The corner is not at the midpoint

The rung below this one left a shortfall, recorded on the day it shipped and stated plainly: it measured the seam corner for the gnomonic face map and for the equal-area one, and not for the conformal one, which conformal.js builds by a different route. The prediction is that it kinks like the equal-area map, since it too is a construction per face; the measurement is one adapter away and is not made.

The adapter took an afternoon. What it produced was a number for the conformal map and, before that, a control that failed — and the failed control is the rung.

The control that failed #

A new measurement gets checked against a known answer before it is believed, and the known answer here was the best-established number in the anchor: the gnomonic face map’s seam corner is exactly zero, on all five Platonic solids, at every obliquity, to 4 × 10⁻¹³ degrees. The rung below states it as a fact about the construction rather than as a measurement, and gives the reason — one central projection restricted to each face in turn, so both faces carry the same map and there is nothing for a seam to disagree about.

Running the conformal measurement needed a crossing somewhere on the edge, and the natural first move was to slide it away from the midpoint to see whether the position mattered. It did. At fifteen degrees along a cube’s edge, at a right-angled crossing, the gnomonic reads

$$21.46^\circ.$$

That is not a small number and it is not noise. It is stable to eleven decimal places across a factor of sixty-four in the resolution used to estimate the tangents, which is what a genuine tangent discontinuity looks like and what a numerical artefact does not.

What the midpoint is #

The measurement in the rung below is correct, and every number it printed stands. What it did not say — because nobody had asked — is that it crossed at the midpoint of the edge, every time, on every solid, at every obliquity. Its own comment records the assumption: everything this rung measures is a function of the obliquity rather than of where along the edge the crossing happens.

That assumption is refuted by the face’s own symmetry, and the refutation also explains it.

Write the Jacobian of a face map at a point on the edge, in coordinates whose first axis is along the edge on both the sphere and the page. The map sends the edge to the edge, so the first column has no second component:

$$J = \begin{pmatrix} a & b \\ 0 & d\end{pmatrix}.$$

Unfolding is a reflection in the page’s edge line composed with the sphere’s own reflection across the edge, so the neighbouring face’s Jacobian at the same point is

$$J' = \begin{pmatrix} a & -b \\ 0 & d\end{pmatrix},$$

and the join is smooth exactly when b = 0. The quantity that decides whether a seam has a corner is the shear — the component along the edge of the image of the direction across it.

Now use the face’s own mirror. A regular n-gon face has a reflection through the midpoint of each edge, every face map here commutes with it, and that reflection reverses the along-edge direction. So b is an odd function of distance along the edge from its midpoint, and an odd function is zero at the origin.

Every face map on this site therefore has no corner at the edge’s midpoint, and the rung below measured all three at the one point where they agree.

What was computed, and how #

Three changes to the machinery, and the first is the one that matters.

The crossing point moved. edgeKink gained a position along the edge, which meant rebuilding the tangent frame at the crossing rather than carrying it from the midpoint. The first version did not, and the symptom was unmissable: the gnomonic read 9.4° at a crossing where the curve was not in fact a great circle through the base point. The control caught it, which is the second time in this rung that a quantity known to be exactly zero has earned its place.

The conformal map arrived through an adapter. polyhedra.js has a solid with numbered faces and a frame per face; conformal.js has one canonical face at the origin of a chart, with its first corner along the chart’s own axis. The rotation between them is the whole adapter, and it has to be a proper rotation: sending the face centre to the chart’s origin and the first vertex to the chart’s first corner forces the third axis to flip sign, and dropping that sign mirrors each face in its own page independently. The two unfolded halves then meet head to tail and every seam reads 180°, which is another failure the gnomonic control would have caught and this one announced itself.

The unfolding’s translation was wrong and is now right. The rigid motion that places the second face was built from a linear projection of the two shared vertices rather than from where the maps actually draw them. The direction was right, because both vertices are the same distance from the face centre and the missing factor is common to them — so every kink number in the rung below is unaffected — and the position was not, so every picture of a seam showed the two halves of the curve separated by a gap they should not have had.

The ordering, reversed #

The gnomonic is the worst of the three, everywhere, by a factor of about three over the equal-area map and by more than an order of magnitude over the conformal one. It is also the face map every published polyhedral globe uses — Fuller’s, Cahill’s, and their modern descendants — chosen because great circles stay straight on each face, which is the property what a face can preserve prices and the globe on a solid opens the anchor with.

The two facts are the same fact. A gnomonic image on the polyhedron is the intersection of the plane of the great circle with the solid’s surface: a planar polygon in three dimensions, straight on each face. A plane section of a polyhedron is not a geodesic of its surface, and unfolding turns a planar section into a chain of straight segments with corners at every edge it crosses. Straightness on each face and smoothness across the seam are different properties, and the construction that guarantees the first is the one that gives up most of the second.

Position, not obliquity #

The rung below swept the obliquity — the angle between the feature and the normal to the edge — and found the equal-area corner peaking at about thirty degrees and falling away again. That sweep was taken at the midpoint, where the corner is zero for every map, so what it measured was a second-order quantity in a place where the first-order one vanishes.

The two parameters are not comparable in size. Along the edge, the gnomonic corner runs from zero to twenty degrees over a cube; across the obliquity at any fixed position it moves by a factor of about two. Position is the first-order effect and obliquity is a modulation of it, which is the reverse of the reading a midpoint sweep supports, and it matters practically because a feature crossing a seam has a position whether or not anybody chose one.

Why the conformal map is the smooth one #

The shear b is what has to vanish, and conformality kills it in one line.

Both faces parameterise the shared edge identically — congruent faces, the same canonical construction — so the tangential derivatives already agree. A conformal map’s Jacobian is a rotation times a scale. If it takes the edge direction to the edge direction, it is a scale times the identity, so b = 0 at every point of the edge and not merely at the midpoint.

Equal-area fixes only the determinant, ad = 1, and leaves b entirely free. That is the difference the table measures, and it is a case of the collection’s standing observation that a condition on a map leaves a whole function free — here the free function is exactly the shear along a seam, and the seam is where it becomes visible.

What this rung can and cannot resolve #

The conformal face map here is a least-squares series solved against a boundary condition it can only satisfy in the limit: at sixteen terms the target polygon’s edge is straight to three parts in a thousand, and pushing to more terms makes the normal equations worse conditioned faster than it improves the boundary.

So the honest statement is that this construction bounds the conformal seam corner at about two degrees rather than measuring it. The bound is already an order of magnitude under the gnomonic’s 21.46° and the equal-area map’s 7.69° at the same crossing, both of which are stable to five decimals, so the ordering survives. What does not survive is any claim about the conformal map’s corner being exactly zero, which is what the argument above predicts and what this measurement cannot confirm.

That is the shortfall this rung inherits and passes on. Confirming it needs the exact conformal map — Lee’s, in elliptic functions — rather than a series fit, and that is a different construction rather than a longer run of this one — the same judgement a conformal map onto a face recorded when it chose a series over elliptic functions in the first place.

Where the model stops #

The face maps are the site’s three and not everybody’s. A polyhedral map may use any map per face, and the two extremes measured here bracket the practical range: one map of the whole sphere restricted to each face, and one canonical construction per face. The condition does not always decide the map applies here as everywhere — there are many equal-area face maps and this is one.

The crossings here are great circles. A feature that is not one has its own bending, which crosses the seam unchanged; what is measured is the extra corner the seam adds.

The measurement stops short of the vertex. Two fifths of the half-length is far enough for the corner to have grown and far enough from the corner that the curve stays inside the two faces. Nearer the vertex all three maps become singular, the conformal one worst, and the quantity being measured stops being a property of the seam.

And a net cuts most of its edges. A solid with F faces keeps F − 1 seams and cuts the rest, so a net that loses the fewest neighbours is choosing which seams a reader will meet. This rung prices what one of them costs; choosing which to keep is that rung’s question, and it now has a number to work with that varies along each edge rather than a single one per edge.

The generalisation #

The rung below ends on a good sentence — continuity across a join is free when the pieces come from one construction and has to be paid for when they do not — and this rung is the correction to it. The gnomonic is one construction, and its join is not free.

The sharper statement is about what a measurement holds fixed. A quantity measured at one point of a symmetric object, on an object chosen for its symmetry, is at risk of measuring the symmetry: the midpoint of an edge is a fixed point of a reflection, the shear is odd under that reflection, and any measurement taken there returns zero regardless of the map. Nothing in the earlier measurement was wrong; what was missing was a parameter, and the parameter was invisible because holding it at zero looked like not having a choice to make.

The collection has met this shape before, from the other direction. Where the worst point is exists because a distortion figure quoted at a projection’s own centre is a measurement of the centre. A projection’s own points of no distortion are the same trap with the sign reversed. A symmetric object has places where every candidate answer agrees, and those are exactly the places a first measurement is most likely to be taken.

Who found it, and when #

That a plane section of a polyhedron is not a geodesic of its surface is elementary and old; unfolding a cube and watching a plane section acquire corners is a classroom exercise. What appears not to be stated anywhere is the consequence for polyhedral map projections, and the reason is probably that the gnomonic’s straightness on each face is such a strong and useful property that the seam is not where anybody looks.

The symmetry argument — that the shear is odd along the edge, so every face map is smooth at the midpoint — follows in two lines from the face’s own mirror and is the sort of thing that is either well known or has never needed saying. It is stated here because it is what makes the rung below’s headline true and narrow at the same time.

What it says about where to put a seam #

The finding is about where along an edge the damage is, and that turns a coarse design question into a finer one.

The existing rule is binary. A polyhedral net is oriented so that its cuts fall in water rather than across land, and the check is whether an edge intersects a feature anybody cares about. That rule treats an edge as uniformly bad, which is what the midpoint result contradicts.

The damage is not uniform along the edge. The shear is odd about the midpoint, so it vanishes there and grows towards the ends — which means a feature crossing a seam halfway along is nearly undamaged and one crossing near a vertex is damaged most. Two orientations that both put a coastline across an edge are not equivalent, and the difference between them is available before either is drawn.

So the design question sharpens from does the seam cross this to where along the seam does it cross. That is a cheaper question than it sounds: the crossing parameter is a number between zero and one, computable from the intersection the existing check already finds, and the damage is a known function of it.

And it makes the vertices the thing to route around, which agrees with where this ladder is going next for a different reason. The edge midpoints are the benign places on a net’s boundary and the vertices are where the sphere’s curvature is concentrated; an orientation search that treated an edge as uniformly costly has been spending its freedom on the wrong parts of the boundary.

One caution, because the result is narrow. The shear vanishing at the midpoint is a statement about that one quantity at that one point, established by a symmetry argument rather than by a general smallness claim. A feature crossing at the midpoint still crosses a seam: it is torn on the net, it lands on two sheets, and its continuity is broken. What the symmetry buys is that the geometry of the crossing is undistorted there, which is worth having and is not the same as the crossing being free.

Where the ladder goes next #

Ten rungs price a polyhedral map at a point, over a face, along a net, across a seam, and now along a seam. The place none of them reaches is the vertex, where three or more faces meet, the surface has a cone point, and the angle deficit is exactly computable. Every face map 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 can be made to vanish — it is the curvature of the sphere, concentrated.