A sphere is a plane plus one point

A sphere is finite. A plane goes on forever. One can be held; the other cannot be drawn, only implied. They are not the same size in any everyday sense, and no amount of stretching turns a beach ball into an infinite sheet.

Remove one point from the sphere, though — any single point — and the two match up exactly. Every point of the punctured sphere corresponds to exactly one point of the plane, and every point of the plane is hit exactly once.

The construction is a single straight line, repeated for every point.

The one-dimensional version is worth checking by hand before trusting the three-dimensional one, since every feature of the general case is already present in it — the bijection, the crowding, and the single exceptional point that spoils an otherwise perfect correspondence.

The construction #

Sit a sphere on a plane so that its south pole touches. Call the top of the sphere the north pole, $N$.

For any point $P$ on the sphere other than $N$, draw the straight line from $N$ through $P$ and continue it until it hits the plane. That landing spot is where $P$ goes.

Points near the south pole land near the origin. Points around the equator land on a circle of moderate size. Points close to $N$ send their rays out almost horizontally, so they land enormously far away — and the closer to $N$, the farther out.

The north pole itself has nowhere to go. Its ray is tangent to the sphere and parallel to the plane, and never lands anywhere at all.

That single exception is the entire price of the construction, and it is the reason the sizes work out. A finite surface with one point deleted has exactly as many points as an infinite plane, which is one of the earlier hints that counting infinite sets requires care.

What survives the flattening #

A map from a sphere to a plane must distort something — this is not a limitation of the construction but a theorem, and the same kind of unavoidable trade as the one that caps the regular solids at five. Areas near the north pole are stretched without bound — Greenland’s fate on a Mercator projection is a relative of this problem. So stereographic projection is not area-preserving and never could be.

What it does preserve is remarkable, and it is why the construction matters rather than merely amusing.

Angles are preserved exactly. Two curves crossing on the sphere at some angle map to two curves crossing at the same angle on the plane. The map is conformal. Shapes are distorted in size but not in local form: a small triangle on the sphere maps to a small triangle of a different size and the same shape.

Circles map to circles. Every circle on the sphere becomes a circle on the plane — with one exception, which is exactly the exception one would predict. A circle passing through $N$ maps to a straight line, because the point that would close the curve off is the one with nowhere to go.

That second fact is the one that reorganises how a line should be thought about. A straight line is not a different kind of object from a circle; it is a circle that happens to pass through the missing point. Once the sphere is allowed to be the real object and the plane merely a chart of it, the distinction between lines and circles dissolves.

Both of those facts have one cause, and it is worth extracting, because it is what conformality actually is. Work out where a point at angle $\varphi$ from the south pole lands and the answer is $r = 2\tan(\varphi/2)$, so a small step $\mathrm{d}\varphi$ along a meridian becomes a step $\sec^2(\varphi/2)\,\mathrm{d}\varphi$ on the plane. Run the same calculation along a circle of latitude and the factor comes out identical. The map magnifies by an amount that depends on where a point is and not on which way it is facing — and a map with that property is locally a similarity. It can enlarge a small shape and it cannot shear one. Angles survive because a similarity preserves them; small circles survive for the same reason.

The factor also says precisely what is being paid. At the south pole it is $1$. At the equator it is $2$: linear sizes doubled, areas quadrupled. Two thirds of the way up it is $4$, and areas are sixteen times too large. Ten degrees short of the pole it is $132$, inflating an area by more than seventeen thousand. The blow-up is neither gentle nor confined to a small cap — the entire northern hemisphere lands outside a circle of radius $2$, and everything beyond that circle, all the way out, is that one hemisphere.

That argument settles the angles and does not settle the circles, which is worth flagging rather than gliding past. A similarity preserves small circles because it preserves everything small. That a circle of finite size on the sphere maps to an exact circle on the plane is a stronger claim, and the local scale factor cannot deliver it: the magnification varies across such a circle, so its image has no particular reason to come out round. It does anyway, for an algebraic reason rather than a geometric one — the projection is a ratio of linear expressions in the coordinates, and maps of that kind carry the equation of a circle to the equation of a circle. The picture makes the first fact obvious and gives no hint that the second one needs a different argument.

The point at infinity #

The natural response to a construction with one exceptional point is to patch it, and the patch is to add a point to the plane whose job is to be the image of $N$. Call it $\infty$.

The plane plus that one extra point is the Riemann sphere, and it is not a metaphor — it is the sphere, with the projection now a perfect correspondence in both directions.

This turns out to be the right setting for complex analysis, and it fixes a number of things that are awkward on the ordinary plane. In the complex numbers, the function $f(z) = 1/z$ is undefined at $z = 0$ and has no value at large $z$ either; on the Riemann sphere it is a perfectly well-behaved map that swaps $0$ and $\infty$ and is defined everywhere. Parallel lines, which are an annoying special case in the plane, meet at $\infty$ like any other pair. Whole classes of “except when the denominator vanishes” caveats disappear.

The pattern — a construction that works everywhere except at one place, repaired by adding the missing place — recurs throughout mathematics, and it is worth recognising as a pattern rather than as a series of unrelated repairs. Projective geometry adds a whole line at infinity so that any two lines meet. The extended real line adds $\pm\infty$ so that every monotone sequence converges. In each case the addition is not a fudge; it is the recognition that the original space was a version of a more natural object with a piece removed.

Compare Euclid’s algorithm, where the process that fails to terminate is not patched but recognised — its failure is the discovery of irrationality. Sometimes the exception is the answer, and sometimes it is a missing point. Telling them apart is a matter of asking whether filling the gap makes the theory simpler or merely hides a real phenomenon.

One point is a great deal #

The correspondence is exact, which invites the conclusion that the punctured sphere and the plane are interchangeable and that the deleted point is a technicality. The opposite is closer to the truth: almost everything that distinguishes the sphere from the plane is carried by that one point.

The sphere is compact. Every infinite collection of points on it has somewhere to accumulate — a sequence of points cannot escape, because there is nowhere to escape to. The plane is not compact, and neither is the punctured sphere: a sequence marching toward the deleted pole has nowhere to land, having had its destination removed.

That difference is not abstract bookkeeping. It decides which theorems are available.

On a compact surface, every continuous function is bounded and attains a largest value somewhere. On the plane it need not: $f(x, y) = x$ is continuous and has no maximum. So the ordinary guarantee that an optimisation problem has an answer is a property the sphere possesses and the plane does not, and one point is the whole of the difference. Delete the pole and the guarantee goes with it; put the pole back and it returns.

The most consequential instance is the fundamental theorem of algebra. A polynomial is a perfectly nice map of the plane to itself and offers no reason to expect a root. Extend it to the Riemann sphere — which it does extend to, since a polynomial sends $\infty$ to $\infty$ — and it becomes a continuous map of a compact surface to itself. Such maps have a degree, degrees are additive, and a map of degree $n$ must hit every value $n$ times over. Every non-constant polynomial has a root, and the reason it does is that the sphere is closed up and the plane is not. The argument does not exist on the plane, because on the plane the object it argues about does not exist.

So the picture’s most reassuring feature — the tidy bijection — is also its most misleading. A bijection matches points and promises nothing about the properties the points collectively have. The plane and the punctured sphere have the same points arranged the same way; the plane and the whole sphere have one point’s difference and are not remotely the same kind of object.

The projection that sends fractions to fractions #

There is a second payoff, and it belongs to a field the construction has no obvious business in.

Write the projection out in coordinates and it is a ratio of polynomials with whole-number coefficients, in both directions. Which means: a point at a rational distance along the line corresponds to a point on the circle whose coordinates are both rational, and every rational point on the circle arises this way from a rational point on the line. Nothing is lost and nothing extra is gained.

That is a complete solution to a problem that looks much harder than it is. Find all the points on a circle with rational coordinates is a question about a curve; the projection converts it into list the fractions, which is not a question at all. Clearing denominators then turns each rational point on the unit circle into a triple of whole numbers satisfying $a^2 + b^2 = c^2$ — so the Pythagorean theorem’s whole-number solutions are the fractions in disguise, and there are exactly as many as there are fractions. The parametrisation deserves its own essay and its own picture, and gets one further along that ladder; what belongs here is the observation that the machine producing it is this projection and nothing else.

The requirement is that the curve already have one rational point to project from. That is a real condition and not a formality: the circle $x^2 + y^2 = 3$ has no rational points at all, so there is no pole to stand at, and the method does not start. A construction invented to flatten a globe turns out to answer a question about whole numbers, and to fail on exactly the curves where the question is hard.

What the picture cannot show #

Every drawing here has an edge, and the plane does not. The rays that leave the pole most shallowly land farthest out, and precisely those are the ones cropped off the page — so the figures show the well-behaved middle of the correspondence and hide the part that makes it interesting.

That crop matters for the central claim. “A finite surface minus one point has exactly as many points as an infinite plane” is a statement about a bijection, and the bijection’s whole character lives in the region the picture cannot contain. A reader could look at these figures and reasonably conclude the plane is a large disc.

Nor can the drawing establish conformality. That angles are preserved exactly is the property that makes the projection useful, and it is a statement about infinitesimal neighbourhoods — invisible at any scale a page can render, and proved with a derivative rather than a diagram.

The ladder from here #

Rungs above: the projection formula in coordinates, and the conformality proof. Circles to circles, proved rather than asserted. The Riemann sphere as the natural home of complex functions, with $1/z$ as a rotation of the sphere. Möbius transformations as the sphere’s rigid motions. Inversive geometry. Cardinality, and the shock that a segment has as many points as a line — Cantor’s territory, where counting arguments replace pictures entirely. Map projections in general, and the theorem that no flat map can preserve both area and angle. The astrolabe as an engineered instance. And the one-point compactification, which is this construction stated for any space at all.

Where it is actually used #

Stereographic projection is around two thousand years old and was in practical use long before it was understood theoretically. Hipparchus is generally credited with it in the second century BC — contemporary, roughly, with Apollonius on the conics — and it is the projection on which the astrolabe is built — because it is conformal, and because it sends the circles traced by stars in the sky to circles on a brass plate, which can then be engraved with a compass. An instrument that had to be manufactured by hand needed a projection whose output could be drawn with the tools available, and this is that projection.

It remains standard in crystallography, in seismology for plotting the orientation of faults, and in complex analysis for the reasons above. Cartographers still use it for polar regions, where the distortion is placed on the far side of the world and does the least harm. That is the general principle of map projection in one sentence: distortion cannot be removed, only moved, so the craft consists of putting it where nobody is looking.

There is a theorem behind the resignation. No flat map can preserve both areas and angles, because a sphere has curvature and a plane does not — a fact Gauss called remarkable and proved by showing that curvature is intrinsic, detectable by measurements made entirely within the surface, with no reference to any surrounding space. A projection must therefore give something up. Mercator keeps angles and wrecks areas at high latitude. Equal-area projections do the reverse. Stereographic keeps angles, keeps circles, and accepts that the far hemisphere sprawls across the whole plane. The same trade appears whenever a curved thing is flattened, which is most of cartography and a good deal of geometry.

Not bad for a construction that is one straight line, drawn again and again, with a single point left over.