowest. The other valleys sit a few parts in a million above it, and nothing in the arrangement a search returns says whether it is the bottom.
Named alongside it
The objects these essays reach for when they reach for this one.
DimensionEuler characteristicGroupProjectionAreaConformalHomeomorphismHypercubeInversionJordan curveKnotMobius transformation