Two opposite points that agree twice

One line that halves them both proved that a single straight cut halves any two shapes, by running one argument on a circle: a quantity that reverses sign between opposite points must be zero somewhere. It mentioned, as an aside, that the same theorem one dimension up puts two opposite points on the Earth with the same temperature and the same pressure.

That aside deserves its own figures, because the circle argument does not simply carry over. On a circle, a single reading reverses sign and the intermediate value theorem finishes the job. On a sphere there are two readings to match at once, and no single quantity changes sign along a path. What replaces the sign change is a count of windings, and it has to be odd.

Two readings, one pair of points #

Take any continuous temperature and pressure on a sphere. For each point $x$, form the two differences between the readings at $x$ and at the opposite point $-x$: $T(x) - T(-x)$ and $P(x) - P(-x)$. The Borsuk–Ulam theorem says there is a point where both differences are zero at once — a pair of opposite points that agree on both readings.

The globe in the figure has a temperature and pressure built from seeded random polynomials of degree three in the coordinates. The orange curve is where the temperature difference vanishes, and on its own that curve is no surprise: one equation on a two-dimensional surface usually picks out a curve. The blue curve is where the pressure difference vanishes. The theorem is the statement that the two curves must cross, and they do, at 11.9°N 44.6°E and at the point exactly opposite, 11.9°S 135.4°W. At both points the temperature reads 0.3305 and the pressure 0.0285, to nine decimal places after Newton’s method refines the crossing found on the grid.

The crossings come in pairs for a structural reason. If $x$ has both differences zero, then so does $-x$, because swapping a point with its opposite turns each difference into its negative, and the negative of zero is zero. Each curve is also symmetric under that swap, which is why the orange and blue curves each look the same after a half-turn of the globe and a flip from north to south.

Borsuk proved the theorem in 1933, answering a question of Ulam’s, and the statement holds in every dimension: a continuous map from the $n$-dimensional sphere to $n$-dimensional space sends some pair of opposite points to the same value.

Walking the equator #

The proof works by walking the equator. At each point of it, plot the pair of differences as a single point of the plane: temperature difference across, pressure difference up. Walking once round the equator traces a closed curve. For this globe the curve stays at least 0.566 from the centre, because neither agreeing point lies on the equator, and it winds once round the centre.

Walking half-way round the equator carries every point to its opposite, and that turns both differences into their negatives. So the second half of the curve is the first half reflected through the centre, and the figure checks that symmetry at all 720 sampled points. A closed curve built that way cannot wind round the centre an even number of times: each half contributes an odd number of half-turns, because the half starts at a point and ends at its reflection, and two odd numbers of half-turns make an odd number of whole turns. That is exactly the argument one line that halves them both used for maps of the circle sending opposite points to opposite points.

Now suppose the differences were never both zero anywhere on the northern half of the globe. Then the curve traced by the equator could be shrunk, by sliding the equator northwards across the hemisphere to the pole, down to the single point the pole gives, without ever passing through the centre. A curve that shrinks to a point without crossing the centre winds round it zero times. Zero is even; the winding is odd; so somewhere on the northern hemisphere, both differences vanish. That is the whole proof, and the winding count doing the work is the same one a loop that cannot miss the middle uses to find the roots of a polynomial.

Always an odd number of pairs #

The winding argument says more than that a pair exists. For a typical field — one whose curves cross cleanly rather than touching — each crossing on the northern hemisphere changes the winding of a shrinking loop by one, so the number of agreeing pairs has the same parity as the equator’s winding. The number of pairs is odd.

Ten globes with seeded random fields of degree five bear that out: six have one agreeing pair, three have three, and one has five, and every equator winds exactly once, clockwise or anticlockwise. Fields with more wiggles can have more pairs, but never an even number, and never none. A degree-five polynomial is still a very smooth field; a real atmosphere has far more structure, and the theorem does not care.

As the weather changes #

The field on a real globe changes from hour to hour, and the agreeing pair moves with it. Because the readings change continuously, the crossing of the two curves slides continuously too, and the pair wanders over the surface rather than jumping. New pairs can appear, but only in twos: two curves that were separate touch and then cross twice, creating two new pairs at once with opposite local windings, or two crossings meet and vanish together. Pairs are born and die in pairs of pairs, so an odd count stays odd through every change of weather, and at no moment is the count zero.

That is the same bookkeeping that keeps the equator’s winding fixed. Each simple crossing contributes a winding of plus one or minus one; creating or destroying two at once adds one of each, which changes the total by nothing, and the total is what the equator measures.

The sandwich in space #

The theorem on the ordinary sphere is exactly what proves the ham-sandwich theorem in three dimensions, which the halving-line essay stated without proof. Take three solids — bread, ham and cheese, placed anywhere. Every direction in space is a point on a sphere, and for each direction there is one plane perpendicular to it that halves the bread. That plane leaves the ham out of balance by some amount, and the cheese by another: two readings attached to each point of the sphere of directions.

Reversing the direction gives the same plane but swaps which side is which, so both imbalances change sign — the readings at opposite points of the sphere are exact negatives. Borsuk–Ulam supplies opposite directions with equal readings, and a reading equal to its own negative is zero. At that direction the plane halves the bread, the ham and the cheese at once. The theorem this essay draws on a globe of weather is, one step removed, the theorem that one straight cut halves a sandwich.

Three regions, and a pair in one of them #

The theorem has a second form, found three years before Borsuk’s by Lusternik and Schnirelmann in 1930. Cover the sphere with three closed regions, and one of them contains a pair of opposite points. The figure divides the globe into three regions, each made of the places nearest to some of nine seeded centres, and marks every sample point whose opposite point falls in its own region: 36.8 per cent of the 16,200 points, mostly in two of the three regions, but in each of them at least a little.

The two forms say the same thing. Given three closed regions, measure each point’s distance to the first region and to the second: that is a pair of continuous readings, and Borsuk–Ulam finds opposite points $x$ and $-x$ with equal distances to both. If both distances are zero, $x$ and $-x$ both lie in the first region. If the first is positive, neither is in the first region; if the second is positive, neither is in the second; and then both must lie in the third. In every case some region holds both.

The regions can be as ragged as anyone likes — a country, an ocean, a scattering of islands counted as one region — as long as they are closed and together cover everything. And the number three is exactly right for the two-dimensional sphere, as the next figure shows.

Four regions that avoid it #

With four regions the guarantee fails. Put four points on the sphere spaced like the corners of a regular tetrahedron and give each the places nearest to it; the result is the four faces of a tetrahedron blown up onto the sphere. Each region reaches 70.53° from its centre, which is $\arccos(1/3)$. The opposite point of anything in the region lies at least $180° - 70.53° = 109.47°$ from that centre, beyond the region’s reach, so no region ever contains an opposite pair — and no sample point in the figure shares a region with its opposite.

So the count is sharp: three closed regions always hold an opposite pair on the ordinary sphere, and four need not. In $n$ dimensions the numbers are $n + 1$ and $n + 2$, with the faces of an $(n+1)$-dimensional simplex providing the covering that escapes. The theorem is spending exactly one region per dimension, the same parameter count that let a line halve two shapes but not three.

A version with labels instead of readings #

The theorem also has a form with no continuity in it at all. Cover the sphere with a fine mesh of triangles that is symmetric under the swap of opposite points, and label every corner of the mesh with one of $+1$, $-1$, $+2$ or $-2$, subject to one rule: opposite corners get opposite labels. Tucker’s lemma, from 1946, says that some edge of the mesh then joins two corners labelled $+i$ and $-i$ for the same $i$ — a complementary edge.

The lemma is proved by counting, in the style of Sperner’s lemma for coloured triangles, and Borsuk–Ulam follows from it by a limit. Label each corner by which of the two readings’ differences is larger in size, with the sign of that difference. If no pair of opposite points agreed, the differences would never both be small, and on a fine enough mesh no edge could be complementary — contradicting the lemma. The continuous theorem is the limit of a finite count, just as the fixed point of something always stays put is the limit of Sperner’s coloured triangles, and the finite version is what algorithms actually search.

The same theorem in a problem about colours #

The antipodal theorem reaches far outside geometry. In 1955 Martin Kneser asked about a family of networks: take all ways of choosing $k$ items from $n$, and join two choices when they share no item. How many colours are needed so that joined choices always differ? An easy colouring uses $n - 2k + 2$ colours, and Kneser conjectured that no fewer would do.

László Lovász proved it in 1978, and the proof put points on a sphere, recorded which choices of items lay in open hemispheres around them, and used the Borsuk–Ulam theorem to show that a colouring with fewer colours would force two opposite points into the same colour class in an impossible way. It was the start of what is now called topological combinatorics. A question about counting the colourings of a network turned out to need opposite points on a sphere, and for many such questions nothing simpler has been found.

What the theorem does not give #

It does not say where. The pair in the hero figure was found by searching a grid and refining, not by the theorem; for a real atmosphere the theorem guarantees the pair exists at every moment and says nothing about finding it, which is the same gap between existence and location that one line that halves them both noted for the halving line.

It needs continuity everywhere. A field with a single jump — a temperature that changes discontinuously across a front drawn as a line — can escape the theorem, just as a jump let a quantity on a circle skip over zero.

And it is about opposite points only. The theorem singles out the swap that sends every point to the one directly opposite. Pairs of points related by some other symmetry, such as a quarter-turn about the axis, have no such guarantee; the argument needs the property that doing the swap twice returns every point to itself while moving each one as far as possible, which is what made the equator’s curve its own reflection.

Brouwer, and the combing of spheres #

The winding count that proves Borsuk–Ulam is the same invariant behind two neighbouring results. Something always stays put proves that a continuous map of a disc to itself has a fixed point, and Borsuk–Ulam implies it: a disc with no fixed point would give a map of the sphere to the plane sending no opposite pair to the same place. Nothing can be combed flat proves that a continuous field of directions on a sphere must vanish somewhere, by another count of turnings.

All three are statements that some invariant count cannot be zero. Here the count is forced to be odd by the antipodal symmetry, and oddness is a property that survives every continuous deformation.

Tame fields, clean crossings and sampled points #

The globes are smooth polynomials. A degree-three or degree-five field is a very tame surface; the theorem applies to any continuous field, including ones with structure at every scale, for which no grid could find the agreeing pairs reliably.

The odd count is checked, not proved, for typical fields. The parity argument needs the curves to cross cleanly; a field in which the two curves are tangent can have its pairs merge, and the count then needs multiplicity to stay odd. The ten random globes happen to be typical.

And the covering figures sample points. The dots and counts come from 16,200 sample points; a region’s boundary is closed but the samples near it are approximations, and the tetrahedral gap of 38.94° is what makes the zero count there trustworthy rather than a sampling accident.

Still open: a square on every closed curve #

A different question about continuous curves has resisted every argument of this kind. Draw any closed curve in the plane that does not cross itself. Are there always four points on it forming the corners of a square? Otto Toeplitz asked this in 1911. The answer is yes for smooth curves, for polygons and for many other classes, by arguments that count how often configurations of points cross each other — close relatives of the winding counts here. For an arbitrary continuous closed curve, the square peg problem is still open, even though every loop is a circle in disguise topologically, and even though a rectangle of any given proportions is now known to fit on every smooth closed curve.

An odd count cannot vanish #

The halving line needed one quantity to change sign. Matching two readings on a sphere needs more, and what supplies it is the winding of the equator’s differences round zero: forced to be odd by the fact that opposite points swap, and forced to be zero if the differences never vanished. Every smooth globe drawn has its agreeing pair, every covering by three closed regions has an opposite pair in one of them, and four regions are one too many for the guarantee.

When a symmetry makes a count odd, the thing the count measures cannot be absent. Parity is the cheapest invariant there is, and it proves existence in exactly the places where no construction is available.