There is no equal-area lattice on a sphere

The first two rungs held the sample fixed and asked what an estimator does with it. This one asks what the sample itself costs, and the answer starts from a fact this collection has already proved in a different context: a sphere cannot be covered by equal, regularly arranged pieces.

Hexagons cannot tile the sphere settles that for cells, by Euler’s formula: any tiling of a sphere by hexagons and pentagons needs exactly twelve pentagons, so a hexagonal grid on a sphere has twelve defects in it and no arrangement removes them. The same obstruction applies to points. There is no set of points on a sphere that is simultaneously equal-area in every direction and equally spaced, at any count above a handful, and every sampler in use is a decision about which of those two to give up.

The three samplers #

The graticule. Equal steps in latitude and in longitude, with each point weighted by the cosine of its latitude so that the weights compensate for the crowding. It is what almost every distortion measurement in the literature uses, because it is what a nested pair of loops produces. Its points are badly distributed and its weights are correct, which is a combination that works better than it looks.

Equal-area rings. Rings at equal steps in the sine of latitude, so each ring carries the same area, with the number of points in a ring proportional to its circumference. It is what measure.js has used here since the collection’s early essays, on the reasoning that a sampler that is equal-area by construction needs no weights and cannot be accused of over-representing anywhere.

The Fibonacci lattice. Points at equal steps in the sine of latitude with longitudes advancing by the golden angle, 137.508°, so that successive points land as far as possible from all the previous ones. It is the standard construction for a near-uniform point set on a sphere and it is what a numerical analyst would reach for.

What “equal-area” turns out to mean #

The ring sampler is equal-area in a specific and narrow sense: each ring carries the same area. That says nothing about how the area inside a ring is divided, and nothing about any region that is not a ring.

The honest measurement is the cap discrepancy: take a set of spherical caps, compare the share of the point set inside each one with the share of the area it covers, and report the largest gap. A point set that were genuinely equal-area against every region would score zero. None does, and none can — a finite set has a largest gap, always, because a cap can be shrunk until it contains one point and a millionth of the area.

The numbers do not fall smoothly and they should not be expected to. A discrepancy is a maximum over a finite set of test regions, which makes it the estimator the first rung of this ladder is about, applied to a sampler rather than to a map — so it is a lower bound on the true discrepancy, it jumps when a point crosses a cap boundary, and it is not monotone in the sample size. Reporting it as though it were a smooth convergence would be the same mistake this ladder exists to name.

What the table does establish is the ordering of magnitudes: all three are in the same range, none is an order of magnitude better than the others, and the ring sampler’s construction buys much less than its name suggests.

The integral, which is what the sampler is for #

Discrepancy is a property of the point set. What actually matters is what happens to a number computed on it, and that is a different question with a cleaner answer.

Fitted rates of −1.05, −0.87 and −0.95 in the point count. All three are first order in the count, which is second order in the spacing — the midpoint rule’s rate, in two dimensions, as it should be.

The ring sampler is the worst of the three at every size tried, by about a factor of six at the top end, and that is the finding. It was adopted here because it is equal-area and therefore needs no weights; the graticule sampler is not equal-area and carries weights that correct for it exactly. Carrying the correction turns out to be better than avoiding the need for one, because the cosine weight is exact and the ring construction’s uniformity is not.

Running it again on a second projection and a second quantity gives the same ordering and the same rates, so the result belongs to the samplers. That matters because distortion over a region computes exactly these quantities for every projection in the library, and the sampler underneath it is the one that comes last in this table.

The Fibonacci lattice does not improve monotonically #

One row of that table is out of order and it is not noise. At 500 points the Fibonacci lattice’s error is 4.7×10⁻³; at 1,200 points it is 2.0×10⁻², four times worse with more than twice as many points.

The reason is the construction. The golden-angle advance makes the lattice nearly optimal when the point count is close to a Fibonacci number, because that is when the successive longitudes come closest to filling the circle evenly; at counts between Fibonacci numbers the pattern is less even and the error is correspondingly larger. So the error oscillates within an envelope that falls at the rate above.

That is a genuine hazard rather than a curiosity, because it defeats the standard test. Doubling the sample and watching the answer settle is how everybody checks a quadrature, and on a Fibonacci lattice the answer can settle, then move, then settle somewhere else. A sampler whose error is not monotone in its size cannot be validated by refining it, which is the fourth rung of this ladder arriving early and by a different route.

What this collection does about it #

Three changes, and none of them is expensive.

The default sampler for a box region already carries the cosine weight, which is the graticule case and is the better of the two. Nothing needs to change there, and the fact that it was never the sampler anybody worried about is the usual shape of these audits.

The cap and ellipse regions use rings, and the ring construction stays, because changing it would move every number distortion over a region has published without improving any of them by more than a factor of six at a cost nobody is paying. What changes is the reporting: a maximum taken over a ring sampler is now stated with the refinement beside it, which is the first rung’s repair applied to the sampler that needs it most.

And the reference for any new integral is a rule of a different kind. Doubling a sampler and watching the answer settle is the test everyone runs and the one this ladder is about to take apart.

The half-step at the rim #

The ring sampler has one further property, and it is the reason this rung was written rather than merely the reason it is interesting.

A ring sampler over a cap of angular radius $\rho$ with $n$ rings places its rings at equal steps in $\cos c$, at the centres of those steps:

$$\cos c_i = 1 - (1 - \cos\rho)\,\frac{i - \tfrac12}{n}.$$

So the outermost ring is at $i = n$, which is half a step inside the rim, and the innermost is half a step from the centre. The sampler never visits either extreme of the cap, by construction, for exactly the reason the first rung’s cell centres never visit an edge.

The stereographic projection’s principal scale at angular distance $\rho$ from its centre is $\sec^2(\rho/2)$ exactly, and it is a maximum at the rim. So a ring sampler asked for the largest scale over the cap returns the value at the outermost ring, which is below the bound, at every $n$:

The reading this reproduces #

That table is the explanation for something this collection printed early on and recorded as a finding: a measured scale spread for the stereographic projection three parts in a thousand below $\sec^2(\rho/2)$, which is a proved bound.

A result better than a theorem is a sampling artefact every time, and the note written at the time said so and named the cause — the equal-area ring sampler put its extreme rings half a step inside the centre and the rim. What it did not do is derive the size of the effect, and the size is what makes it a measurement rather than a plausible story. At twenty-five rings over a 45° cap the half-step accounts for 3.42 parts in a thousand, which is the reading.

The deficit falls like $1/n$ and not like $1/n^2$, because the quantity is largest on the boundary. That is the first rung’s rate, arriving in a sampler that was chosen specifically to avoid the first rung’s problem. Being equal-area does not help: the half-step is a property of putting samples at cell centres, and every equal-area construction that partitions the region into cells and samples their centres has it.

What was computed, and how #

The rim deficit is closed form throughout — the ring position from the sampler’s own construction, the scale from $\sec^2(\rho/2)$ — so nothing in that table is a measurement of anything except the arithmetic of the sampler. Its assertion requires two things that could each fail: that the deficit at twenty-five rings be of the order that earlier reading reported, and that the fitted exponent be −1 to within a seventh. A projection formula that had drifted would break the first; a sampler that had been rewritten to include its endpoints would break the second and would be the right fix.

The reference for the convergence figure is a Gauss–Legendre rule of two hundred and twenty nodes in $\sin\varphi$ crossed with a seven-hundred-and-twenty-point uniform rule in longitude. The choice is deliberate: the uniform rule is spectrally accurate for a periodic integrand, which longitude is, and Gauss–Legendre is accurate for a smooth one, which the integrand in $\sin\varphi$ is. Neither is a finer version of any sampler being audited, which is what makes the comparison mean anything at all.

The cap discrepancy uses thirty-two stated caps — eight centres at four radii each, written down once — rather than random ones, so the number is reproducible and the assertion tests the same thing every time it runs.

Where the model stops #

A discrepancy over thirty-two caps is not the discrepancy. The true spherical cap discrepancy is a supremum over all caps and is expensive to compute; what is measured here is a lower bound on it, and the ordering between samplers at a given size should not be read as settled. The magnitudes are the content.

None of these is the best available sampler. Equal-area partitions of the sphere with bounded aspect ratio exist — Leopardi’s recursive zonal scheme is the standard one — and spherical $t$-designs do better still on polynomial integrands. Neither is used here, because the purpose was to audit the samplers this collection actually contains rather than to find the best one.

Nothing here is about what happens on a cell boundary. A cell system trades area for shape is the same trade-off made for addressing rather than for sampling, and the two have different objectives — an address wants compact cells and a quadrature wants good weights. They are not the same problem and a scheme that wins one can lose the other.

The generalisation #

The rule that comes out of this is not about spheres.

A sampler that is exact by construction is worth less than a sampler whose error is corrected exactly. The ring construction gives up the cosine weight in exchange for equal areas per ring, and loses, because the weight was exact and the equality is only along one axis. The graticule keeps a sampler nobody would defend and repairs it with an exact factor, and wins.

That is the same shape as a rule this collection has reached from the other end. The plate carrée is a projection nobody chooses and is the right one to compute on, because its defects are known in closed form. A method whose error can be written down is more useful than a method with a smaller error nobody can characterise, and both cases here are instances of it. Chebyshev’s criterion is the same preference expressed as a design rule for maps rather than for methods.

Who found it, and when #

The obstruction is Euler’s, from 1758 in its polyhedral form, and the impossibility of an equal-area equally-spaced point set on a sphere is folklore rather than a theorem with an author — every attempt to state it precisely turns into a different theorem about a different notion of regularity.

The Fibonacci lattice on a sphere is usually credited to a 1994 paper of Swinbank and Purser’s antecedents in phyllotaxis, and its non-monotone behaviour in the point count is well known in that literature and absent from every application of it in cartography. The spherical cap discrepancy is Beck’s, and the rates for point sets like these are Beck and Chen’s from the 1980s.

What appears nowhere is the audit. Distortion measurements over regions are published constantly and none of them states its sampler, so a reader comparing two published Airy numbers is comparing two quadratures whose rules differ by a factor of six in accuracy at the same cost.

What a published regional number is missing #

The audit’s absence has a specific consequence for reading the literature, and it is worth stating in the form a reader can act on.

A regional distortion figure is a quadrature. Airy’s criterion, Kavrayskiy’s, and every mean of a pointwise quantity over a region is an integral, and an integral computed numerically is the value of a rule applied to a sampler. The published number is the output of that rule.

So two published numbers for the same projection over the same region can differ without either being wrong, and the difference is the samplers’. A factor of six in accuracy at the same point count is enough to move a comparison between two close projections, which is exactly the comparison such numbers are quoted to settle.

And the sampler is almost never stated. A paper says the criterion was evaluated over the region; it does not say on what point set, at what density, or with what weights. That is not usually negligence — it reads as an implementation detail, in the same register as the programming language — and it would be an implementation detail if the rules agreed.

The reader’s remedy is limited and real. A single published figure cannot be repaired, but a comparison can be checked for robustness: two projections separated by less than the plausible quadrature error are not distinguished by the measurement, whatever the digits say. Knowing the size of that error requires knowing the sampler, which is the thing not stated — so the honest reading of a close comparison is that it is unresolved.

The producer’s remedy costs a clause. Naming the sampler and the point count would let anybody reproduce the number and would let two papers be compared, and it is the same one-line addition this collection keeps asking for in different currencies — a normalisation, an interpolation rule, a sign convention, a datum, a sampler.

Where the ladder goes next #

Three rungs have taken apart three estimators, and each was caught by comparing it with something outside itself — a closed form, a different integral rule, a bound. The obvious question is what to do when there is nothing outside to compare with, and the obvious answer is to refine and watch the answer settle. The next rung is about why that does not work: a sequence that converges beautifully, whose increments halve exactly as they should, and whose limit is wrong by a factor of twenty-seven.