The sphere is not the plane at small counts

How many sheets an atlas needs answers a question with two halves. The first half is geometry with a proof behind it: a tolerance on scale error fixes a cap radius, through Chebyshev’s bound sec²(ρ/2) − 1 = tolerance, and the site computes it exactly. The second half is a covering problem — how many caps of that radius are needed to cover the sphere — and there the essay multiplied the ideal count by a constant:

$$\frac{2\pi}{\sqrt{27}} = 1.2092$$

which is the thinnest covering density of the plane by equal discs, proved by Kershner in 1939. The essay recorded the borrowing at the time:

sheetsForTolerance multiplies the ideal count by 2π/√27, the thinnest covering density of the PLANE by equal discs. The sphere’s covering numbers are not the plane’s at any finite count, and for the small counts a coarse tolerance produces — thirteen sheets at one part in ten — the plane’s density is a poor guide.

It is a poor guide, and not in the direction the note assumed.

Four counts with proofs #

A numerical search over cap positions proves nothing: failing to find a covering does not show that none exists. So the comparison that matters is made only where the optimum is a theorem, and there are four such counts small enough to matter.

The radii are exact: arccos(−1/3)/2 for the tetrahedron, and the corresponding expressions for the other two, which are the arcs from a solid’s centre to a face’s own centre. They are the same numbers the polyhedral ladder uses for face radii, which is not a coincidence — a covering by caps centred on a solid’s vertices is tight exactly when the caps reach the centres of the faces.

At two caps the density is 1.0000, which is less than the plane’s 1.2092 and is the case a plane cannot imitate: two hemispheres cover a sphere with no waste at all, and no two discs cover a plane.

At four, six and twelve the densities are 1.3333, 1.2679 and 1.2321 — above the plane’s number, falling towards it, and reaching it only in the limit.

Why the direction changes #

The two errors have different causes and it is worth separating them, because only one of them goes away.

Above four caps the sphere is worse than the plane because a covering of the sphere cannot be locally hexagonal everywhere. The plane’s optimal covering is the hexagonal arrangement, in which each disc’s Voronoi cell is a regular hexagon; on the sphere, Euler’s formula forbids an all-hexagonal tiling — hexagons cannot tile the sphere is the site’s own proof of it, and the shortfall is always exactly twelve pentagons. Twelve defects among twelve cells is the whole arrangement; twelve among a thousand is a correction; so the density falls towards the plane’s as the count rises and never reaches it.

At two and three caps the sphere is better than the plane for a different reason entirely: a spherical cap can be large. At 90° radius it is a hemisphere and two of them tile the sphere exactly. The plane has no such degenerate case, because a disc is always small compared with the plane.

So the plane’s density is not an approximation that gets better; it is the limit of a sequence that approaches it from above, sitting above a small-count regime where the sphere behaves qualitatively differently.

The two hemispheres, which are the case with no waste #

The n = 2 row deserves a paragraph of its own because it is the one a plane cannot have and because it is the cleanest possible statement of the difference.

Two caps of 90° radius are two hemispheres. Their union is the sphere; their overlap is a great circle, which has no area. The density is exactly 1, which is the theoretical floor for any covering of anything, and it is reached — on the sphere, at n = 2, exactly.

The tolerance that a 90° cap meets is 1.000 — a scale spread of a factor of two across each sheet — which is far too coarse for any real map. So the case is not practically useful and it is logically decisive: one row where the plane’s number is 21 per cent too high settles the question of whether it may be quoted as a bound.

Where the search is used, and how far it is trusted #

Between and beyond the proved counts there are no theorems, so the site runs an optimisation: relax the cap centres by moving each to the minimax centre of its own Voronoi cell, restart from several deterministic spirals, and shake-and-relax to escape a basin. Every number it returns is an upper bound — n caps of that radius do cover the sphere — and none of them is an optimum.

Two things make the bound worth printing.

It is checked where checking is possible. At the four proved counts the search comes within 0.2, 0.6, 1.2 and 3.4 per cent of the theorem. A search that overshot at those counts would have nothing to say at the others.

Its own evaluation is exact. The covering radius of a set of centres is attained at a vertex of their Voronoi diagram — a point equidistant from three of them with no fourth nearer — so it can be computed from circumcentres of triples rather than by sampling. Measuring it on a probe set instead reported a covering radius below the proved optimum at four caps, which is the arithmetic saying that the probes were too coarse and not that the theorem was wrong. The exact evaluation removed that failure mode entirely.

The honest weakness is that the bound loosens as the count rises: by twenty caps the search’s density is worse than its own value at twelve, which cannot be true of the optimal arrangement. That is a statement about the optimiser and is drawn as one.

What this changes about an atlas #

Less than the size of the errors suggests, and the reason is worth being clear about.

At the tolerances a real atlas works to — a part in a thousand or finer — the sheet count is in the hundreds or thousands, and there the plane’s density is an excellent approximation: the twelve pentagons are lost among a thousand hexagons. The site’s earlier essay computes 1,210 sheets at one part in a thousand, and that number is not disturbed by anything here.

What changes is the coarse end, which is where a thematic atlas lives — a set of a dozen plates covering the world at one part in ten. There the plane’s estimate is out by up to a fifth, and it is out downwards at every count above three, which is the dangerous direction: a plan for five sheets that needs six.

What a covering is, and what it is not #

Three ideas keep being confused in this area and the arithmetic is different for each.

A covering asks that every point be inside at least one cap, and it is what an atlas needs: no place may be missing from every sheet. Its density is at least 1 and the interesting question is how close to 1 it can get.

A packing asks that no two caps overlap, and it is what a set of non-interfering transmitters needs. Its density is at most 1 and the interesting question is how close to 1 it can get. The plane’s answer is 0.9069, again hexagonal, and the two constants are often confused because they are both about hexagons.

A tiling asks for both at once, and on the sphere it exists only for the counts a Platonic or Archimedean arrangement allows — which is the argument hexagons cannot tile the sphere makes about the hexagonal case and the cells ladder builds its schemes around.

An atlas is a covering, because the alternative to overlap is a gap. That is worth stating plainly since a reader coming from the cell-scheme side of this site has spent three essays on tilings, where overlap is forbidden and the arithmetic is the other one.

What was computed, and how #

The tolerance-to-radius step is Chebyshev’s bound and is exact. sec²(ρ/2) − 1 = tol inverts to ρ = 2 arccos(1/√(1 + tol)), with no fitting anywhere in it, and it is the same expression the earlier essay uses.

The proved radii are quoted from the covering literature and checked against the solids. Each is the arc from a Platonic solid’s centre to a face centre, computed from the site’s own vertex coordinates rather than typed in, and they agree with the published values to the digits given.

The search is deterministic. A seeded generator drives the shake, so the same arrangement comes out every time the figure is drawn; an unseeded one would make the covering a different measurement every time anybody looked, which is a species of the reproducibility problem a published coordinate is a result is about.

Density is n(1 − cos ρ)/2, which is n cap areas over the sphere’s area, and it is the quantity the plane’s 2π/√27 is comparable with — both are dimensionless ratios of covered area to area.

The other constant in the same expression #

Correcting the covering density draws attention to the term next to it, which is not a constant at all.

The ideal count is 4π / (2π(1 − cos ρ)), the sphere’s area over one cap’s, and it is exact. The covering density multiplies it. But the tolerance-to-radius step assumes the sheet is drawn with the best possible conformal projection for its own cap, which is the stereographic one centred on it — and a real atlas uses one projection for every sheet, so most sheets are not centred on their own good region.

Where the model stops #

No lower bounds are computed. Everything here is either a theorem quoted or an upper bound found. Proving that thirteen caps of some radius cannot cover the sphere is a different kind of argument — the published ones use area counting with a bound on how much two caps can overlap — and none is attempted.

Equal caps only. A real atlas does not use identical sheets: it uses larger sheets near the equator, or a graticule-aligned layout, or sheets whose sizes come from a paper standard. The equal-cap problem is the idealisation that makes the count a pure geometry question, and it is a lower bound on the real thing in the sense that any equal-cap covering can be realised by sheets and not every sheet layout is an equal-cap covering.

The caps are the projection’s own good region. A sheet is a cap only because Chebyshev’s criterion says the optimal conformal map of a cap has its scale spread controlled by the cap’s radius. A sheet drawn with the wrong projection has a good region that is not a cap at all, and the atlas essay’s second half measures that penalty separately — it is a factor of 56 for Mercator over a 10° cap, which dwarfs everything in this essay.

Who found it, and when #

Kershner proved the plane’s thinnest covering density in 1939 and the answer had been conjectured since Thue’s work on packings at the turn of the century. The sphere’s small-count coverings are a separate literature: Fejes Tóth established the optimal coverings for small n in the 1940s and 50s, and the cases with clean answers are exactly the ones with a Platonic solid behind them — n = 2, 4, 6 and 12. Beyond twelve the proved cases are sparse and most published values are upper bounds from searches, exactly as here.

The cartographic literature has generally not asked. Sheet counts in practice come from a graticule-aligned layout and a paper size, which is a different and more constrained problem; the covering formulation appears in the discussion of how many charts an atlas of a manifold needs, which is topology, and in satellite-constellation design, where the caps are footprints.

What a bound from a search is worth #

Most of the covering numbers past twelve are upper bounds from searches rather than proved optima, here as in the literature, and it is worth being clear about what that does and does not weaken.

An upper bound is exactly what the application needs. The question an atlas asks is how few sheets suffice, and a covering exhibited by a search answers it: here is an arrangement, it covers, therefore this many suffice. Nothing about the answer’s usefulness depends on whether a better arrangement exists.

What a search cannot support is a claim of impossibility. No arrangement does it in fewer is a statement about everything untried, and a search that failed to find one is evidence about the search. So a sheet count derived this way is a genuine achievement and not a floor, and any argument resting on the floor — that a series cannot be shortened, that a design is optimal — needs the proved cases.

The proved cases are the ones with a solid behind them, which is a satisfying pattern rather than a coincidence: a covering whose optimality can be proved is usually one whose symmetry supplies the proof, and the symmetric arrangements on a sphere are the Platonic ones. Beyond them the arrangements get irregular and the proofs get hard at the same point and for the same reason.

Why the plane’s constant is the wrong instrument here #

The whole rung turns on a substitution nobody would defend if it were stated plainly, and it is worth setting out why the substitution is so tempting.

The plane’s covering density is a beautiful constant and it is asymptotic. Kershner’s result describes how thinly discs can cover the plane, and the plane has no boundary, no curvature, and no smallest interesting number of discs. Every statement about it is a statement about the limit of many discs.

A small count is the opposite regime. Covering a sphere with four caps is a question about a particular arrangement of four particular objects, and the answer is decided by the geometry of the tetrahedron rather than by any density. There is no sense in which four is close to infinity.

And the sphere’s curvature works in the direction that helps. A cap of angular radius $\rho$ has less area than a flat disc of the same geodesic radius, but the sphere it must cover is finite and closes on itself, so the arrangements available at small counts are better than the plane’s asymptotic density suggests — which is why the direction of the error changes with the count rather than staying on one side.

So the borrowed constant is wrong twice: it is an asymptotic figure applied to a small case, and it is a flat figure applied to a curved surface. Either alone would be a reason to check; together they are the reason the answers come out wrong in both directions at different counts.

The reason it survives is that it is the only number available in closed form. A sphere’s covering numbers are a table of separate results with no formula behind them, and a table is harder to quote than a constant — so an estimate needing a number reaches for the one that exists.

Which is why the honest presentation gives both statuses. A table of covering numbers in which some entries are theorems and some are the best anybody has found is more useful than one that hides the distinction, because the two support different arguments — and a reader planning a series wants the first kind and a reader claiming a limit needs the second.

Where the ladder goes next #

The choosing field is now free of borrowed constants: the criterion is solved rather than quoted, the aspect is searched in the dimensions it has, and the covering numbers are the sphere’s own.

Every one of those is about drawing the world. The other half of this site’s applied work is about indexing it, where the unit is not a sheet but a cell — and where the same trap waits in a different disguise, because the cost of a query against a cell system is also usually estimated with a plane’s arithmetic.