Two opposite places on the same spot

The first rung of this ladder establishes that a projection cannot be defined, continuous and one to one on the whole sphere at once, and counts which of the three each library member gives up. That is a statement about what is impossible. It is silent about the shape of the failure, and the shape turns out to be far more constrained than the theorem needed it to be.

Suppose a projection keeps continuity. It is a genuine function of the place — not of a longitude that jumps, not of a coordinate the poles do not possess, but of the point itself — and it is continuous at every place on the Earth. It must then fail to be one to one, and the obvious guess is that where it fails is a design decision: a fold here, a doubled patch there, wherever the construction happens to put it.

It is not a design decision. Every continuous map of the whole sphere into a page sends some pair of antipodal places to the same point. Whichever map, however it was built, whatever it was built for.

The theorem, and the trick that makes it computable #

Borsuk–Ulam, in the form that applies here: for any continuous map f from the sphere to the plane there is a point p with f(p) = f(−p).

The proof is short and the shortness is what makes it a measurement rather than a citation. Define

$$g(p) \;=\; f(p) - f(-p).$$

Then g is continuous, g maps the sphere into the plane, and g is odd: g(−p) = −g(p) identically, by construction, with no assumption about f at all. Borsuk–Ulam is exactly the statement that a continuous odd map of the sphere into the plane has a zero, and a zero of g is a glued antipodal pair.

So the search is a two-dimensional root find. Sample g over the sphere, take the best point of the grid, and run Newton in the tangent plane. There is nothing approximate about what comes back: a zero either converges to machine precision or it does not exist, and that difference is what makes the same procedure usable as a refusal later in this essay.

Five maps, five pairs #

The theorem says nothing about projections. It says something about continuous maps, and a projection is only one kind. So the measurement is run over five, of which two are maps of the world and three are not maps of anything.

Every one of them is written in the three Cartesian components of the unit vector for the place, which is the condition that matters: a function of the place has no antimeridian in it, no pole line, and nothing to be discontinuous at. That is precisely what the library’s projections are not — they are functions of a longitude and a latitude, and longitude is a circle numbered by an interval, so they carry the interval’s endpoint with them wherever they go.

The pairs themselves are worth reading. The orthographic glues the two ends of its own view axis, which is the minimum the theorem permits: exactly one antipodal pair and no other. The obliquely viewed orthographic glues the two ends of its axis, at 130.1° west and 40.1° south, which is nowhere special and is not a place either the map or the theorem chose. The odd cubic pair glues the south pole, and it does so for a reason that is worth stating — both of its components are odd functions of the place, so g = 2f, and the glued pair is a zero of the map itself rather than of a difference.

What continuity actually costs #

The theorem names one pair, and one pair is not what a continuous map really pays.

An orthographic view is two to one almost everywhere: the near hemisphere and the far one are drawn on top of each other, so half the world is under the other half. The theorem’s contribution is that among all those doubled pairs, at least one is antipodal — which is a small-sounding statement about a very large fold.

So the honest summary of the trade is this. A map that gives up continuity pays a curve — the seam, whose two page edges the first rung measures across the library at between 4.4 and 45.5 units apart, for a ground curve of 20,004 kilometres in every case. A map that gives up injectivity pays a region, and the theorem’s guaranteed minimum is a detail of it rather than a description of it.

That asymmetry is why every world map anybody actually uses is cut. It is not a convention, and it is not because cut maps look better. A cut costs a set of dimension one and a fold costs a set of dimension two, and the first rung’s point about incommensurable units does not apply here, because for once the two quantities are the same kind of thing and one is unambiguously smaller.

The refusal #

An assertion that cannot fail is not an assertion, and every claim on this site is required to have an input it must reject. The input here is a projection that has been cut.

Mollweide is defined at every place, single-valued at every place, and discontinuous across the antimeridian, so it is not a continuous map of the sphere and Borsuk–Ulam has nothing to say about it. If the search finds a glued antipodal pair anyway, the machinery is finding something that is not there.

The floor is the width of the map. Two antipodal places on a cut map are separated by roughly half of it in longitude and the cut prevents them ever meeting, so the best the search can do is a number of order one rather than of order 10⁻¹⁴. Six refinements do not move it. The refusal is not marginal and does not depend on where the tolerance was set.

This is where the mechanism becomes visible rather than merely proved. The reason a cut map escapes is not that it is cleverer than the theorem. It is that the seam is a discontinuity in the function of the place — one ground curve, two page curves — and the intermediate-value argument at the heart of Borsuk–Ulam runs on continuous functions and stops at a jump.

Every continuous pair of quantities, not just position #

The version of Borsuk–Ulam used here happens to take the two page coordinates as its pair of quantities, but the theorem does not know that they are coordinates. Any two continuous functions on the sphere will do, which turns it into a family of statements that are surprising in different ways.

Take a and b, the two principal scale factors of some projection, as a continuous pair. On any projection where both are continuous over the whole sphere there is an antipodal pair of places whose indicatrices have exactly the same two semi-axes. Take the areal factor and the angular deformation: there is an antipodal pair with identical distortion by both measures at once. Take temperature and pressure, which is the form of the theorem usually quoted: there are two opposite places on the Earth with the same temperature and the same barometric pressure, at every instant, whatever the weather is doing.

None of those is a coincidence and none of them is about the quantity. All three are the same statement about the sphere, and the reason it is stated in this ladder rather than in the distortion field is that it survives any change to what is being measured. Change the projection and a and b change; the antipodal pair with equal a and b does not stop existing.

Why a cut is the one anybody ships #

A fold and a cut both destroy injectivity. They destroy it differently, and the difference is not aesthetic.

On a cut map, a page point names exactly one place, everywhere except on the seam. A reader who lands on the seam is in genuine trouble — an outline that crosses it becomes two pieces or one enormous one, and whether a point is inside a ring stops having an answer — but the trouble is confined to a curve and can be described, guarded and tested for.

On a folded map, a page point names two places over half the sheet, and nothing on the page says which. Every operation this site has priced becomes ill-posed at once: a nearest-feature query has two candidate answers, a pixel’s ground footprint is two footprints, an area is double-counted. A cut breaks a curve’s worth of questions; a fold breaks half the questions there are.

So the trade is not between two equally awkward failures. It is between a failure of dimension one and a failure of dimension two, and every world map anyone has ever printed has taken the first. The orthographic escapes this only by being used as a picture of a hemisphere rather than as a map of the world: nobody reads the far side off it, because nobody draws the far side on it, and declining to draw it is the first rung’s other escape rather than this one.

The globe, and the gore #

There is a physical version of all of this and it is older than any of the mathematics.

A globe gore is a long thin lens of paper that goes onto a sphere without stretching, and a globe is made of a dozen or so of them. Every gore is bounded by two meridians, and the boundaries are cuts: a place on the join between two gores is printed on both, once at the right edge of one and once at the left edge of the next. That is the seam of this essay, made of paper, and the polyhedral ladder counts exactly the same structure when it enumerates the spanning trees of a solid’s face graph and finds that the cut edges of a net are one.

What the gore maker never does is fold. There is no globe anywhere made by printing two hemispheres on top of each other, and the reason is the one above rather than a limitation of printing. A cut can be hidden along a line nobody reads across; a fold has nowhere to be hidden.

The same choice appears at every scale this site works at. Interrupted projections buy shape by adding more cuts and the essay prices exactly how much. A polyhedral map is a cut taken to its logical end. None of them is a fold, and none of the twenty projections that would be improved by folding has ever been drawn.

Where the model stops #

Continuity is required over the whole sphere, and that is a real restriction. Only two of the eighteen projections in this site’s library are continuous as functions of the place — the orthographic and the gnomonic — and both of them are, because both are undefined on a region rather than cut. A projection that omits ground is not a map of the sphere at all, so the theorem applies to the orthographic only in the extended sense used here, where the far hemisphere is drawn rather than declined. The five maps in the table were chosen to be genuinely continuous, and the choosing is not a convenience: continuous whole-sphere maps into a page are rare, and they are rare because of what they have to pay.

The fold share depends on the sampling. The 47 per cent for the orthographic is a count of grid cells receiving two distant places, at one grid resolution and one page tolerance, and the true answer for an exactly two-to-one map is 50. The gap is grid alignment rather than anything about the map, and the figure states the count rather than the ideal because the count is what was done.

The search finds a pair, not all of them. Newton converges to the zero of g nearest its start, and a map with several glued antipodal pairs would report only one. The theorem guarantees at least one and this machinery confirms at least one; the question of how many is a different question and this essay does not answer it.

Who found it, and when #

Stanisław Ulam conjectured the theorem and Karol Borsuk proved it in 1933, in a paper about spheres and antipodes with no application in mind. The weather version — two antipodal places with the same temperature and pressure — was in circulation almost immediately and has been the way the result is introduced ever since, which is unfortunate, because it makes a theorem about topology sound like a curiosity about meteorology.

The cartographic reading appears to be new here, and it is the one that makes the theorem do work rather than entertain. Set beside the other two rungs it completes a description of what a page can hold. A sphere will not go into a page one to one — that is compactness, and it is the first rung. A sphere will not carry a consistent direction — that is Poincaré–Hopf, and it is the second. And a sphere that insists on continuity will glue two opposite places — that is Borsuk–Ulam, and it is this one. Three theorems, none of which mentions distance, and between them they account for every edge, every cut, every pole line and every rim in the whole library.

Why the weather version does the theorem a disservice #

The observation about how Borsuk–Ulam is usually introduced deserves following up, because the framing decides what people think the result is for.

The temperature-and-pressure version is true and it is a party trick. Two antipodal places with the same temperature and pressure is a striking sentence, it requires no background, and it is what almost everybody who has heard of the theorem has heard. It also suggests that the theorem is a fact about the atmosphere, which it is not, and that its interest lies in the surprise, which it does not.

What the theorem is actually about is what a continuous pair of numbers can do on a sphere, and the pair does not have to be physical. A projection assigns two numbers — a page coordinate and another page coordinate — to every place on the Earth, continuously, and that is exactly the hypothesis. The conclusion is that two opposite places get the same pair, which is to say the same point on the page.

So the cartographic reading is the theorem’s own subject rather than an application of it, and the weather version is the analogy. Getting that the wrong way round makes a structural constraint on every world map look like a curiosity about isobars.

And the framing has a cost. A result introduced as a surprise gets remembered as a surprise, and nobody goes looking for it in a place where it would do work. The theorem has been available since 1933 and the maps it constrains have been drawn for four centuries; what kept them apart is that one of them was filed under amusing consequences.

Where this leaves the field #

The impossibility field carried one anchor for fourteen phases and it was curvature: eleven essays which between them establish that no map is an isometry, that the obstruction is Gaussian curvature, that curvature is computable by two independent routes, and — most recently — that the obstruction is a statement about smoothness and evaporates if the map is allowed only one derivative.

Every one of those arguments admits a tolerance. Make the region smaller and the areal error falls; accept a wiggle and the isometry comes back. None of the three theorems on this ladder admits one. There is no small version of a missing hemisphere, no partial credit for a field that nearly has no zeros, and no map that glues slightly fewer than one antipodal pair.

That is the boundary between the field’s two anchors and it is worth having stated once: curvature is how wrong a map is, and topology is what a map cannot be. The first is a measurement and has units. The second is a count and does not.