Two charts are enough, and one is not

No map of the whole sphere is one to one settles the question this field opens with: no single continuous injective map carries the sphere into a plane, and the argument is compactness rather than cleverness.

That is a statement about one map. It leaves the interesting question open, and the interesting question is the one an atlas actually asks: how many maps does it take?

The answer is two, and the answer costs something.

One chart, minus a point #

Before counting to two it is worth being precise about what one chart cannot do, because the usual statement is stronger than the theorem.

The stereographic projection is a homeomorphism from the sphere minus one point onto the whole plane. It is continuous, injective, surjective, conformal, and its inverse is continuous. Everything is in it except the pole opposite its centre.

So a single chart is not merely close to enough. It carries every place on Earth but one, and if that one is chosen in the middle of an ocean nobody would notice its absence from the coverage.

That is what “one is not enough” actually means, and it is not a topological failure at all — the topology is perfect right up to the missing point. It is a failure of paper. A one-chart world map exists and its far hemisphere occupies a region that grows without limit, so anybody who prints it has to crop it, and cropping is where the second chart comes from.

Two charts, and a floor #

Two caps cover a sphere exactly when each has angular radius at least 90°. That is one line of spherical geometry and it fixes something nobody chose.

The worst point of a two-chart atlas is at 90° from a chart’s centre, and no arrangement moves it. Whichever two charts are used, some place is 90° from the nearest centre, because the two centres are at most 180° apart and a place halfway between them is 90° from both.

So the counting alone forces a distortion floor:

A factor of four in area, or thirty-nine degrees of shape. Both are closed forms — $\sec^4(\rho/2)$ at $\rho = 90°$ is 4, and the equal-area azimuthal’s principal scales at 90° are $\sqrt2$ and $1/\sqrt2$, giving $2\arcsin\frac13 = 38.9424°$ — and there is no third option, because no map is faithful and a map that were both would be an isometry.

That last observation is worth a sentence of its own. In most of mathematics an overlap between charts is free — it is what makes an atlas an atlas, and enlarging it costs nothing. Here it costs, because a chart that reaches further has a worse rim, and the rim is the worst point. A minimal atlas is minimal in radius as well as in count, and the two minimisations agree, which they had no obligation to.

The seam #

Where two charts overlap there is a change of coordinates, and it is a map with properties of its own.

For two stereographic charts centred on opposite poles the transition is the inversion $r \mapsto 4R^2/r$ composed with a reflection. It is conformal in modulus and orientation-reversing, because each chart looks at the sphere from its own side — so the two sheets of the minimal atlas disagree about handedness, and one of them has to be reflected before a reader can lay them side by side and have east run the same way.

The invariant $r\cdot r' = 4R^2$ holds to $10^{-9}$ at every row of that figure, which is the check that the transition really is the inversion rather than something that resembles one over a small range.

The practical consequence is the one every published atlas has: a feature that straddles the seam is drawn twice, at two different scales, and the two drawings cannot be made to agree by any amount of care. Two opposite places on the same spot is the same seam looked at as a gluing rather than as a cut.

The other count #

An atlas in the ordinary sense does not have two sheets. It has hundreds, and the reason has nothing to do with any of the above.

How many sheets an atlas needs derives the second count: a tolerance fixes a cap radius through $\sec^2(\rho/2) - 1 = \text{tol}$, and the sphere’s area divided by the cap’s, times the covering density, is the number of sheets.

The two counts meet at a tolerance of about 100 per cent — a map that may be wrong by a factor of two in scale — which is exactly the floor the previous section computed. That is not a coincidence: the two-chart atlas is the atlas whose tolerance is that floor, so the tolerance formula, evaluated at the floor, has to return two. It does, and that agreement is the only thing tying the two counts together.

Everywhere else they are answers to different questions. Every sheet past the second is bought by accuracy, not by topology, and at the hundredth of a per cent a national series works to that is a hundred and twenty thousand minus two of them. The scale factor was chosen is the same trade made one sheet at a time.

What the minimal atlas is actually good for #

Two sheets that between them show every point of the Earth, each conformal, with a worst areal factor of four, is not a bad object. It is worth saying what it is and is not for.

It is a complete reference. Nothing is missing and nothing is duplicated except in a stated overlap, which is more than can be said for any single world map: every whole-world projection has a cut or a point drawn as a line, and the two-chart atlas has neither. The seam is a shared circle rather than a tear, and both sheets contain it.

It is a poor picture of anything. A factor of four in area across a hemisphere is worse than Mercator manages over most of its range, and the reader is looking at the world down two funnels. That is why nobody publishes one: the object that is topologically ideal is visually terrible, and the visually acceptable objects are the compromise projections that give up the completeness.

And it is the right structure for a computation. A two-chart atlas is exactly how a numerical scheme on a sphere avoids the pole problem — work in whichever chart the point is further from the edge of, and use the transition to move between them. The floor of four does not matter to a computation, because a computation does not look at the paper. North cannot be up everywhere is the obstruction that makes such a scheme necessary, and the two-chart atlas is the standard answer to it.

What was computed, and how #

Everything here is a closed form. The single chart’s areal factor is $\sec^4(\rho/2)$, the equal-area azimuthal’s angular deformation at $\rho$ is $2\arcsin\frac{a-b}{a+b}$ with $a = \sec(\rho/2)$ and $b = \cos(\rho/2)$, and the sheet count comes from measure.js’s own tolerance formula unchanged.

Three assertions carry the rung, and each rejects something different.

Two charts of radius 90° must cover the sphere and two of radius 89.99° must not. That is the counting stated as a check, and it is the one that would break if the covering condition had been written with the wrong inequality.

The conformal floor must be exactly four and the equal-area floor exactly 38.9424°. Both are closed forms rather than measurements, so an agreement to nine decimal places is a check on the arithmetic and not evidence about maps.

And the accuracy-driven count must exceed the topological one by orders of magnitude at a working tolerance, while falling to it at a loose one. A version where the two counts never met would mean they were measuring different things badly; a version where they always agreed would mean the rung has nothing to say.

The count nobody quotes #

One more number is worth extracting from the pair, because it is the one a producer would actually use.

At what tolerance does an atlas need more than ten sheets? From the table, about 12 per cent — a map that may be wrong by an eighth in scale. Ten sheets is roughly the plate count of a small world atlas, and the tolerance that buys it is far looser than anybody would state and far tighter than the two-chart floor. That is the whole working range of the subject compressed into one decade of tolerance, which is why sheet counts vary so wildly between products that look similar.

Where the model stops #

Closed sheets, not open charts. The differential-geometric definition of an atlas uses open sets, and two open caps of radius exactly 90° do not cover the sphere — their union misses the shared circle. Every chart here is closed, which is what a printed sheet is, and the counting is the counting for closed covers. The open version needs radius strictly greater than 90°, which raises the floor by an arbitrarily small amount and changes nothing.

Two centres, chosen antipodal. Nothing here proves that antipodal centres are optimal, only that they achieve the floor the counting forces. Any two centres less than 180° apart leave a worst point further than 90° from the nearer one, so antipodal is optimal, but that argument is a sentence rather than a measurement and is stated rather than drawn.

And the sheet count assumes discs. How many sheets an atlas needs already prices the difference between a disc covering and a real series of rectangular sheets, and finds the covering density is the smaller of the two effects. Nothing here revisits that.

The one number the two counts share #

They meet at a tolerance of about one, and that meeting is not a coincidence worth passing over.

The two-chart atlas’s floor is a scale factor of two — the conformal case’s areal factor of four is a linear scale of two at the worst point — and the tolerance formula, asked what radius a scale tolerance of one buys, returns 90°. Two caps of 90° are exactly the minimal atlas. So the two counts are one function evaluated at two ends of its range: the topological minimum is the tolerance formula’s answer at the loosest tolerance any map could be said to hold to.

Below that tolerance the formula is the whole story and topology never enters again, which is why nobody in cartography has ever needed to count charts.

The generalisation #

The rule is one this collection reaches from several directions and which this rung states in its cleanest form: a topological minimum is not a design target.

Two is the answer to how few pieces can cover this? and nobody has ever wanted that answer. What everybody wants is the answer to how few pieces will hold my tolerance?, and that is a completely different quantity governed by a completely different mechanism — an area ratio rather than a covering argument.

The two are related in exactly one place, which is that the topological minimum has a distortion floor and so is the atlas of one specific tolerance. The floor is the join, and it is what makes the pair of counts a single story rather than two unrelated numbers.

This is the same relation as the one between the square costs the poles and a tile pyramid: a structural decision fixes a bound, and the bound is what the engineering then negotiates against. A minimum that carries no bound with it is a curiosity; a minimum that fixes a price is a design constraint, and this one fixes a factor of four.

Who found it, and when #

That the sphere needs two charts and admits two is standard from the moment the word manifold was defined, which is Weyl’s in 1913 and Whitney’s in the 1930s. The stereographic pair is the first example in every textbook, and the transition map $z \mapsto 1/\bar z$ is the first transition map anybody computes.

The cartographic side is much older and has never used the language. Globe gores, hemispherical plates, and the two-sheet world maps of the seventeenth century are all two-chart atlases built for reasons of paper, and the makers knew that two hemispheres cover the world without knowing there was a theorem in it.

What appears in neither literature is the floor. The differential geometry does not care what the charts cost, because a chart is a chart; the cartography does not count charts, because it counts sheets and its sheet count is set by scale. Multiplying the two together — the minimum count, and what that count forces on the distortion — is one line of arithmetic that nobody has an obvious reason to write down.

A line of arithmetic nobody’s job required #

The observation that the floor appears in neither literature is the rung’s own account of why it exists, and the mechanism is worth naming because it is this collection’s most common finding.

Each field has half the ingredients and no use for the other half. Differential geometry supplies the minimum chart count and treats charts as free — a chart has no cost in the theory, so asking what one costs is not a question the subject poses. Cartography supplies the distortion arithmetic and counts sheets rather than charts, because a sheet count comes from a scale and a paper size and has nothing to do with topology.

The product of the two is a statement neither is looking for. Two charts are forced, and two charts force at least this much distortion needs the topological floor and the metric price in the same sentence, and there is no reader for whom both halves are routine.

Which is why the gap is not evidence of anybody’s oversight. A fact goes unwritten when it sits at the boundary of two competences and answers a question neither field asks — not because it is hard, and not because somebody missed it, but because nothing brings the halves together.

The alternative explanation — that the fact is too trivial to be worth writing — does not survive the arithmetic, since the floor it produces is a real constraint on a real design and not a restatement of either input.

And it says where to look for more of them. The productive place is not the frontier of either subject but the seam between them, and the marker is a quantity that is elementary in one field and unavailable in the other. This collection’s habit of computing both halves with one piece of machinery is what makes those seams visible: when the same code measures a chart count and a distortion, putting them in one sentence is the obvious next step rather than an interdisciplinary undertaking.

Where this field goes next #

Six rungs have established that no map of the sphere is one to one, that no field of directions can be, that antipodal pairs are glued by every continuous map, that a map has a degree, that four colours suffice, and now that two charts do. All of them are statements about the sphere as a topological object, with the metric appearing only when a price is being computed.

What none of them has asked is what happens when the object is not a sphere. A body that is not an ellipsoid is still topologically a sphere and everything here holds; a body with a hole in it is not, and its atlas has a different minimum, a different four-colour bound, and a different degree theory. Whether any real body is one is a question for the neighbouring field, and the answer is closer to yes than it looks.