One chart is never enough

The first rung matches the sphere with the plane by rays from the north pole, and one point is left over. The third repairs it by adding a point to the plane and reading the result as the complex numbers with infinity attached.

That is one repair and it is not the general one. The general repair does not touch the plane at all: it uses a second copy of the plane, projected from the other pole, and lets the two overlap.

What a chart is, and what an atlas is #

A chart on a surface is a patch of it matched with a patch of the plane, continuously and reversibly. It is a coordinate system that works somewhere and need not work everywhere.

An atlas is a collection of charts covering the whole surface. Where two charts overlap, each point has two sets of coordinates, and the rule taking one to the other is the transition map. A surface with an atlas is a manifold, and essentially all of geometry is written in that language.

The sphere’s atlas is the smallest interesting one. Two charts — projection from the north pole, projection from the south — and they cover everything, since each misses only its own pole and the poles are different points.

The transition map is as clean as such maps ever get. On the overlap, which is the sphere minus both poles, the coordinate given by one chart is $1/z$ in the other’s coordinate, with a conjugation for the orientation. That is checked at four hundred points in the figure and the disagreement is at the level of arithmetic error.

Why one chart cannot do it #

The missing point is not an artefact of choosing the north pole, and it is worth being clear that no cleverness removes it.

Suppose a single chart covered the whole sphere: a continuous, reversible map from the sphere onto a region of the plane. The sphere is compact — closed and bounded, so every sequence of points has a subsequence converging inside it — and a continuous image of a compact set is compact. So the region would be closed and bounded, and therefore would have a boundary point.

Now look at that boundary point’s preimage on the sphere. Every point of the sphere has a neighbourhood that is a small disc; its image would have to be a neighbourhood of the boundary point in the plane, and a boundary point has no neighbourhood inside a closed bounded region. Contradiction.

It is worth checking that the argument does not prove too much. A plane certainly is covered by one chart — itself — and the argument does not apply, because the plane is not compact and its image need not be closed. An open disc is covered by one chart and is not compact either. The hypothesis is doing exactly the work it appears to, and the way to be sure of an argument like this is to find the case it must not exclude and see where it fails.

So the failure is compactness, and it applies to every compact surface. A torus needs several charts, a genus-two surface needs several, and only the non-compact surfaces — the plane, the cylinder, the open disc — can be covered by one.

What the transition map has to be like #

The atlas language has a purpose, and the purpose is that everything about the surface can be checked one chart at a time provided the transition maps are well behaved.

A property is defined on the manifold when it is defined in each chart and the two definitions agree on the overlap. Whether that agreement holds depends entirely on the transition map, so the map’s regularity is what decides which properties the manifold has.

Continuous transition maps give a topological surface. Smooth ones give a smooth surface, on which differentiation makes sense. And holomorphic ones — complex-differentiable — give a Riemann surface, on which complex analysis makes sense.

The hierarchy is strict and each level is a real restriction. There are topological surfaces with no smooth structure at all, in high enough dimensions; there are smooth surfaces with many inequivalent complex structures — the torus has a whole family of them, one for each shape of parallelogram it is built from. The manifold is the same set of points at every level and a different object, and which level one is working at is decided entirely by what the transition maps are required to preserve.

The sphere is the exception that makes the point: it has exactly one complex structure, so for the sphere the levels collapse and the distinction is invisible. Meeting the theory on the sphere first is convenient and slightly misleading.

The sphere’s transition map is $z \mapsto 1/z$, which is holomorphic everywhere on the overlap, since the overlap excludes the origin. So the sphere is a Riemann surface, and it is the Riemann sphere of the rung below — the same object, arrived at by a route that never mentions adding a point to anything.

The transition map, worked #

The reciprocal is easy to state and worth deriving once, because the derivation is three lines and it explains where the conjugation comes from.

Put the sphere of radius one at the origin. The north chart sends a point $(x, y, z)$ to $(x, y)/(1 - z)$; the south chart sends it to $(x, y)/(1 + z)$, with the second coordinate negated so that both charts agree on which way round is positive.

Multiply the two magnitudes. The north chart’s distance from the origin is $r/(1-z)$ where $r^2 = x^2 + y^2 = 1 - z^2$, and the south chart’s is $r/(1+z)$. Their product is

$$\frac{r^2}{(1-z)(1+z)} \;=\; \frac{1 - z^2}{1 - z^2} \;=\; 1.$$

So the two distances are reciprocals, exactly, at every point. And the two directions are the same angle measured in opposite senses, because the second chart is viewed from the other side — which is the conjugation.

Written as complex numbers, $w = 1/\bar{z}$ with the flip, and $w = 1/z$ once the orientation is fixed by negating one coordinate. The map is an inversion in the unit circle, which is the transformation that trades circles for lines, and the appearance here is not a coincidence: stereographic projection is built from rays through a point, and so is inversion.

That identity — the product of the two distances is one — is what the figure checks numerically, and the derivation says it is exact rather than approximate.

What each chart gets wrong #

Neither chart is faithful, and knowing what each distorts is what makes an atlas usable rather than merely correct.

The blow-up is quantitative and easy to state. A patch at angular distance $\theta$ from the pole a chart is projected from is magnified by a factor of about $1/\theta^2$ in area, so a patch a degree from the pole is drawn about three thousand times too large and one a tenth of a degree away three hundred thousand times. There is no bounded distortion anywhere near the pole and no radius at which it becomes acceptable.

The complementary distortion is the point of having two charts. Anything near the north pole is badly served by the north chart and well served by the south, and a computation on the sphere is done in whichever chart is well conditioned for the region it is looking at. That is not a metaphor: numerical work on spherical domains does exactly this, switching charts to avoid the coordinate blow-up, and the transition map is what makes switching legitimate.

The same reasoning is why an atlas of the Earth has many sheets rather than one. A single sheet of the whole globe distorts something enormously — area, or angle, or distance — and every projection makes a different sacrifice. Several sheets, each nearly faithful over its own region, with a stated rule for matching them at the edges, is the same construction with the same justification.

The rest of the definition #

Two more requirements make the definition work, and both are usually passed over.

The atlas must be maximal, or the manifold depends on the atlas. Two different atlases can describe the same surface — projecting from two different pairs of antipodal points, say — and the object should not depend on which is chosen. The fix is to take all charts compatible with a given atlas, which makes the choice irrelevant.

And the transition maps must agree in threes. With three charts overlapping, going from the first to the second to the third must be the same as going from the first to the third directly. That is automatic when the maps are all restrictions of one underlying correspondence, as here, and it is a real condition when charts are glued abstractly — which is how most manifolds are built.

The same construction, one dimension up #

The three-sphere gets the same treatment and it is where the atlas stops being a formality.

A three-sphere is the set of points at distance one from the origin in four-dimensional space, and nobody sees it. Stereographic projection from one of its points carries it into ordinary three-dimensional space, faithfully except at that point, and the image is something a picture can hold.

The second chart, from the antipodal point, covers what the first misses, and the transition map is again an inversion. So the three-sphere is a manifold by the same two-chart argument, and the argument did not have to be re-invented — which is the point of having a method rather than a repair.

Every statement in this rung is dimension-independent except the pictures. Compactness forbids one chart, two suffice, the transition is an inversion, and the surviving properties are the ones the inversion preserves. That is why the circles filling a three-sphere can be drawn at all: they are drawn in a chart, and the chart is honest everywhere but one point.

Where the count of charts matters #

How many charts a surface needs is a real invariant and it is not the same as any other.

The sphere needs two. So does the torus in the smooth sense, though the usual atlas uses four squares. The projective plane needs three, and the reason is worth a sentence: it is not orientable, so no two charts can be arranged to agree on which way round is positive, and a third is needed to carry the disagreement. And the general fact is that a compact surface of any kind needs at least two and at most a small number, with the exact minimum a question that has been answered and is not very interesting.

What is interesting is that the count is a lower bound with a topological proof, and the proof is the compactness argument above in every case. A statement about coordinates turns out to be a statement about the shape, and the transition from one to the other is the whole reason the manifold language was invented.

The higher-dimensional versions are the same. The three-sphere needs two charts and stereographic projection provides them, which is what the rung above uses to draw an object nobody can see.

Two charts, one object, and the price #

There is a cost to the atlas language and it is worth naming, because it is the reason people resist it.

Working in one chart, a point is a pair of numbers and everything is concrete. Working in an atlas, a point is an equivalence class of pairs of numbers, one per chart containing it, and every statement has to be checked to be independent of the chart. That is genuine extra work and it is why a first course does the sphere with spherical coordinates and a blind spot rather than with two charts and a transition map.

What the work buys is that nothing is ever wrong. Every quantity defined chart by chart, with the transition checked, is defined on the object; every quantity defined in one chart and assumed to carry over is a claim that has not been made.

The discipline shows up most clearly in what it refuses. A formula for the sphere written in one chart may be perfectly correct and still not define anything on the sphere, if it disagrees with itself under the transition map — and the disagreement is not visible from inside either chart. Checking the overlap is the only way to see it, which is why the check is part of the definition rather than an afterthought. Spherical coordinates have a singularity at the poles, and every formula written in them acquires a special case there — a special case that is invisible in the algebra and appears as a division by zero at run time. A chart’s blind spot does not go away by being ignored; it turns into a bug.

That trade — more bookkeeping, no special cases — is the same trade adding one point to the plane makes on the rung below, and the two are worth comparing. Adding a point removes the special case for one construction. An atlas removes it for every construction, at the price of never being allowed to forget which chart one is in.

What the pictures cannot show #

The transition map is checked at four hundred points, sampled along a spiral, and agrees to the last bits of the arithmetic. It is an identity and the check is a check on the implementation.

The compactness argument is given and not drawn. A picture of a continuous map failing to be open at a boundary point would be a diagram of an argument rather than a picture of an object.

The maximality condition is stated and never used. Every figure here works with the two named charts, and enlarging the atlas to all compatible charts changes nothing a picture would show.

And the atlas is drawn as two caps and a band, which is a schematic. The charts are not regions of the sphere but maps from regions to the plane, and what the figure shows is their domains rather than the charts themselves.

Where the ladder goes next #

Named here as debts. The general one-point compactification, which adds a point to any locally compact space and gives a sphere only in this case, and which was named as a debt on the rung below and remains one. And the atlas on a torus, where four charts are the usual answer and two are enough with more care.

Sideways, the projection this doubles is the first rung, the conformality that makes the transition holomorphic is the second, the complex structure it produces is the third, and the circles that fill a three-sphere are what the two charts make visible.

What is worth carrying away #

When a construction leaves out exactly one point, the useful response is to ask whether the omission is removable or structural.

Stereographic projection misses a pole, and adding a point to the plane repairs it for that projection. Using a second projection repairs it without touching the plane, and the second repair generalises to every compact surface while the first does not — because the first is a fact about the sphere and the second is a method.

The habit worth taking is to ask what compactness forbids. A great many “one point is missing” situations are compactness in disguise, and recognising it says immediately that no rearrangement will help and that a second chart will.

The corollary is about what a transition map is for. The interesting content of an atlas is never the charts, which are arbitrary; it is the maps between them, which carry all the structure. A surface is not a collection of coordinate systems but the rules for changing between them, and every property worth having is a property preserved by those rules.