The seam falls faster than the paper

The map is finer than the paper it is folded from ended with two ways down and followed only one of them. A hand-folded net is accurate to about a degree of dihedral angle, and a degree of dihedral error moves the far edge of a face by that face’s own angular radius times a degree. So the assembly has a floor, and the floor falls in exactly two ways: fold better, or make the faces smaller.

Among the five regular solids the second way stops at twenty faces. The icosahedron has the smallest faces there are, and the verdict it gave was that the gnomonic face map — the one that draws every great circle straight — carries a seam corner of 9.19° against a floor of 0.65°. That is fourteen times the paper, on the best solid, and it made the gnomonic look beyond saving on any folded globe.

Subdividing does not stop at twenty. Split each of the icosahedron’s triangles into $\nu^2$ smaller ones and push the new corners out to the sphere, and the result is a geodesic solid of $20\nu^2$ faces, every one still flat and every one still foldable. The question the paper left was whether the gain from smaller faces survives the cost of folding more of them. It does, but for a reason the paper account did not predict. The two quantities being compared shrink at different rates, and the defect shrinks faster than the floor it is judged against.

Two things shrink, and not at the same rate #

The fold’s error is the easy one. The earlier account defined it as a face’s own angular radius times the dihedral error, and a face of a solid subdivided ν ways has about $1/\nu$ the radius of the icosahedron’s. The fitted exponent over the subdivided solids is −0.972. It is not exactly −1, because the new faces are not all the same size. Pushing points out to a sphere stretches the faces in the middle of each old triangle more than the ones near its corners, so the worst face at ν = 6 reaches 7.20° from its centre where a uniform one would reach 6.23°.

The seam corner is the interesting one. A gnomonic seam corner is what a great circle does when it crosses from one face to its neighbour. On each face it is drawn straight, since that is what a gnomonic map is for. Unfold the two faces into one plane and the two straight pieces meet at an angle, because the two faces were tangent planes at different points of the sphere.

That angle needs two things at once. It needs the two faces to be tilted against each other: two faces lying in one plane would be one face, and a line would cross between them without turning. And it needs the crossing to be away from the middle of the shared edge. The corner is not at the midpoint showed that a face’s own mirror symmetry forces the corner to vanish exactly there, at any tilt and any obliquity.

Both of those shrink as $1/\nu$. The angle between neighbouring faces is the angle between their normals, which is about the distance between their centres, and that is a face width. The distance from an edge’s midpoint to a point two fifths of the way to its end is also a fraction of a face width. So the corner, being roughly their product, should shrink as $1/\nu^2$.

It does. Measured on every edge of every solid and taking the worst edge each time, the corner’s fitted exponent is −1.971. At twelve divisions the corner is 0.87 per cent of its twenty-face value, while the fold’s error is 9.7 per cent of its own.

That difference in rate is the whole result. Any defect that falls faster than the floor it is judged against must eventually go under it. The only open question is where, and whether that place is a solid anybody would make.

Why the square, drawn rather than argued #

The argument above is dimensional, and dimensional arguments are easy to fool, so it is worth testing directly. Take, for each frequency, the steepest fold between two neighbouring faces — 180° minus their dihedral angle — and multiply it by the distance, in radians, from an edge’s midpoint to the point where the crossing is read. If the corner is the product of those two things, the points must lie on a straight line through the origin.

They lie on it, with a slope that comes out at 1.00 to two figures, and the one point that strays is the one expected to stray. The plain icosahedron’s faces meet at 138.2°, so they fold through 41.8°, and an angle that size is not small enough for the first-order picture to be exact. By ν = 3 the steepest fold is 14.1° and by ν = 12 it is 3.6°. Once both factors are small, the corner is their product and nothing else.

A slope of exactly one needs a sentence, because it looks like more than it is. A great circle crossing a hinge sees the two faces tilted against each other by the fold angle, and a point that far along the edge sees the same tilt rotate the direction of the line by the fold angle times the distance. For small angles on a unit sphere those two combine with no leftover constant. The figure’s content is that the approximation already holds by the second subdivision. Its slope is a check on the geometry, not a discovery.

Where the corner goes under the paper #

On a 100 mm globe the corner reaches the budget between ν = 5 and ν = 6. At 500 faces the corner is 0.451° against a budget of 0.332°. At 720 faces it is 0.315° against 0.321°, just underneath. By 1,280 faces it is 0.58 of the budget and by 2,880 a quarter of it.

That is the frequency on the globe above, and it is not exotic. Geodesic domes are commonly built at frequencies from two to six. A 720-face paper globe is a patient afternoon rather than an impossibility, though it is a long way from the twenty-piece icosahedral map that polyhedral globes are usually folded from.

The crossing is narrow and should be read as a region rather than a point. At ν = 6 the corner is 0.98 of the budget, and a folder at 1.1° rather than 1° would move it back above. What is robust is the direction. Every added division takes the corner down faster than it takes the floor down, so the margin widens with every step past the crossing. By ν = 8 it is 0.58 and by ν = 10 it is 0.38. A result that sits on its threshold at one frequency and is safely past it at the next is a result about the rate, and the rate is what the exponents measure.

The floor stops being the fold #

Something happens on the way to the crossing that the Platonic solids never showed. The paper budget has four terms and on the plain icosahedron the fold dominates them: 0.652° against the crease’s 0.286°. The crease, the cut and the print are fixed widths of paper, a quarter of a millimetre of bend radius and a tenth of kerf and eight hundredths of registration, and they do not care how many faces there are.

So by ν = 3 the fold’s error, at 0.242°, has fallen below the crease’s 0.286° and the crease has become the largest term. From there the budget barely moves: 0.382° at 180 faces, 0.321° at 720, 0.302° at 2,880. It is converging on the crease, which subdividing cannot touch.

This changes the answer to the question the earlier account asked. It asked whether the total error turns round as the faces shrink — whether the rising crease count eventually makes a finer solid worse. In displacement terms it does not. Nothing in the budget grows with the frequency. The crease’s width is the same on every fold, and a point near a crease is disturbed by one crease’s width whether the solid has twenty folds or two thousand.

What happens instead is that subdividing stops buying accuracy. Past ν = 3 on a 100 mm globe the floor is the crease, and more faces lower the floor by less each time, until they lower it by nothing. The gnomonic corner is the only thing still falling. That is why the gnomonic crossing falls at a definite frequency and not at infinity: there is a floor for it to go under, and the floor has stopped moving.

What it costs, in a currency the budget does not count #

If the total does not turn round, the cost of subdividing must be paid somewhere other than in displacement, and it is. It is paid in area.

A crease is not a line. In 120 gsm paper a fold has a bend radius of about a quarter of a millimetre, so about half a millimetre of the sheet is curved through the fold and lies on neither face. The map printed on that strip is on neither of the two planes the projection assumed. It is not displaced by a small amount the way a misfolded face is. It is removed from the picture, into a narrow gutter that the eye reads as the seam.

A net of $F$ faces keeps $F - 1$ of its edges as folds and cuts the rest; the net that loses the fewest neighbours is about choosing which, and past the regular solids a net can land on top of itself if the choice is careless. The number of folds therefore grows as $\nu^2$, each is $1/\nu$ as long, and the total length of crease grows as ν. Divided by a sphere’s surface, the share of the sheet folded out of sight rises linearly with the frequency. The fitted exponent is 0.986.

On a 100 mm globe that share is one per cent on twenty faces, 5.8 per cent at the gnomonic crossing, and 11.5 per cent at 2,880 faces. So the price of an invisible gnomonic seam is a map a twentieth of which sits in gutters. The gutters are visible, and they sit exactly where the seams used to be.

The trade has the shape more faces, less distortion, more cutting found for the ideal solid: distortion down as the reciprocal of the face count, cut length up as its square root. The paper version has the same shape in different units. The defect a reader could see goes down as $1/\nu^2$, and the paper a reader cannot see goes up as ν. What a cut buys priced a cut by the distortion it removes. The crease is now priced by the map it hides, and neither price was on the books before the paper had a thickness.

The count depends on the globe, and then stops depending on it #

The crossing at 720 faces belongs to a 100 mm globe. The crease is a fixed width of paper, so on a larger globe it is a smaller angle, the floor is lower, and the corner has further to fall.

On a 25 mm pocket globe the crossing comes at ν = 3, 180 faces. On 200 mm it is ν = 10 and 2,000 faces, on 800 mm ν = 14 and 3,920. At 1.6 m it is ν = 16, 5,120 faces. That is also where it stays for any larger globe, because there the crease is negligible and the floor is the fold alone. The fold’s error does not depend on size, and the corner crosses it for good between ν = 14 and ν = 16. At ν = 16 the corner is 0.0451° against a fold error of 0.0477°.

So there is a ceiling to the face count, and it is a clean statement: a gnomonic icosahedral globe of any size has an invisible seam by 5,120 faces. Below that the answer depends on the size. Above it the answer is always yes.

The crease share at each crossing runs the opposite way, and it is the more practical number. On the pocket globe the crossing costs 11.6 per cent of the sheet, because the globe is small and a half-millimetre gutter is a large angle on it. On the 1.6 m globe it costs one per cent. The globes that most need a small face count to be practical are the ones where subdividing hides the most map, which is the sort of arithmetic that decides what gets built.

What a globe maker would take from this #

The paper account ended with an ordering: choose the solid for the floor, choose the face map for the corner, stop refining the face map once its corner is under the floor. Subdividing adds a fourth decision and changes the second.

The face map is no longer fixed by the solid. On the five Platonic solids the gnomonic was above the paper on every one, by factors of 14 to 30 against the fold alone. That verdict was about face size, not about the gnomonic. Given enough faces, the one face map that keeps every great circle straight across a face also keeps its seam under the paper, which is a property no other face map in the account has.

The frequency is set by the globe’s size, from ν = 3 in the pocket to ν = 16 on a wall-sized globe, and never past sixteen.

The price is a share of the sheet, and it falls with the globe’s size. Reckon it in gutters rather than in degrees. On a small globe an invisible gnomonic seam costs a ninth of the map folded out of sight; on a large one, a hundredth.

And the conformal face map does not need any of this. The exact map says the seam is smooth puts its corner at four thousandths of a degree on the plain icosahedron, 165 times under the floor. A globe maker who wants invisible seams and can accept curved great circles has them at twenty faces and a one per cent crease share. The gnomonic buys straight great circles, and 720 faces is what they cost on a desk globe.

The quantities, and the controls on each #

At twenty faces the new measurement must reproduce the old. The subdivided solid at ν = 1 is the icosahedron itself, and its worst seam corner must equal the 9.1852° the Platonic seam measurement gave for that solid, the same construction on the same edges. It does to four decimals, and all thirty edges agree, as the icosahedron’s symmetry says they must.

Each rate must be a rate and not a pair of points. Both exponents are fitted over eight frequencies from two to twelve, and each must land within a tenth of its predicted value: −0.972 for the fold against −1, −1.971 for the corner against −2.

The corner must be above the budget at the start and below it somewhere, so that the crossing is found and not assumed. It is 12.8 times the budget on twenty faces and 0.26 of it at ν = 12.

The crease share must rise. A finding that subdividing is free would be the finding this whole account exists to rule out, and the share goes from 1.04 to 11.5 per cent.

The worst edge is what is reported, not a typical one. A subdivided solid’s edges are not all alike, since the faces near an original corner are smaller, and a reader sees the worst seam. The mean over all edges runs about 15 per cent below the worst: 0.266° against 0.315° at the crossing.

What the crossing does not cover #

The tolerances are the same stated values as before. A quarter of a millimetre of crease radius, a tenth of kerf, eight hundredths of registration and a degree of fold. The crossing frequency depends on all of them, the fold most. The large-globe ceiling is where the corner meets the fold alone, and the fold’s error is proportional to the folder’s: a folder at half a degree lowers the floor and pushes the ceiling out to about ν = 30, and one at two degrees brings it in to about ν = 8. A better folder needs more faces, because the corner has further to fall.

A degree of fold is assumed to stay a degree as the faces shrink. Folding 5,120 small triangles is not obviously done to the same angular tolerance as folding twenty large ones. A smaller face is a shorter lever, and a hand that places an edge to a tenth of a millimetre places a short one to a worse angle. If the fold’s error grows as the face shrinks, the floor falls slower than $1/\nu$ and the crossing moves out, perhaps a long way. This is the softest input and the one most worth measuring on real nets.

The crease share is area, and area is not the same kind of thing as a displacement. The budget and the share are reported side by side because they cannot be added. Whether a reader would rather have a seam corner of a third of a degree or five per cent of the map in gutters is a question of taste and purpose. The account prices both and does not choose.

Only the gnomonic was carried to the subdivided solid. The conformal face map needs a series fitted to a regular face, and a subdivided solid’s faces are not regular; its corner there is unmeasured, though on the plain icosahedron it is already under the floor. The equal-area face map ran into a finding of its own that stops it being carried over as it stands. On the plain icosahedron its seam corner is 1.77° on twenty-eight of the thirty edges and 2.31° on the other two, although the solid’s symmetry says all thirty should agree. The Platonic verdict measured one edge and reported 1.77°. Until that disagreement is explained, a worst-edge figure for the equal-area map on a subdivided solid would be a number about the construction rather than about the map.

And the crossing is read at two fifths of each edge, the place every earlier seam measurement used. The corner that is the curvature showed the corner rising toward a vertex, to a finite limit. On a subdivided solid the vertices are many and small, and the corner nearest one is larger than the figure here. It still falls with the frequency, since the limit is set by the deficit and the deficit falls as $1/\nu^2$, but the crossing it gives would come later.

Still open: whether a smaller face is folded as accurately #

Everything above leans on one number that was not measured: the dihedral error of a fold, taken as a degree whatever the face’s size. It is the input the crossing is most sensitive to, and it is the one subdividing is most likely to change.

The mechanism is ordinary. A fold is made by aligning an edge, and a hand aligns an edge to some distance, not to some angle. On a face 40 mm across a tenth of a millimetre of misplacement is a seventh of a degree; on a face 4 mm across it is a degree and a half. If folding is limited by distance rather than angle, the fold’s error in angle rises as $1/\nu$. Multiplied by a face radius falling as $1/\nu$, the floor stops falling at all, and the gnomonic crossing is decided by the crease alone.

Whether that is how paper actually behaves — whether a folder’s error is closer to a fixed angle or a fixed distance, and at what face size one gives way to the other — decides whether the ceiling of 5,120 faces is real or only a property of the stated tolerance. That is a measurement of folded nets rather than of geometry, and a cell system trades area for shape is the nearest place in this subject where the same subdivided solid is used without any paper at all.