The corner that is the curvature

Twelve rungs of this anchor measure 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, every face map is singular there, and the angle deficit is exactly computable and is not zero. Two rungs of this anchor have named it as the place to go next and neither went.

The expectation, and why it was wrong #

A face map takes a spherical face to a plane polygon. At the vertex the sphere has a full 2π of angle round the point and the plane polygon has less, so something has to be discontinuous, and the natural guess is that the corner a crossing feature acquires grows without bound as the crossing approaches.

It grows and it stops. On a cube: 25.33° halfway along the edge, 47.25° nine-tenths of the way, 52.53 at ninety-nine hundredths, 53.07, 53.12, 53.1295. On a tetrahedron: 98.2121. On an icosahedron: 24.8723.

The reason is that the corner is a property of the two faces sharing that edge, and those two faces are still perfectly ordinary surfaces right up to the vertex. What is singular at the vertex is the third face, and the fourth, and the fifth — the ones the crossing never enters. A curve crossing a single seam near a vertex sees a well-behaved join between two well-behaved maps, and the join’s corner converges to whatever the two maps do at the common endpoint.

The deficit is a statement about all k faces together, and a single seam crossing only ever sees two of them.

Establishing a limit rather than reporting one #

A number that has stopped moving in the digits printed is not a limit. It is a number measured to too few digits, and this collection has a habit for telling the two apart.

The gnomonic’s first-order approach is what it should be. Its face map is a closed form — a radial projection onto a tangent plane — so the only error in the measurement is the chord artefact the previous rung is about, and a chord’s departure from a tangent is first order in the span, and the span here shrinks with the gap.

That the equal-area map does not follow it is a fact about the construction rather than about the vertex, and it is worth stating rather than smoothing. What a face can preserve builds that map by integrating dθ′/dθ with a shooting method to close it after a full turn, and its own note records that a tabulated inversion was accurate in the value and not in the derivative. The corner is a derivative.

So the gnomonic’s limit is a measurement and the equal-area map’s is a good estimate, and this rung says which is which. On all five solids the gnomonic’s fitted exponent is between 0.994 and 1.006, and the equal-area map’s between 0.77 and 1.91.

The walk, and the span that has to shrink faster than the gap #

Two parameters move together in the hero figure and getting their relation wrong would make the whole measurement meaningless.

The gap is how far the crossing is from the vertex, and it shrinks geometrically: half the edge, then nine-tenths, then 0.99, 0.999, 0.9999, 0.99999 of the way. On a cube’s 35.264-degree half-edge the last of those is 0.00035 degrees from the corner, which is about forty metres on an Earth-sized cube.

The span is the arc the tangent is read over, and the previous rung establishes why it matters: a tangent read over a finite arc is a chord direction, and a chord differs from a tangent in proportion to the arc. Here it must shrink faster than the gap, and not merely for accuracy — if the arc reached past the vertex, the curve would leave the two faces whose join is being measured and enter a third, and the number would be about a different join.

So the span is set to three-tenths of the remaining gap, capped at a hundredth of the edge. At the last row that is a hundredth of a thousandth of a degree, and the curve is sampled at two thousand and one points across it.

That construction is why the residual figure is a straight line rather than a floor. Nothing in the measurement stops improving as the crossing approaches, so the limit is established by extrapolation rather than by the numbers ceasing to move.

What the deficit is, and why no map can move it #

The other number in the hero figure is the one that matters, and it is not measured at all in the sense the corner is.

Descartes’s theorem is the discrete Gauss–Bonnet: the deficits at a convex polyhedron’s vertices sum to 4π, exactly, whatever the polyhedron. The globe on a solid is the rung of this anchor that establishes it — π at each of the tetrahedron’s four vertices, π/5 at each of the dodecahedron’s twenty, and always 4π in total, which is exactly the total curvature of the sphere it replaced.

The consequence for this rung is a conservation statement rather than a measurement. A face map can choose where the missing angle is taken from — sharply, at the seams, or smoothly, spread through the curvature of arcs inside the faces — and it cannot choose how much. Ninety degrees is missing at a cube’s corner under the gnomonic, under the equal-area map, under the exact conformal map of the previous two rungs, and under any map anybody ever writes.

Why the conformal map is missing from these figures #

Two face maps appear in the walk and the third does not, and the omission is deliberate rather than an oversight.

The conformal face map available on the four non-square solids is a fitted series, and the span ladder, run on all five establishes what that means: its seam corner is real, it is the fit’s own error made visible, and it does not settle as terms are added because the normal equations become ill-conditioned faster than the boundary improves. Near a vertex all of that gets worse, since a vertex is exactly where the face’s boundary has a corner and a polynomial series is least able to follow one.

Running the walk on it produces numbers, and the numbers are about the fit. On a cube the six readings are 0.147, 0.643, 10.271, 27.714, 29.768 and 29.977 degrees — a rise of a factor of sixteen between the second and the third, then a settle at thirty degrees.

Reported flat, that says the conformal map’s vertex corner is 29.98°, which would be a real result and is not one. It is a series whose seam corner two-fifths of the way along the edge is three quarters of a degree turning into thirty degrees at the vertex, and a construction that changes by a factor of forty over the last tenth of an edge is a construction leaving its range rather than a map with a large corner. The exact map of a square dihedron would settle it and the four other solids have no exact map, which is the same wall the rung below this one runs into.

So the conformal column is left out of the vertex table and its readings are quoted here instead, as evidence about the fit.

The trade this makes explicit #

Eleven rungs of this anchor have compared face maps on their seam behaviour and the ordering has been the same every time: the gnomonic’s corner is largest, the equal-area map’s is next, the conformal map’s is smallest. The corner is not at the midpoint establishes it on a cube and the span ladder, run on all five confirms it on all five.

Read as a ranking, that says the conformal map wins. Read against the deficit, it says something less comfortable.

A conformal face map has a small seam corner because conformality fixes the whole Jacobian from its tangential part, so two faces agreeing along an edge as parameterised curves agree in their derivatives too. It does not have a small deficit, because the deficit is not a property of the map. So the angle a conformal map does not lose at its seam is angle it loses somewhere else — in the curvature of every feature inside every face, distributed rather than concentrated.

Which is a genuine choice rather than a strict improvement. A polyhedral globe drawn with the gnomonic has visible kinks at the folds and straight great circles inside the faces. One drawn conformally has smooth folds and no straight lines anywhere. The total angle each of them fails to account for is identical, and where it goes is what a reader notices.

What every net puts on its boundary #

The vertex is not a place a reader has to go looking for. It is on the outline of every net ever printed, several times over.

A cube has eight vertices and its net has eight notches; an icosahedron has twelve and its net has twelve; a tetrahedron has four. Every one of them is a place where the outline turns by the deficit as well as by whatever the faces’ own corners do, and every one is where two cut edges have to be brought together when the model is assembled.

So a reader following a coastline round a folded cube crosses eight of these, and at each one the feature turns by 53 degrees under the gnomonic before the fold and by whatever the fold restores after it. That is not a subtle effect and it is the most visible thing about a polyhedral globe.

The net that loses the fewest neighbours scores nets by how far apart they put places that touch on the globe, and every one of the cube’s 384 nets scores between 11.30 and 17.97 on that measure. What none of them can change is how many vertices sit on the boundary or what each of them is worth, because both are fixed by the solid.

What the numbers say about a folded model #

The practical reading is short and it inverts the anchor’s usual advice about subdividing.

A vertex corner of 53 degrees on a cube or 98 on a tetrahedron is not a subtlety. It is a feature visibly doubling back at a fold, on the scale of the fold itself, and no face map removes it — the equal-area map reduces it to 13 and 33 and a conformal map reduces it further and none reaches zero, because the deficit does not.

Subdividing does reduce it, and by exactly the amount the deficit falls. An icosahedron’s deficit is 60 degrees against a cube’s 90 and a tetrahedron’s 180, and its gnomonic vertex corner is 24.87 against 53.13 and 98.21. More faces, less distortion, more cutting prices the same trade for distortion within a face, and the vertex behaves the same way for the same reason: 4π of total curvature spread over more vertices is less at each.

What subdividing cannot do is get rid of the vertices. Descartes fixes the total at 4π however many there are, so a solid with a thousand faces has a thousand-odd tiny cone points rather than twelve substantial ones, and a feature crossing near any of them still bends. The failure is redistributed rather than removed, which is the same statement no map is faithful makes about the sphere and the plane, arriving on a polyhedron as a finite sum rather than as an integral.

The one exact-looking number #

Among the five limits there is a coincidence worth reporting and not explaining, because this collection’s habit is to say when something is unproved.

The octahedron’s gnomonic vertex corner comes out 59.9992 degrees against a deficit of exactly 120. That is 0.5000 of the deficit to four figures, and the eight ten-thousandths it misses by is the same size as the chord artefact at the last row of the ladder.

None of the other four does anything like it. The cube reads 0.590 of its deficit, the tetrahedron 0.546, the dodecahedron 0.631 and the icosahedron 0.415. There is no pattern in those and no reason to expect one, since the corner depends on the face’s shape, the number of faces meeting, and the angular size of the edge, none of which combine into anything simple.

So the octahedron’s half is recorded as an observation with no account behind it. It may be exact — the octahedron is the one solid here whose vertex figure is a square, and squares have a way of producing exact halves — and it may be a coincidence of four digits. Nothing in this rung settles it, and a rung that reported it as a result would be reporting a hope.

Where the vertex sits in a net #

One consequence of the deficit deserves separating out because it decides what a reader of a folded model actually sees.

A net is the solid cut open along a spanning tree of its face graph, and the cut edges all meet at vertices. Every vertex of the solid is therefore either interior to the net — surrounded by faces that are all joined — or on its boundary, and the deficit shows up in the two cases completely differently.

At an interior vertex the faces surround the point and their angles sum to 2π − δ, which is less than a full turn, so the net cannot be flat there. It is not: an interior vertex of a net is where the paper would have to crumple, and a net has none. Every vertex of a net is on its boundary, and that is not a coincidence but a consequence — the cut has to go somewhere counts the ways of choosing the cut and every one of them opens every vertex.

At a boundary vertex the deficit is simply the gap between two edges of the outline, and it is what closes when the model is folded. A cube’s net has a 90-degree notch at each of its eight corners; a tetrahedron’s has a 180-degree one, which is why a tetrahedral net’s corners are straight lines rather than notches.

So the deficit a reader meets is not a defect in the drawing. It is the shape of the outline they cut round.

The two things a vertex is, kept apart #

Reading the rung back, two quantities have been in play throughout and running them together is the error it exists to prevent.

The deficit is a property of the solid. It is 180, 90, 120, 36 and 60 degrees on the five, it comes from the face angles by Descartes’s theorem, it is identical under every face map, and it is the sphere’s curvature at that point rather than any map’s failure. Nothing a cartographer does moves it.

The corner is a property of the map. It is what a feature visibly does at one fold, it depends on which face map was chosen, and it is between four and sixty-three per cent of the deficit depending on the map and the solid. It is a design decision.

The two are related by a conservation law and are not the same number, and the figures here keep them in different inks for that reason. A polyhedral globe cannot have less deficit; it can have the deficit arriving in a different place.

That distinction is the anchor’s own version of the one this collection makes everywhere. No map is faithful separates the theorem from the engineering: the impossibility is fixed and where the error goes is chosen. On a polyhedron the impossibility is a finite sum of angles at a finite set of points, which is the least abstract form it takes anywhere on this site — a reader can cut one out and see the notches.

Where the ladder has not been: paper, and the folding of it #

Thirteen rungs price a polyhedral map at a point, over a face, along a net, across a seam, along five seams and now at a vertex. Every one of those measurements is geometric: an angle, an area, a length, a count.

None of them is about paper. A polyhedral map is printed, cut and folded, and the folding is a physical operation with a tolerance — a crease is not a line, a cut is not infinitely thin, and two faces brought together at a fold meet to within whatever the person doing it manages. The question of how accurately a net can actually be assembled, and of whether the geometric quantities in these thirteen rungs are above or below that threshold, is the one the anchor’s own rung eleven named as next and nothing here has touched.