The map is finer than the paper it is folded from

The corner that is the curvature is the thirteenth measurement of a polyhedral map and the thirteenth to be geometric. A point, a face, a net, a seam, five seams, a vertex: an angle, an area, a length, a count. Every one of them is a property of an ideal object.

A polyhedral map is not an ideal object. It is printed, cut and folded, and each of those is a physical operation with a tolerance. A crease is not a line — it has a radius, and the paper inside it lies on neither face. A cut is not infinitely thin. And two faces brought together at a fold meet to within whatever the person doing it manages.

So the question the eleventh measurement named and the thirteenth repeated is whether any of the thirteen numbers is larger than the paper. For one of the three face maps, on four of the five solids, it is not.

What a folded net is accurate to, beside what the map it carries is wrong by. For a cube globe of 100 mm radius: each tolerance of cutting and folding a paper net, as a displacement on the assembled sphere, and the seam corner of each of the three face maps beside it. The fold's own error is the largest at 0.955° — 106 km at Earth scale — because a degree of dihedral error moves the far edge of a face by a degree of the face's own radius, which for a cube is 54.7°. The conformal face map's corner is 0.769°, below the whole budget of 1.000°; the gnomonic's is 20.2°, twenty times above it. — Fig. 1 For a cube globe of 100 mm radius: each tolerance of cutting and folding a paper net, as a displacement on the assembled sphere, and the seam corner of each of the three face maps beside it. The fold’s own error is the largest at 0.955° — 106 km at Earth scale — because a degree of dihedral error moves the far edge of a face by a degree of the face’s own radius, which for a cube is 54.7°. The conformal face map’s corner is 0.769°, below the whole budget of 1.000°; the gnomonic’s is 20.2°, twenty times above it.

Putting two kinds of number in one currency

Before any of it can be compared, the two quantities have to be made the same kind of thing, and they are not obviously so.

A seam corner is an angle in the image of a curve: a great circle crossing a seam is drawn as two straight pieces meeting at some angle, and the corner is that angle. It is a property of the map and it is scale-free — a cube’s gnomonic seam turns 20.2° on a globe of any size.

A folding error is a displacement of a point: the face is rotated slightly about its crease, so everything on it is in slightly the wrong place on the finished sphere. It is measured in degrees of arc on the assembled globe, which converts to kilometres at Earth scale.

Both are read off the same object by the same reader, and the conversion that makes them comparable is to ask how far each moves something the reader is looking at. A degree of arc is a degree of arc whether it arrived as a bend in a drawn line or as a face set slightly askew. That is a judgement rather than a theorem, and it is the one this whole measurement rests on.

Four tolerances, and one of them is not like the others

The four are taken as stated values for a hand-assembled net on ordinary paper: a crease radius of a quarter of a millimetre in 120 gsm stock, a cut or laser kerf of a tenth, print registration to the cut line of eight hundredths, and a dihedral angle brought together by hand to one degree.

Three of them are widths of paper. A quarter of a millimetre is a quarter of a millimetre whatever globe it is folded into, so in angular terms it shrinks as the globe grows: on a 25 mm pocket globe a crease occupies 1.15° of arc and on a 1.6 m globe 0.018°.

The fourth is an angle, and it does not shrink at all.

The fold's error is nothing at the crease and everything at the far edge. How far a misfolded face displaces a point, against how far that point is from the crease, for three qualities of folding. A dihedral error of one degree displaces a point on the crease by nothing and one at the far edge of a cube's face — 54.7° away — by 0.955°, which is 106 km at Earth scale. That is the term the crease's width, the cut's and the print's have to be compared against, and at 100 mm they are 0.30, 0.06 and 0.05 of it. A very careful folder at half a degree halves the whole budget; a careless one at two degrees doubles it. — Fig. 2 How far a misfolded face displaces a point, against how far that point is from the crease, for three qualities of folding. A dihedral error of one degree displaces a point on the crease by nothing and one at the far edge of a cube’s face — 54.7° away — by 0.955°, which is 106 km at Earth scale. At 100 mm the crease, the cut and the print are 0.30, 0.06 and 0.05 of it.

A face rotated about its crease by a degree moves every point on it, by an amount proportional to how far that point is from the crease. At the crease, nothing; at the far edge of a cube’s face, 54.7° of arc times one degree in radians, which is 0.955° of arc. Neither term in that product knows how large the globe is.

Three of the four shrink with the globe and the one that matters does not. Each tolerance against the globe's radius, from a 25 mm pocket globe to a 1.6 m one. The crease, the cut and the print are fixed widths of paper, so in angular terms they fall as the reciprocal of the radius — the crease is 1.146° on a 25 mm globe and 1.79e-2° on a 1.6 m one. The fold's error does not move at all: it is 0.955° at every size, because a degree of dihedral error is a degree whatever the globe is made of. So the total falls to a floor and stops, and the floor is the fold. — Fig. 3 Each tolerance against the globe’s radius, from a 25 mm pocket globe to a 1.6 m one. The crease is 1.146° on a 25 mm globe and 0.0179° on a 1.6 m one. The fold’s error does not move at all: it is 0.955° at every size. So the total falls to a floor and stops, and the floor is the fold.

That is the structural fact the rest depends on. Making the globe bigger buys accuracy from three of the four tolerances and none from the fourth, so the assembly budget falls to a floor and stays there. A polyhedral globe cannot be made more accurate than its folding by making it larger.

A tolerance is the right shape of answer

The word tolerance is doing real work here and it is worth saying why this is the form the question had to take.

Nothing in the thirteen geometric measurements is wrong, and nothing in them is a bound on what a reader experiences. They are properties of a map, and a map is an intermediate object between a sphere and a thing somebody holds. What decides whether a difference between two maps reaches the reader is whether it survives the operations between them — printing, cutting, folding — and those have tolerances that the map does not.

That is the ordinary structure of an error budget: a chain of operations, each with its own contribution, and a total in which the largest term decides. A published coordinate is a result makes the same point at the other end of this subject, where a coordinate’s quoted accuracy is a property of the whole chain that produced it and not of the last calculation in it.

What is unusual here is that the chain’s largest term is not in the map at all. Thirteen measurements refined the first link and the last one is a degree of dihedral angle.

What can be done about it, and it is not folding better

More faces, smaller faces, a smaller floor. The fold's own error for each of the five solids, on a 100 mm globe, with the face's own angular radius printed beside it. The error is that radius times the dihedral error, so it falls with the face: 1.231° on the tetrahedron, whose faces reach 70.5° from their centres, down to 0.652° on the icosahedron at 37.4°. That is the only way a net's assembly can be made more accurate without a better folder: cut it into more pieces. — Fig. 4 The fold’s own error for each of the five solids, on a 100 mm globe, with the face’s own angular radius printed beside it. It is that radius times the dihedral error, so it falls with the face: 1.231° on the tetrahedron, whose faces reach 70.5° from their centres, down to 0.652° on the icosahedron at 37.4°.

The floor is the face’s own radius times the folding error, so there are exactly two ways down: fold better, or use smaller faces.

Folding better is a person. A very careful assembler at half a degree halves the whole budget and a careless one at two degrees doubles it, and nothing about the map changes either way.

Smaller faces is a choice about the solid, and it is the one the map maker has. The tetrahedron’s faces reach 70.5° from their centres and the icosahedron’s 37.4°, so the icosahedron’s floor is 0.652° against the tetrahedron’s 1.231° — a factor of 1.9 bought by cutting the sphere into twenty pieces instead of four.

Fig. 5 The fold’s error against the face’s own angular radius, for the five solids. It is a straight line through the origin because the error is the radius times the dihedral error, in radians — so this figure has no content beyond the definition, and that is why it is here: it is the check that the five numbers above are five readings of one relation. The dodecahedron and icosahedron share a point, as do the cube and the octahedron, because duals have the same face radius.

That line is worth drawing precisely because it contains nothing. If the five solids did not sit on it, the five numbers would not be five readings of one relation and something in the conversion would be wrong.

What the crease actually does

The crease deserves a paragraph because it is the only term that removes map rather than displacing it.

A fold in 120 gsm paper has a radius of about a quarter of a millimetre, so roughly half a millimetre of the sheet is bent through the dihedral angle rather than lying flat on either face. The map printed on that strip is on neither of the two planes the projection assumed, and it is compressed: the strip is drawn as though flat and occupies less than its flat length once folded.

On a 100 mm cube globe that is 0.287° of arc — 32 km at Earth scale — along every one of the twelve edges. It is not the largest term, but it is the one a reader can actually see, because it is a visible band at every seam rather than a displacement of a whole face that nothing is there to compare against.

What the net heuristic cannot find counts cuts and chooses a net by how few neighbours it separates; the crease count is the complement of the cut count, since every edge is one or the other. So a net chosen to minimise cuts is a net that maximises creases, and the two tolerances pull against each other in a way neither measurement priced.

The verdict

Below the line the map's own corner cannot be seen on a folded net. Each face map's seam corner divided by the fold's own error, for the five solids in order of face count. Above one the map's defect is larger than the assembly's and a reader could in principle see it; below one it cannot be seen at all. The gnomonic sits 21 to 30 times above the line on every solid and the equal-area 2.7 to 15. The conformal crosses below it at the cube and keeps falling: on the icosahedron it is 6.1e-3 of the floor, which is a corner 165 times finer than the paper it is printed on can be folded to. The tetrahedron is the exception, at 3.3. — Fig. 6 Each face map’s seam corner divided by the fold’s own error, for the five solids in order of face count. Above one the map’s defect is larger than the assembly’s; below one it cannot be seen at all. The gnomonic sits 21 to 30 times above the line on every solid and the equal-area 2.7 to 15. The conformal crosses below it at the cube and keeps falling: on the icosahedron it is 6.1 × 10⁻³ of the floor, a corner 165 times finer than the paper can be folded to. The tetrahedron is the exception, at 3.3.

Divided by the folding floor, the three face maps separate completely and the answer to eleven measurements’ worth of comparison is short.

A gnomonic face map’s seam corner is 21 to 30 times the folding floor on every solid. It is the largest defect on the finished object by an order of magnitude, and a reader holding the globe is looking at it rather than at any fault of the assembly.

An equal-area face map’s is 2.7 to 15 times above. Also visible, on every solid.

A conformal face map’s crosses below the floor at the cube — 0.769° against 0.955° — and keeps falling: 0.376 on the octahedron, 0.161 on the dodecahedron, and 0.0040 on the icosahedron, where the corner is a hundred and sixty-five times finer than the sheet can be folded to. The exact map says the seam is smooth measured that corner and found it bounded rather than zero; on a folded net, bounded at four thousandths of a degree and zero are the same thing.

The tetrahedron is the exception and its reason is the same one that governs everything else here: its faces are the largest of the five, so its seam carries the most curvature and its folding floor is the highest. Both go up together, and the corner goes up faster.

On a globe of any size the corner is smaller than the paper. For an icosahedron globe at three sizes, the whole assembly budget and the conformal face map's seam corner beside it, with the corner also given as a length on the printed sheet. On a 25 mm globe the corner is 1.7e-3 mm of paper; on a 400 mm one, 0.028 mm. It is below the budget at every size, by a factor of 341 at 25 mm and 166 at 400 mm — the ratio improving with size only because three of the four tolerances shrink while the corner does not. — Fig. 7 For an icosahedron globe at three sizes, the whole assembly budget and the conformal face map’s seam corner beside it, with the corner also given as a length on the printed sheet. On a 25 mm globe the corner is 1.7 × 10⁻³ mm of paper; on a 400 mm one, 0.028 mm. It is below the budget at every size, by a factor of 341 at 25 mm and 166 at 400 mm.

And the verdict does not depend on the globe’s size, which is the point of the floor. On an icosahedron the conformal corner is below the whole assembly budget by a factor of 341 on a pocket globe and 166 on a 400 mm one — the ratio worsening with size only because three of the four tolerances shrink and the corner does not.

What this says about eleven measurements

It does not say they were wasted, and it is worth being exact about why.

The measurement of every solid’s seam at every span and the ones around it established which face map has a real corner and which has a bounded one, and the verdict above uses those numbers as its input. A comparison of face maps that could not tell them apart would be no use here either.

What it does say is that the comparison has a floor under it, and the floor decides which differences can matter to a reader. Between a gnomonic net and an equal-area one there is a real, visible difference on the assembled object. Between an equal-area net and a conformal one there is also one. Between a conformal icosahedral net and a hypothetical perfect one there is nothing whatever — the difference is four thousandths of a degree against a floor of six-tenths, and no folder alive closes that gap.

So the honest ordering for somebody actually making a globe is: choose the solid for the floor, choose the face map for the corner, and stop refining the face map once its corner is under the floor. On an icosahedron that happens some way before the conformal map.

What a globe maker would take from this

Three decisions, in the order the floor puts them.

The solid decides the floor. Twenty faces against four buys a factor of 1.9, and it costs thirty creases against six — which is the trade the next section leaves open and is the first thing to settle for a real object.

The face map decides whether the map is under the floor. Conformal on anything but a tetrahedron is; equal-area and gnomonic are not, by factors of 2.7 to 30. A globe maker who wants the seam invisible has one choice of face map and it is not the one that draws great circles straight.

And past that, refining the face map buys nothing. On an icosahedron the conformal corner is a hundred and sixty-five times under the floor, so an exactly-zero-corner map — if one existed, which the exact map says the seam is smooth establishes it nearly does — would be indistinguishable from it on the finished object.

That last is the practical content of the whole measurement, and it is a stopping rule rather than a result: there is a point past which a better map is not a better globe, and here it is.

What each number was checked against

Doubling the globe must halve the three widths and leave the fold alone. The crease, the cut and the print each halve to nine decimal places when the radius goes from 100 mm to 200; the fold’s error is identical to 10⁻¹². A construction that scaled the fold would be converting twice, and every ratio here would be wrong by a factor of the radius.

The fold must be the largest term on an ordinary desk globe, or the budget is a statement about printing rather than about assembly. At 100 mm it is 0.955° against the crease’s 0.287°.

More faces must mean a smaller floor. The icosahedron’s is 0.652° against the tetrahedron’s 1.231°, and all five sit on one line through the origin.

The gnomonic and equal-area corners must be above the floor on every solid, and they are — the smallest of the ten is the icosahedron’s equal-area at 2.7 times.

The conformal corner must be below it on every solid but the tetrahedron, and the tetrahedron must be checked as an exception rather than left out — so that a change to it is caught rather than quietly widening the claim.

What a stated tolerance is not

Four numbers, chosen and not measured. A quarter of a millimetre of crease radius, a tenth of kerf, eight hundredths of registration and one degree of fold are plausible figures for a hand-assembled paper net and they are not measurements of anybody’s net. Everything scales with them: halve the folding error and the floor halves, and the conformal map’s corner on a cube crosses back above it.

One degree of fold is the whole story and it is the softest input. Three of the four terms could be wrong by a factor of two without changing any verdict; the fold could not. A serious version of this would measure the dihedral error of assembled nets rather than take it on trust.

The errors are added in quadrature. They are treated as independent, which the crease and the fold are not — a wider crease makes the dihedral angle harder to control. The correlation would raise the total and would not move the floor, which is the fold alone.

The seam corner is not the only geometric quantity. The thirteen measurements price several — the overlap of a net, the spread of its pieces, the deficit at a vertex — and only the seam corner is compared here, because it is the one eleven of them converged on and the one that is a defect of the map rather than of the cut. A cell system trades area for shape prices a different family of quantities on a related object, and none of them is in this budget.

And the comparison is corner against displacement. A seam corner is an angle in the image of a curve and a folding error is a displacement of a point; setting them side by side is a judgement that both are read by the same reader looking at the same object. It is not a theorem.

Still open: whether a subdivided solid keeps the advantage

The two ways down are fold better or use smaller faces, and the second has an obvious continuation this account stops short of. A geodesic solid — an icosahedron with each face divided into four, or sixteen, or sixty-four — has faces a fraction of the size and a floor to match, and the same subdivision is what a modern polyhedral globe is actually made of.

But the count of creases rises with it, and each crease is a place where a quarter millimetre of paper is not on either face. The net that loses the fewest neighbours chooses an unfolding by counting what its cuts separate, and what a cut buys prices a cut in the distortion it removes; neither counts what a crease costs, because until the paper was given a thickness a crease cost nothing. The floor falls as the face shrinks while the crease term is paid more often, and whether the total falls or turns round is not determined by anything measured here.

Where that minimum is, whether it lands on a subdivision anybody uses, and whether a conformal face map’s corner stays under the floor as the faces shrink — it should, since the corner falls with the face too, but the two rates are different — are questions five Platonic solids cannot ask.

The map is finer than the paper it is folded from

Putting two kinds of number in one currency