What a face can preserve

The first two rungs of this ladder chose a solid and decided where to cut it. Both left one thing assumed: the map from the sphere onto a face, which has been the gnomonic throughout because it is the obvious construction and because great circles come out straight on it.

Nothing requires that. A face is a region of the sphere and a flat polygon, and any map between them that takes the boundary to the boundary will do — the pieces fit together on the strength of the boundary and nothing else.

What the gnomonic does, and what it costs #

The gnomonic map onto a face is the ray from the centre of the sphere: a place is drawn where the line from the Earth’s centre through it meets the face’s plane.

Its exact property is straightness. A great circle lies in a plane through the centre, that plane cuts the face’s plane in a straight line, and so every geodesic within a face is drawn as a ruled line. Measured with the flexion machinery that is a flexion of zero to the noise floor of the arithmetic, and it is the only exact zero the whole library contains.

Its price is that the corners of the face are further from the centre of the sphere than the middle is. Across one icosahedron face the areal factor runs from 1.00 at the centre to 1.52 at a corner, and the angular deformation reaches 8.0°.

An area-preserving face, built from its own differential equation #

John Snyder published an equal-area polyhedral projection in 1992, and its principle is short enough to rebuild rather than transcribe.

Work in polar coordinates about the face’s centre. Let ρ_b(θ) be the spherical boundary radius at azimuth θ and r_b(θ′) the plane boundary radius at azimuth θ′. The area of a thin spherical wedge out to the boundary is proportional to 1 − cos ρ_b(θ); the plane’s is ½ r_b(θ′)². Requiring the cumulative areas to match gives a relation between the two azimuths, and requiring the areas to match again within a wedge gives the radius:

$$r = r_b(\theta') \sqrt{\frac{1 - \cos\rho}{1 - \cos\rho_b(\theta)}}$$

which is exactly area-preserving by construction and takes the boundary to the boundary, so the faces still meet.

The azimuth relation is where the arithmetic lives. It is an ordinary differential equation — dθ′/dθ = k·g(θ)/G(θ′) — and treating it as one rather than as a pair of tables is what makes the construction exact. The first implementation here tabulated both cumulative areas and inverted by linear interpolation, which is accurate in θ′ and not accurate in dθ′/dθ, and the areal factor depends on the derivative: the measured area spread was 0.6 per cent at 720 samples and fell only as the first power of the step. Reaching this site’s equal-area tolerance of one part in a million that way would have needed a million samples.

Integrating with Runge–Kutta and interpolating with a cubic whose slopes are the exact right-hand side gives a continuous derivative correct to third order, and the same 720 samples then hold the areal factor to a few parts in a billion.

The condition that had to be solved for #

One detail is worth extracting because it is the sort of thing that quietly ruins a construction.

The constant k in that equation is not, quite, the ratio of the two total areas. It has to be the value that makes θ′ come back to exactly 2π after a full turn round the face — otherwise the map’s azimuths do not join up and the face is drawn with a seam through it — and the ratio of totals is that value only in the limit of an exact quadrature. At 180 samples the closure error was 6.6 × 10⁻⁶, which is small and is a map that does not close.

So k is solved for, by a secant iteration on the closure, and the assertion the library carries is that the loop closes to better than 10⁻⁹. That is a condition on the construction rather than a tolerance on an answer, and getting it wrong produces a picture that looks entirely convincing.

What each face map preserves, measured #

Two things in that table are worth reading carefully.

The last column is the claim being made and it is exact. Under one part in a billion on four of the five solids, and one part in a million on the cube, whose square face has boundary corners the interpolation handles slightly less smoothly. Against this site’s equal-area tolerance of 10⁻⁶ that is an equal-area projection by exactly the standard the projection that shows true size holds the cylindrical ones to — and the four-orders-of-magnitude margin on the triangular solids is what says the construction is exact and the residual is arithmetic.

The two ω columns do not move together. On the triangular-faced solids the area-preserving map is worse in shape than the gnomonic; on the cube and dodecahedron it is better. That is not a paradox — the two maps are different functions with different failures, and there is no reason for one to dominate — but it does dispose of the tempting summary that equal-area costs shape. On the tetrahedron, the octahedron and the icosahedron it costs shape; on the cube and the dodecahedron it buys shape as well as area. Only a measurement says which, and the pattern that emerges from the measurement is not one anybody would guess from the shapes of the faces: it splits by whether the plane face is a triangle.

The gnomonic keeps one thing, and it is not nothing #

The table above makes the gnomonic look like a poor choice on the cube and the dodecahedron, where it loses on both counts. It keeps something neither column shows.

Straight great circles are the property polyhedral maps were built for in the first place, and they are the reason Fuller’s map is used to argue about air routes. On a gnomonic-faced polyhedral map, a great-circle route within a face is a ruled line, and across a face boundary it is a ruled line that changes direction at the edge and nowhere else. On an equal-area-faced one it is a curve on every face.

The site’s usual way of putting this is that a criterion has to be named before a projection can be called better, and here there are three candidate criteria — area, shape, straightness — and three different winners. The trade-off is two lines says two of them cannot be had at once; nothing says which of the three a reader wants.

A whole map with the areas right #

That last clause is the property worth having and it is not obvious. An equal-area face map guarantees nothing about the relation between faces; what makes the whole net equal-area is that the solid is regular, so all twenty faces cover exactly the same solid angle and are drawn at exactly the same plane area. A polyhedral map on an irregular solid with equal-area faces would still be a patchwork of scales.

Do the pieces actually fit? #

Taking the boundary to the boundary is necessary for the faces to join and it is not obviously sufficient. Two adjacent faces each map the shared edge onto the shared edge — but each does it with its own parameterisation, and if face A puts a point a third of the way along the edge where face B puts it two fifths of the way, the net has a seam that no amount of careful drawing closes.

The measurement is direct: take points along a shared edge, map each one from both faces, and compare how far along the edge each lands.

The gnomonic’s agreement is exact and needs no argument: a point on the shared edge lies in the plane of both faces, so both projections send it to the same place by the same ray.

The area-preserving map’s agreement is a consequence of the solid’s symmetry rather than of the construction, and that is worth saying plainly. The two faces are congruent and the edge sits in the same position in each, so the azimuth equation is the same equation on both sides and produces the same parameterisation. On an irregular polyhedron with faces of different shapes it would not, and Snyder’s construction would leave seams — which is one reason a polyhedral map is nearly always built on a regular or a subdivided-regular solid.

The check that the whole net is right, not just each face #

An exact property inside each of twenty faces is a claim about twenty separate functions, and the thing a reader wants is a claim about the map. There is one number that carries it.

The net’s total plane area, summed over every face and measured with the shoelace formula on the mapped vertices, must equal 4π — the sphere’s own area — because that is what an equal-area map of the whole sphere means. Measured:

Eight significant figures, on a quantity computed by a route with nothing in common with the construction: the construction works in polar coordinates about each face centre and integrates an ordinary differential equation; this measurement takes the mapped positions of the corners and applies a formula from plane geometry. Two routes, one answer, which is the site’s standing requirement for a number it is going to print.

The gnomonic net’s total area is 15.16 by the same measurement, against the same 4π, which is the 21 per cent of area the straight lines cost.

The third member, which is not built #

There should be a conformal face map beside the other two, and there is not.

A conformal map of a spherical triangle onto a plane triangle exists — the Schwarz–Christoffel machinery constructs it, and Laurence Lee published the conformal tetrahedral projection in 1965 using Dixon elliptic functions. What it needs is a family of special functions this site has no other use for, and importing them for one figure would be a library about complex analysis living inside a library about maps.

So it is named and not built, which is the honest option of the three available. The other two would have been to approximate it and call it conformal — the exact error this whole site exists to refuse — or to leave it unmentioned, which would let a reader conclude that a face cannot be conformal when it can.

What can be said without building it is the impossibility half, and that needs no new machinery. Conformal means the two principal scales are equal; equal-area means their product is one; both at once forces both to one, which is an isometry, which the trade-off is two lines shows cannot exist on any region of a sphere however small. A face is a region of a sphere. So a face map can be conformal or equal-area and not both, and the fact that the region is a twentieth of the world changes nothing about the argument.

What a reader gets from each #

Three properties, three constructions, and the choice is the field’s usual one — name the objective first.

Straight routes are the gnomonic’s, and they are what a polyhedral map is usually printed for: an air route across a Dymaxion map is a straight line within each face. The gnomonic companion is the essay about using that property deliberately.

Honest areas are the equal-area construction’s, and they are what a polyhedral map needs to carry data rather than illustrate: a choropleth on a gnomonic-faced net over-states the corners of every face by a factor of up to 1.5, which is exactly the failure the projection that shows true size is about.

Honest shapes are nobody’s here, because the conformal member is named and not built.

The site’s rule is that no projection is best without an objective, and the polyhedral family makes the rule unusually easy to apply: the three face maps differ only in what they preserve, over the same faces, with the same cuts, so the comparison has nothing else in it.

Where the model stops #

The areal factor of the equal-area map is exact and the map is not unique. Any area-preserving map of the face onto the face would do, and this one is picked out by being radial about the centre in the sense the construction describes. A different rule for spreading the azimuths would give a different equal-area face map with a different shape error, and nothing here says this is the best of them.

The corners of a face are singular for both maps. The gnomonic’s scale is finite there and its distortion is the worst on the face; the equal-area map’s azimuth relation is continuous but its derivative is least smooth exactly at a corner of the plane polygon, which is where the cube’s one-part-in-a-million residual comes from. Neither map has a problem at a corner; both have their worst arithmetic there.

Nothing here measures a face’s second-order behaviour except the gnomonic’s zero. The equal-area face map bends geodesics — it must, since only the gnomonic does not — and by how much is a measurement not made here.

Who found it, and when #

Snyder’s equal-area polyhedral projection is from 1992, and he wrote it for exactly the purpose it is used for here: giving Fuller’s and Cahill’s constructions an exact property they had lacked, so that a polyhedral map could be used for data rather than for illustration.

The gnomonic is very much older — it is the projection Thales is supposed to have used, and the oldest one for which there is any evidence at all. The pairing is a good summary of what a polyhedral map is: the most ancient projection there is, applied to twenty small pieces of the world by a construction from the 1990s, on solids Plato listed.

The pairing deserves one more sentence, because it says something about how this subject accumulates. Nothing was discarded to build the equal-area member: the gnomonic remains exactly as useful as it was, for the property it has, and Snyder’s construction sits beside it for a different property. Two and a half millennia separate them and they are alternatives rather than successive approximations — which is what happens in a subject where the objects are specifications met rather than models of something, since a specification does not become obsolete when a different one is also met.

Two properties, one face, no way to have both #

The pair of face maps is a small instance of the trade this whole site is about, and it is worth stating in that form, because the polyhedral setting makes the trade unusually stark.

A face is a small region, so the trade is small — and it is not zero. The gnomonic draws every great circle straight and pays in area; Snyder’s member gets the area exactly and pays in angle. Over a twentieth of the sphere both prices are modest, which is the whole appeal of the family.

But the trade is the same theorem as everywhere else. A face is a piece of a sphere and a sphere has curvature, so the two conditions conflict on a face for exactly the reason they conflict on a hemisphere. Making the face smaller makes the conflict smaller and never removes it, which is why a subdivided solid improves both properties at once rather than escaping the choice.

And the choice is decided by the use rather than by the map. A reader tracing a route wants great circles straight; a reader counting anything by area wants the area right. Those are different readers, the same net serves neither better than the other, and the family’s literature has historically supplied one member and left the other to be wished for.

Which is what Snyder’s construction actually did. It did not improve a polyhedral map; it added the other side of a trade that had been available on one side only for two and a half thousand years — so a designer choosing a polyhedral map now makes the same decision every other cartographer makes, rather than accepting whichever property the construction happened to have.

Where the ladder goes next #

The polyhedral anchor now carries a solid, a cut and a face map. What it does not carry is the thing every practical use of this construction actually wants: a subdivided solid, with hundreds of faces rather than twenty, where the distortion goes to nothing and the cut length goes up without limit.

That is where a polyhedral map stops being a picture and becomes an indexing scheme, and the argument moves fields — the cells ladder takes it up in the applied field, where the question is not how the world looks but what an identifier for a piece of it means.