The exact map says the seam is smooth

The corner is not at the midpoint measured the seam corner for the gnomonic and the equal-area face maps and could not measure it for the conformal one. The reason was recorded as a shortfall:

The series fit’s boundary residual is three parts in a thousand at sixteen terms and pushing further makes the normal equations worse conditioned faster than it improves the boundary, so the corner wanders between a tenth of a degree and two and never settles. The prediction the geometry gives is that it is exactly zero — conformality fixes the whole Jacobian from its tangential part, so the join is C¹ — and confirming it needs an exact map rather than a longer run of this construction.

There is one, in one case, and it is enough.

The exact map #

A hemisphere maps conformally onto a square, in closed form, by a composition of two steps that are each conformal.

Stereographic, from the opposite pole. It carries the hemisphere onto the unit disc, conformally, with the equator going to the unit circle. Its radial function is $\tan(\rho/2)$, which is one at $\rho = 90°$.

The lemniscatic integral. The Schwarz–Christoffel map from the disc to a square is

$$z(w) = \int_0^{w}\frac{\mathrm{d}t}{\sqrt{1 - t^{4}}},$$

and its four corners are the images of $w = 1, i, -1, -i$, at $z = \pm K$ and $\pm iK$ with $K = \Gamma(\tfrac14)^2/(4\sqrt{\pi}) = 1.3110288$.

The composite is Peirce’s construction of 1879, and its two halves make a net of two squares sharing an edge — a square dihedron, which is a polyhedron with two faces and is the only solid in this collection’s family whose conformal face map has a closed form.

What the quadrature had to do #

The integral is over a straight segment from 0 to $w$, and one thing about it is worth stating because getting it wrong is how this construction usually goes silently astray.

No branch tracking is needed. For $|t| < 1$ the real part of $1 - t^4$ is at least $1 - |t|^4 > 0$, so the integrand’s argument never leaves the open right half-plane and the principal square root is analytic along any path inside the disc.

But the endpoint is singular on the boundary. At $|w| = 1$ — which is the whole equator, and therefore every point the seam measurement needs — the integrand behaves like $(1-s)^{-1/2}$ at the far end of the segment. A graded mesh converges on that only as the square root of the panel count, and a first version returned $K$ to three digits and stopped improving. The substitution $s = 1 - u^2$ turns the integral into $\int 2u\,g(1-u^2)\,\mathrm{d}u$, whose integrand is analytic at $u = 0$, and forty Gauss–Legendre nodes then give thirteen figures.

What the exactness buys #

The series construction’s conformality is limited by its own boundary residual, which is three parts in a thousand at sixteen terms. This map’s is limited by the arithmetic.

Measured over a grid of ninety-nine points, the worst angular deformation is 2 × 10⁻⁵ degrees, and that number is the finite difference’s floor rather than the map’s. Which is itself an instructive detail, and it is the sampling anchor’s finding arriving one level down.

jacobian divides a difference by its step, so it amplifies whatever noise the forward map carries by the step’s reciprocal — and this forward map is a quadrature, smooth to about eleven digits rather than to sixteen. At the site’s default step of $10^{-4}$ the noise reaches $2\times10^{-3}$ degrees of spurious angular deformation; at $10^{-2}$ it is $9\times10^{-6}$, and the truncation the larger step buys back is $10^{-8}$. The optimum step moves when the function being differentiated is computed rather than evaluated, which is exactly what the step under everything else is about.

The seam, and what the earlier rungs were measuring #

The two squares are unfolded — the southern one placed beside the northern along their shared quarter of the equator — and a great circle crossing at a stated obliquity is drawn by each chart in turn.

The corner that comes out is not zero at any finite measurement span, and it is not a corner.

Every halving of the span halves it. The fitted exponent is 0.990, which is the first power, and the first power is exactly what a chord’s departure from a tangent does when the curve it is read from is curved.

So the number is the instrument. A tangent estimated from three points spanning a finite arc is a chord direction, and the image of a great circle under any of these maps is a curve, so a perfectly smooth join reports a corner proportional to how far away the tangent was read from.

What the series construction was doing #

The series map this anchor has used since a conformal map onto a face is a truncated expansion fitted by least squares to hold a constant boundary scale, and the shortfall’s diagnosis of it is confirmed rather than replaced by this rung.

Its boundary residual at sixteen terms is three parts in a thousand. A corner is a derivative quantity read off the boundary, and a derivative inherits a residual amplified by the reciprocal of whatever arc it is read over — so a boundary held to three parts in a thousand supports a corner estimate good to a fraction of a degree at best, and to nothing at all when the corner itself is smaller than that.

Adding terms does not help, because the normal equations become worse conditioned faster than the boundary improves — which is the nodes were evenly spaced’s subject in the neighbouring anchor, and which that rung shows is a property of where the collocation points are rather than of how many there are. The two shortfalls turn out to have the same cause and were paid separately.

So the wandering the shortfall recorded — a tenth of a degree to two — was two artefacts superposed: the fit’s own residual, and the chord artefact this rung measures at between 0.09° and 2.8° depending on the span. Neither is a corner, and neither could be told from one without an exact map or a span ladder.

The control that makes it a measurement #

A quantity that falls to zero under refinement is only evidence if the same refinement leaves a real quantity alone.

The gnomonic face map on a cube, fifteen degrees along its edge, has a genuine corner: the previous rung measured it at 21.46°. Run down the same ladder it reports 21.4572° at twenty-four degrees of span, at twelve, at six and at three — the same number to four decimal places, with a spread of 1.00000000.

That is the rejecting form of the whole rung. If the ladder shrank both numbers, it would be measuring the instrument and nothing else; if it shrank neither, the conformal seam would have a corner. It shrinks exactly one.

What was computed, and how #

The map is a composition of two closed forms and one quadrature, checked three ways.

The square is a square. The four corners come back at radius $K$ to a part in a million, and the equator between two corners is straight to $2\times10^{-9}$ of the side’s length.

The map is conformal. Worst angular deformation $2\times10^{-5}$ degrees over the sampled graticule, at a derivative step chosen for a computed rather than an evaluated function.

And the unfolding is an unfolding. The southern square’s centre must land on the far side of the shared edge from the northern one, and the two halves of the drawn curve must meet. Both are checked at every row, and the first caught a real mistake: composing a further reflection onto the placement — which a two-chart atlas seems to need, since the two charts see the sphere from opposite sides — turns the sheet over twice and lays the southern square exactly on top of the northern one. That version reported a corner of twice the crossing angle at every obliquity, which is a perfectly smooth curve seen through a wrong rigid motion.

What it says about the other solids #

The dihedron is the case with a closed form; the cases anybody would draw are the Platonic solids, and the argument transfers although the map does not.

The geometry is the same. Two conformal charts agreeing on a shared curve agree in their full derivatives along it, whatever solid the curve is an edge of. Nothing in that argument uses the number of faces.

The construction is not. More faces, less distortion, more cutting is the trade a polyhedral map is chosen for, and every member of it past the dihedron needs either a numerical fit or an elliptic-function construction. Lee’s tetrahedral map exists in closed form and needs Dixon’s functions; Adams’s cube does not.

And the span ladder does not need either. Running the four-span ladder on the series face map would separate a corner from an artefact there too, at the cost of four evaluations, without waiting for an exact map that may never be written. That is the practical outcome of this rung and it is available immediately.

One further link is worth making explicit. The cut has to go somewhere establishes that every polyhedral map has cut edges and joined ones, and this rung is about the joined ones only: a cut edge has no join to be smooth across, and the two sides of it are simply two different places on the page.

Where the model stops #

The two-square net is not the quincuncial arrangement. Peirce’s own layout cuts the second hemisphere into four triangles and packs them round the first square, which tiles the plane and is what the projection is famous for. The two-square net used here is Guyou’s, and it is the one with a single shared edge, which is what a seam measurement needs.

One solid, and a degenerate one. A square dihedron has two faces and no vertices in the ordinary sense, so it is the easiest polyhedral net there is. The Platonic solids have no closed-form conformal face map — Lee’s tetrahedral map needs Dixon’s elliptic functions and Adams’s cube is a numerical construction — so nothing here directly measures a cube’s conformal seam. What it establishes is the geometry: conformality forces the join to be C¹, and the number the series construction reported was its own residual.

The prediction was already available. Conformality fixes the whole Jacobian from its tangential part — a conformal map’s derivative is a scalar times a rotation, so knowing what it does along a curve determines what it does across it — and two conformal maps agreeing on a curve therefore agree in their full derivatives there. This rung does not discover that; it confirms it in the one case where confirming is possible.

And the ladder is a slope rather than a limit. The corner at the shortest span tried is 0.092°, not zero. What is established is that the sequence falls like the first power of the span with fitted exponent 0.990, which is an artefact’s signature, and that the control does not.

The generalisation #

The rule is one this collection keeps meeting and has not stated in this form: a measurement whose answer depends on how it was made needs its dependence measured, not its answer refined.

Rungs 9 and 10 both reported kinks at a fixed measurement span, and both were right to, because the quantities they were measuring are far above the artefact — 21° against a fraction of a degree. Rung 10’s shortfall arose exactly where they are not: the conformal seam’s corner is at or near zero, the artefact is the same size as anything a longer series could resolve, and refining the fit was chasing a number the instrument was producing.

The way out was not a better fit but a different question. Instead of what is the corner, ask how does the corner behave when the instrument changes — and the answer separates a corner from an artefact in four evaluations, with no exact map required at all.

That is worth carrying, because the exact map is a piece of luck. Most of the seams this anchor cares about have no closed form and never will, and the span ladder works on all of them.

Who found it, and when #

Peirce published the quincuncial projection in 1879, and the construction is his: the stereographic projection followed by an elliptic integral, chosen because the map is conformal everywhere except at four points and tiles the plane. Guyou’s rearrangement of 1887 puts the two hemispheres in two squares, which is the net measured here. Adams extended the family to other polygons in the 1920s and Lee to the tetrahedron in 1965, with Dixon’s elliptic functions.

That the join between two conformal charts is C¹ is elementary complex analysis and is in no cartographic treatment, because no cartographic treatment asks about the join — an interrupted map is understood to be interrupted, and whether a river drawn across a seam has a visible kink is a question about drawing rather than about construction.

The chord artefact is not a discovery either. It is what a finite-difference tangent does, and every numerical analyst knows it. What is new is only the pairing: a seam whose corner might be zero, an instrument that reports a nonzero number for a zero corner, and a control that separates them.

Why an exact case was worth building #

It is fair to ask whether an exact map of a two-faced solid earns its place in an anchor about real polyhedral projections. Two things say it does.

It converts a prediction into a measurement. The geometry’s answer was available before this rung and was recorded as a prediction, which is not the same as a number. What a face can preserve is this anchor’s habit stated plainly: a property is not a property here until the machinery has computed it and could have come out the other way.

And it calibrated the instrument. The span ladder was written for this rung and it works everywhere, including on the maps that have no closed form. That is the more useful of the two outcomes: the exact map is a single case and the ladder is a method.

There is a general form of that second point worth stating, because it applies to every numerical result in this collection. A method run only on cases with no closed form produces numbers that cannot be checked: a disagreement between the method and the truth is indistinguishable from the truth being surprising, and there is nothing to attribute it to. An exact case separates the two — it says what the method costs when the answer is known, so that when the method is pointed at a case where the answer is not known, the residual can be assigned to the geometry rather than to the quadrature. The exact map is therefore not a small result among large ones; it is the thing that licenses the large ones to be read as measurements at all.

That is why the two-faced solid was built rather than found: nothing in the family of solids anybody prints has a closed form to calibrate against.

Where the ladder goes next #

Eleven rungs have taken the polyhedral family from a globe on a solid to the exact conformal map of one face and the join between two. Every one of them has measured a property of the net — its cuts, its neighbours, its overlaps, its seams. What none of them has measured is what a reader does with one: a polyhedral map is folded, and the folding is a physical operation with a tolerance, so the question of how accurately two faces can actually be brought together is a question about paper rather than about conformality.