More faces, less distortion, more cutting

The polyhedral ladder has spent three rungs on the five Platonic solids, and the five have a property that is easy to mistake for a design constraint: there are only five of them. That is a theorem about regular polyhedra and it says nothing whatever about mapping.

Nothing in the construction needs regularity. What a polyhedral map needs is a convex solid whose faces are flat, and any convex solid’s faces are flat. Subdivide each face of an icosahedron into ν² triangles and push the new vertices out to the sphere: the result has 20ν² faces, every one still flat, every one still unrolling without stretching, and every one carrying a face map whose distortion is computable exactly as before.

So the family does not stop at twenty. It runs to any number, and it makes the same trade at every point along it.

The two exponents, and where they come from #

Both are predictable and both are worth predicting before measuring, because a prediction that matches is a check on the machinery and a prediction that does not is a finding.

A face’s angular size goes as N^(−1/2). The sphere’s area is fixed and there are N faces, so each face has area 4π/N and its radius goes as the square root of that.

The gnomonic map’s angular deformation goes as the square of the face radius. That follows from the closed form: for the gnomonic map at arc ρ from the centre, the principal scales are sec²ρ and sec ρ, so ω is 2 arcsin((sec²ρ − sec ρ)/(sec²ρ + sec ρ)), which is ρ²/2 to leading order.

Together: distortion goes as N^(-1). Measured: −0.947.

The number of cut edges is one fewer than the number of corners. A net is the solid opened along enough edges to lie flat, so the kept edges form a spanning tree of the face graph and the cut edges a spanning tree of the skeleton — the argument the cut has to go somewhere makes, with the counts equal both ways. For a triangulated solid that is about N/2 edges, and each is about N^(−1/2) long.

Together: cut length goes as N^(1/2). Measured: +0.509.

So halving the distortion costs √2 times the cutting, for ever. No amount of subdividing escapes it and no amount of subdividing makes it worse than that.

What subdividing costs that the table does not show #

The faces stop being identical, and that has three consequences worth separating.

Every quantity becomes a distribution. The plain icosahedron has one face, twenty times over; a three-frequency geodesic solid has faces whose circumradius runs from 11.4° to 13.9°. The ladder reports the worst, because a map is held to its worst face, and the mean would flatter it by about a fifth.

The corners stop being alike. The original twelve vertices of the icosahedron still have five faces meeting at them and still carry an angle deficit of π/5; every vertex created by subdivision has six faces meeting at it and carries a deficit near zero — but not zero, and that is where the curvature has gone.

Descartes’ theorem is unchanged. Whatever the subdivision, the deficits total 4π. The curvature that used to sit in twelve big spikes is now spread over hundreds of small ones, which is the discrete version of what happens to a sphere’s curvature when it is not concentrated at all.

A finer solid, drawn #

The picture is worth putting beside the numbers because it makes the second column visible. At this face count the map’s distortion is negligible in any practical sense; what is not negligible is that the object is a mesh rather than a sheet, and every edge in it is a place where the map could be torn open. Which of those edges actually are torn is the net’s business, and the count of ways to choose is the determinant the second rung computes.

Where a real map sits on the curve #

Every published polyhedral map is a point on the trade-off, and the interesting thing is how few of them are far along it.

Fuller’s Dymaxion is the icosahedron: 20 faces, 8.0° of worst angular deformation measured over the interior of a face, 11 cuts. Cahill’s butterfly is the octahedron: 8 faces, 20.7°, 5 cuts. Snyder’s equal-area polyhedral projection is defined for any of the solids and is usually drawn on the icosahedron.

Nobody prints a 320-face map, and the reason is in the second column rather than the first. At 320 faces the worst distortion inside a face is 1.03°, which is better than any world map in the library by an order of magnitude — and the map has 161 cuts and 48 sphere radii of tear. It is not a map of the world; it is a pile of triangles.

The net, at a face count anybody would print #

Twenty faces and eleven cuts is a map. Three hundred and twenty faces and 161 cuts is a mesh. Nothing in the geometry marks the boundary between those two descriptions — the exponents run smoothly through it — and where a reader puts it is the only thing that decides how far along the curve a real map may go.

What was computed, and how #

The solids are built rather than tabulated. Each face of the icosahedron is subdivided in the plane and every new vertex normalised to the sphere — the construction a geodesic dome uses. Vertices are deduplicated by rounded coordinates so that shared edges are shared, which is what makes the edge count come out at 30ν² rather than 60ν².

Euler’s formula is checked on every solid, and it is a real check rather than a formality: a subdivision that failed to identify two copies of the same vertex would produce a mesh that looks right, draws right, and has the wrong Euler characteristic. V − E + F = 2 on every row.

The distortion is taken from the closed form, and the closed form is checked against the site’s own measurement. Sampling every face of a 1,280-face solid with the full distortion() routine would be a hundred thousand Jacobians; instead ω is computed from each face’s own circumradius by the gnomonic expression, and that expression is required to match distortion() applied to a real face projection to 3 × 10⁻¹⁰ degrees. One check licenses the shortcut everywhere.

The cut length is the number of cut edges times the mean edge length, in units of the sphere’s radius. The number of cut edges is E − F + 1, which is exact and is V − 1 by Euler.

Why the exponents are the interesting part #

A table of six rows can be read off and forgotten. Two exponents cannot, because they say what happens outside the table.

The measured pair says that to halve the worst distortion, a design must quadruple the face count and accept twice the cutting. Extrapolating: the 8.0° of a Dymaxion map becomes 0.5° at about 320 faces and 0.1° at about 1,600, at which point the cut length is 96 and 215 sphere radii respectively — six and thirteen times the world’s circumference in tears.

Nothing else in this site’s library behaves that way. Every other family trades one distortion against another: the trade-off is two lines shows conformality against equal area, and every compromise projection is a choice about how to divide a fixed failure. A polyhedral map buys distortion down with a currency that is not distortion at all, and the exponents say what the exchange rate is.

Why the worst face and not the mean #

Two numbers could describe a solid’s distortion and the ladder reports one of them, which is a choice worth defending.

The mean over the faces is smaller and, for a solid whose faces are nearly alike, not much smaller: on the plain icosahedron the mean face circumradius and the worst are the same number, because the faces are identical. On a six-frequency geodesic solid the mean is 6.0° against a worst of 7.2°, so the mean would flatter the distortion by about a third once it is squared.

The worst is the right number for the same reason it is the right number for a national grid’s scale factor: a map is used everywhere it covers, and a reader standing at the corner of a face is not comforted by the average. Where the worst point is makes the general form of that argument, and the polyhedral case is its cleanest instance, because the worst point is always the same place — a corner — and it is always the place a cut passes through.

The dual, which is where the cells come from #

Subdividing an icosahedron and taking the dual of the result gives a Goldberg polyhedron: hexagons everywhere except for exactly twelve pentagons, which is the tiling hexagons cannot tile the sphere is about.

That is the same construction seen from the other side, and it is why the polyhedral ladder and the cells ladder keep meeting. A geodesic solid is a triangulation of the sphere, useful when faces are things to draw on; its dual is a tiling, useful when faces are things to count in. The face counts are related by Euler: a solid with 20ν² triangles has a dual with 10ν² + 2 cells, twelve of which are pentagons however large ν gets.

Where the model stops #

The face map is the gnomonic throughout. The area-preserving and conformal face maps of the previous two rungs apply unchanged to a geodesic solid’s faces, and their distortion would follow different exponents — the equal-area map’s angular deformation also goes as the square of the face radius, so the exponent would be the same and the constant smaller. That is asserted here rather than measured, which is the honest description.

The subdivision scheme is one of several. The vertices here are placed by subdividing in the plane and normalising, which is the geodesic-dome construction; subdividing along great circles gives a different vertex placement with the same combinatorics and a slightly different face-size distribution. The choice changes the numbers in the third figure by a few per cent and neither exponent.

Nets are not enumerated past the regular solids. Counting the unfoldings of a 320-face solid is a determinant of a 320 × 320 matrix — Kirchhoff’s theorem, which the ladder’s second rung uses — and the count is astronomically large but perfectly computable. What is not computable at that size is the overlap test that rung ran exhaustively, and any claim that a subdivided solid’s nets do not overlap would be an assumption.

The cutting is measured as length, not as damage. A map’s tears are objectionable in proportion to what they cut through, which is a question about the world rather than about the solid — and it is the freedom Fuller spent his three degrees of orientation on. A cut length of 96 sphere radii falling entirely in ocean is a different object from one of 12 crossing four continents, and nothing in this ladder can tell them apart, because telling them apart needs a coastline and the site has declined one on stated grounds.

The exponents are fitted over a decade and a half of face count. From 20 to 1,280 is a factor of 64, which is enough to separate a reciprocal from an inverse square root and not enough to detect a slowly varying correction. Both fitted values sit about five per cent inside their predicted ones, in the same direction, which is consistent with the leading-order expansion of the gnomonic’s deformation being an underestimate at the coarse end where the faces are large.

Who found it, and when #

Geodesic subdivision is Fuller’s, patented in 1954 for domes rather than for maps, and the mathematics is older: the Goldberg polyhedra were classified by Michael Goldberg in 1937, and the fact that exactly twelve pentagons survive any subdivision is Euler’s, from 1758.

The mapping use came later and from a different direction. Weather and climate models needed a grid on the sphere without the polar convergence a lon/lat grid has, and the icosahedral geodesic grid is what they reached for — Sadourny, Arakawa and Mintz in 1968, and Williamson the same year. Those grids are the dual construction, used for counting rather than for drawing, which is the cells ladder’s subject and where this one hands over.

The exchange rate is not what decides the face count #

The two exponents describe a trade, and a trade invites the reading that a designer sits somewhere along it and picks a point according to taste. That reading is wrong about how polyhedral maps are actually chosen, and the reason is worth stating because it is a general fact about constraints that are thresholds rather than costs.

Distortion is a cost and interruption is a threshold. A reader who is handed a map with two per cent scale error and one with one per cent will take the second and be mildly better off. A reader handed a map whose cut runs through the middle of a continent they care about will reject it outright, and no amount of reduced distortion buys that back. The first is a quantity to be traded and the second is a condition to be satisfied.

So the design does not run along the curve. It runs like this: enumerate the orientations in which the solid’s edges avoid the land, take the finest solid for which such an orientation exists, and accept whatever distortion that gives. The exchange rate never enters the decision, because one side of it is not being traded.

And the supply of such orientations collapses quickly. A net’s cuts are its edges, and a finer solid has more of them and each is shorter — so the edge set becomes denser on the sphere at exactly the rate that makes threading it through the oceans harder. The icosahedron’s thirty edges can be rotated so that all but a few fall in water, which is what Fuller’s Dymaxion arrangement is; a solid one subdivision finer has 120, and the ocean has not grown.

That is why the printed examples cluster at the coarse end and it is not conservatism. The Dymaxion, Cahill’s butterfly, Waterman’s polyhedron, the AuthaGraph — every one of them sits within a factor of a few of twenty faces, on a curve that continues smoothly for four more decades. The curve is real, the maps stop, and the exponents are not the reason.

Where the fine end does get used is exactly where the cuts stop mattering. A geodesic grid used by an atmospheric model is never drawn: its faces are cells in a computation, its edges are shared between neighbours rather than torn apart, and nothing is being cut at all. The interruption exponent is a cost of flattening, and a model on the sphere does not flatten. So the same family that stops at twenty faces for a printed map runs to millions for a grid — one exponent applies to both and the other applies to only one.

That is the honest reading of the pair. The distortion law is a fact about the face map and holds wherever the solid is used; the interruption law is a fact about the page, and the page is a constraint some users have and others do not.

Where the ladder goes next #

The polyhedral ladder is now complete as an argument: the solid decides the cutting, the face map decides the distortion, the two are independent, and the family extends to any face count at a stated exchange rate.

What it has never done is treat a face as a thing to index rather than a thing to draw on — and that is a different question, with a different unit. When the faces are addresses, what matters is not the distortion inside one but how many of them a question touches.