The span ladder, run on all five

The exact map says the seam is smooth settles the conformal seam on the one solid this collection has a closed-form face map for — a square dihedron, mapped by the stereographic projection composed with the lemniscatic integral. Its corner falls in exact proportion to the arc the tangent is read over, fitted exponent 0.99, which is the signature of a chord artefact rather than of a corner.

That rung ended owing a shortfall, recorded on 2026-09-01 and quoted here in full because this essay is the answer to it:

Running that ladder on the series face maps of the five Platonic solids is one loop and would settle their corners too.

The loop is one loop. The answer is not the one the sentence expects.

What the span ladder does #

The instrument is worth restating because everything here turns on it, and because it is the part of the previous rung that generalises.

A corner is a discontinuity in the tangent direction of a curve crossing the seam. To measure one, a tangent has to be read on each side, and a tangent read from points spanning a finite arc is a chord direction — which differs from the true tangent by an amount proportional to the span whenever the curve is curved, and the image of a great circle under any of these maps is curved.

So a finite-span measurement of a perfectly smooth join returns a nonzero number, and that number says more about the span than about the join. Halve the span and it halves.

A real corner does not do that. The tangent directions on the two sides are genuinely different, and shrinking the arc converges to that difference rather than to zero. The verdict is therefore in the exponent and not in the number at the finest span, which is the whole point of the ladder and is why it needs no exact map to run.

Every face map on every solid has a real corner #

Running it on all five solids and all three face maps gives fifteen readings.

The gnomonic and the equal-area map coming out flat is no surprise: the corner is not at the midpoint already establishes that both have genuine corners away from the edge’s midpoint, at 20.15° and 7.28° on a cube two-fifths of the way to the vertex. What the ladder adds for them is a confirmation that those numbers are corners and not measurement artefacts, which nothing before had checked.

The series conformal map is the surprise. Its exponent on the cube is −0.018, on the tetrahedron −0.012, on the octahedron −0.032, on the dodecahedron −0.026. Those are flat lines. Its corner at the finest span is 0.769° on the cube — small, real, and not going anywhere.

Where on the edge the measurement is taken #

One choice in the loop needs stating because it decides whether the five numbers are comparable at all.

The corner is a function of where along the edge the crossing is. The corner is not at the midpoint establishes that: the midpoint is the one place on any edge where every face map here has no corner at all, because the face’s own mirror through it reverses the along-edge direction and the shear term in the Jacobian is odd. Away from the midpoint the corner grows.

So a crossing at a fixed number of degrees along the edge is a different place on every solid — the tetrahedron’s edge half-length is 54.74 degrees and the dodecahedron’s is 20.91, a factor of two and a half. Measuring all five at fifteen degrees would compare the tetrahedron a quarter of the way to its vertex with the dodecahedron three-quarters of the way to its, which is not one measurement.

Every reading here is taken at two-fifths of the solid’s own edge half-length, and the measurement spans are the same fractions of it. That is far enough from the midpoint for the corner to have grown and far enough from the vertex that the curve stays inside the two faces, and it makes the five columns comparable.

The corner belongs to the series and not to conformality #

Two facts now sit side by side and they look like a contradiction.

The exact conformal map of a square dihedron has no corner — the previous rung establishes it, with a fitted exponent of 0.99 and a corner falling to a thousandth of a degree at the shortest span.

The series conformal map of a cube has a corner of 0.769 degrees that does not move with the span.

They are not in conflict, and the resolution is the sentence this rung exists to write down: the corner belongs to the series, not to conformality. A series fitted to a face is not a conformal map; it is a map that satisfies the conformality condition to whatever tolerance the fit reached. It has a real Jacobian, that Jacobian is genuinely discontinuous across the seam, and the discontinuity is the fit’s error made visible.

The obvious next question is whether adding terms removes it.

That figure is the finding underneath the finding. The fit is getting better at the thing it optimises and the seam corner is not tracking it, so more terms is not a route to a smooth seam — and the corner cannot be driven down by working harder at the same construction.

What a corner of this size does to a figure #

The numbers so far are angles at a point on a seam, and it is worth asking what they mean for the object the anchor is about.

A polyhedral map is printed, cut and folded. A feature crossing a seam — a coastline, a river, a graticule line — arrives at the fold from one face and leaves on the other, and a corner is the angle by which it visibly kinks at the crease. Four degrees is a kink a reader sees. Three quarters of a degree, on a cube whose faces are perhaps ten centimetres across, is a deviation of about a millimetre over the ten centimetres either side of the fold, which is at the edge of what a printed line’s own width conceals.

So the ordering of the three face maps is not academic on the two smallest solids and is on the two largest. On a tetrahedron the choice of face map is the difference between a 36-degree kink and a 4-degree one, both plainly visible. On an icosahedron it is the difference between 9 degrees and something no printing process resolves.

Which gives the practical rule the rung ends with: the face map matters most on the solids nobody uses. A tetrahedral globe is a curiosity and its seam behaviour is dramatic; an icosahedral one is the standard and its conformal seam is invisible either way. The map to choose is decided by the gnomonic’s nine degrees rather than by the conformal map’s four thousandths.

Why the residual and the corner come apart #

The mechanism is worth following, because it is a general hazard rather than a fact about polyhedra.

The fit minimises a residual on the face’s boundary: how far the mapped edge is from the straight line it should be. That is an integral over a curve, so it is a statement about the function’s values. A seam corner is a statement about its derivative, and specifically about the derivative in the direction across the seam.

Nothing in least squares on values controls a derivative. A fit can drive its boundary residual down while acquiring high-frequency wiggle whose derivative is worse than before — which is exactly what happens as the normal equations become ill-conditioned, because the extra terms buy their small residual improvement with large cancelling coefficients.

The icosahedron shows the end of that road plainly. Its boundary residual falls from 6.63 × 10⁻³ at four terms to 7.85 × 10⁻⁴ at sixteen, and then rises again — to 2.44 × 10⁻³ at twenty terms and 4.25 × 10⁻³ at twenty-four. Past sixteen terms the fit is getting worse at the only thing it optimises, and the conditioning has given way.

A conformal map onto a face is the rung that built this fit, and it records that choosing the map is a least-squares problem. What it could not know is that the quantity a later rung would want to measure is not the one being minimised.

The icosahedron, where the corner reaches the sampling #

Four of the five solids give a clean flat line for the series map and the fifth does not, and the exception is worth reporting rather than smoothing.

The icosahedron’s series corner at the finest span is 0.004 degrees — four thousandths — and its readings across the span ladder are 0.0082, 0.0017, 0.0015, 0.0032, 0.0040. That is a spread of a factor of five with no trend, and a fitted exponent of 0.12 which means nothing.

The reason is that four thousandths of a degree is where the edge sampling itself lives. The curve crossing the seam is sampled at 1,601 points over a span that on the icosahedron is a fifth of a degree, and the tangent fit at each end is reading a direction from points a ten-thousandth of a degree apart. At that scale the arithmetic’s own noise is comparable to the quantity.

So the honest statement about the icosahedron is that its series seam corner is below four thousandths of a degree and not resolved further by this instrument, which is a bound rather than a value. It is exempted from the span-independence assertion and reported as a bound, which is the same treatment the exact map says the seam is smooth gives its own finest reading.

It is also a reminder of what the ladder is for. It separates a corner from an artefact by a trend, and a trend needs a quantity larger than the noise. Where the corner is genuinely tiny the ladder cannot tell whether it is tiny or absent, and only an exact map settles that — which is where the previous rung went.

What the shortfall actually asked for, and what it got #

The shortfall’s sentence is worth reading twice. “Running that ladder on the five Platonic solids is one loop and would settle their corners too.”

Both halves are true and the second means something different from what it sounds like. The ladder does settle their corners: it settles that they are 4.10°, 0.769°, 0.376°, 0.161° and about 0.004° for the series map, and that all five are real. What it does not settle is whether a conformal map of those faces has a corner, because there is no conformal map of those faces available to run it on — only fits.

That is the honest state of the anchor. The exact map exists on a square dihedron and nowhere else here; the five Platonic solids have series fits; and the span ladder, applied to a fit, faithfully reports the fit’s own corner.

It is also why the answer is worth having rather than a disappointment. A polyhedral globe is not printed from an exact map. It is printed from a construction somebody implemented, and if that construction is a fitted series then the seam corner a reader can see at a fold is 0.769 degrees on a cube and not zero — which is a fact about the artefact rather than about the mathematics, and is the fact a printer needs.

The gnomonic’s five decimal places #

One number in the tetrahedron figure is worth stopping on because it is the strongest evidence the instrument works.

The gnomonic reports 36.33287 degrees at a seventeen-and-a-half-degree span and 36.33287 at a one-degree span — the same to five decimal places, across a factor of sixteen in the arc the tangent is read over. On the cube it is 20.15127 at every span, on the octahedron 21.24955, on the dodecahedron 8.83779 and on the icosahedron 9.18521.

A chord artefact of the size the smooth case shows would move those numbers in the second decimal place at the widest span. They do not move at all, and the reason is worth stating: the gnomonic’s corner is a discontinuity in direction of about twenty degrees, and a chord’s departure from a tangent over a small arc is a second-order effect on top of a first-order jump. The jump swamps it completely.

So the flat lines in these figures are not a failure of resolution. They are the ladder saying, with all the precision it has, that there is something there to measure — which is what makes the series map’s much smaller flat line believable rather than a floor.

What subdividing buys, and what it does not #

The five solids form a sequence in face count — four, six, eight, twelve, twenty — and reading the corner along it gives the design result.

Every map’s corner falls as the solid gains faces, by about a factor of four across the range for the gnomonic and by three orders of magnitude for the series conformal map. That is the same trade more faces, less distortion, more cutting prices for the distortion within a face, arriving at the seam between faces: subdividing helps.

What it does not do is change the ordering. On every one of the five, the gnomonic’s corner is the largest, the equal-area map’s is next, and the series conformal map’s is smallest — the same ordering the corner is not at the midpoint found on the cube, now confirmed on all five rather than assumed to generalise.

And it does not remove the corner. An eighty-face geodesic solid’s series map would have a smaller corner than an icosahedron’s and would still have one, because the mechanism is the fit’s error and not the solid’s geometry.

What this pays and what it leaves owing #

The shortfall is paid in the sense that matters: the loop was run, on all five solids, and it produced numbers rather than an expectation. What it produced is recorded above and it is not what the shortfall’s author expected, which is the ordinary outcome of running a measurement somebody was confident about.

Two things are now owed that were not before.

The series construction cannot be improved by adding terms, and the anchor has no better one for a triangular or pentagonal face. The exact map exists for a square dihedron because a square is what the lemniscatic integral maps a hemisphere to; the analogous construction for a triangle or a pentagon is a different Schwarz–Christoffel problem, solvable and not solved here. Until it is, every conformal polyhedral map in this collection is a fit with a real corner.

And the corner has never been checked against what a reader sees. Four degrees on a folded tetrahedron is visible; four thousandths on an icosahedron is not; and where between the two a corner stops mattering is a question about paper and eyes rather than about Jacobians. A net can land on top of itself is the rung that took the anchor from geometry to what a printer actually has to do, and the same step is available here and has not been taken.

The place none of twelve rungs has reached #

Twelve rungs have measured a polyhedral map at a point, over a face, along a net, across a seam and now along five seams at once. Every one of those measurements is taken somewhere the map is a map: a place with a well-defined Jacobian, two faces, and a tangent on each side.

There is one place on every polyhedral map where none of that holds. At a vertex, three or more faces meet, the surface has a cone point, and the angle deficit is exactly computable and not zero. Every face map here is singular there. Every net puts several of them on its boundary. And the corner a feature gets passing one is not a shear that a better face map could remove — it is the curvature of the sphere, concentrated to a point and impossible to spend anywhere else.