The family is a symmetry, not a shape

Cylinders, cones and planes opens this ladder with the classification every textbook gives: wrap the sphere in a cylinder, a cone or a plane, project onto it, and unroll. Seven rungs work through what that buys and what it does not, and the seventh dissolves it: eleven named projections fitted to one two-line perspective construction, six reproduced to arithmetic noise, five refused at any projection centre. The developable surface turns out to describe how the projections were discovered rather than what they are.

That rung leaves a hole where the classification was. Every atlas, every library, every textbook chapter is organised by cylinders, cones and planes; the taxonomy is not arbitrary, the three classes really do behave differently, and something must be doing the work. If it is not the shape, what is it?

It is a symmetry, and it can be measured from the map with no formula at all.

The test #

Turn the sphere. Look at what the picture does.

On a cylindrical map the picture slides sideways — the same map, translated. On an azimuthal map it turns about the centre — the same map, rotated. On a conic map it turns about the apex, through a different angle. In every case the page has moved rigidly: nothing has been stretched, nothing bent, and the two pictures are congruent.

That is a property of the projection and it is checkable without knowing anything about how it was built. Sample the sphere, rotate, project both sets, fit the best rotation-and-translation of the page, and measure what is left over relative to the size of the picture. A projection with the symmetry leaves nothing; a projection without it leaves something.

The scale is held at one. That is the difference between this test and the similarity fit the identification ladder uses, and it is the whole discipline of the measurement: a fit that could shrink the picture would let a projection which halves everything pass as symmetric.

The axis is found, not assumed #

A rotation needs an axis, and giving each projection its own would be assuming the answer. So the axis is searched for.

Nothing about the projection’s family, formula or name enters that search. What comes back is exactly what the construction would have said: the polar axis for every cylindrical and every conic, and the line through its own centre for every azimuthal.

And that is the aspect, measured. Rung two of this ladder establishes that a projection’s aspect is a free choice — rotate the sphere first and any projection can be centred anywhere. What this rung adds is that the aspect is the symmetry axis: choosing an aspect is choosing where to point the one axis a projection has, and the axis is recoverable from the finished map.

The three classes are three values of one number #

Once the axis is known, the page’s response is a rotation through some angle, and the interesting quantity is the ratio between that angle and the sphere’s.

So the taxonomy is a ratio, it runs continuously from zero to one, and the two named classes at the ends are the two values a cone constant is not allowed to take: zero, where the cone has become a cylinder, and one, where it has become a plane. The conic is the whole family reaches the same conclusion from the formulae and finds the two limits by taking the cone constant to its ends; this is the same statement made from the pictures.

Why there is exactly one kind of symmetry available #

The reason the three classes exhaust the possibilities is a fact about the rotation group and it is worth stating, because it explains why nobody has ever found a fourth family.

A projection with a continuous symmetry is invariant under a one-parameter subgroup of the sphere’s rotations. Every one-parameter subgroup of the rotation group is a rotation about some axis — there are no others — and any two are conjugate, which is to say that they differ only by where the axis points. So up to aspect there is exactly one continuous symmetry a projection can have.

A projection invariant under more than one such subgroup would be invariant under the whole rotation group, and a map invariant under every rotation of the sphere would have the same distortion everywhere, which is an isometry, which Gauss’s theorem forbids.

So the possibilities are: no continuous symmetry, or exactly one. The three classes are the three ways of realising the one, and there is nothing else for a fourth family to be.

The failure is not a matter of degree #

The obvious objection is that every smooth map is nearly affine over a small enough patch, so a pseudocylindrical should look symmetric if the rotation is small enough.

It does not.

Mollweide’s rate is 0.0930, 0.0925, 0.0930 and 0.0927 of the picture per radian at ten, five, two and a half and one and a quarter degrees. That is a constant, it is the derivative of the failure at zero rotation, and it does not go away. There is no rotation small enough to make a Mollweide cylindrical.

What “pseudo” turns out to mean #

The prefix is a cartographer’s word and it has never had a definition beyond a family resemblance: a pseudocylindrical has straight horizontal parallels like a cylindrical and curved meridians unlike one; a pseudoconic and a pseudoazimuthal are described the same way, by listing what they keep and what they drop.

The measurement gives it one. A pseudo-family is a projection that keeps the shape of a family and loses its symmetry. The parallels are still horizontal lines, so it looks cylindrical; a rotation of the sphere no longer slides the picture, so it is not.

And the definition has consequences the descriptive one does not. It says why an interrupted projection is nearly always pseudocylindrical: cutting at meridians is only possible if the meridians are distinguishable from one another, which is exactly what a broken longitudinal symmetry provides. It says why a pseudocylindrical has a “central meridian” as a parameter while a cylindrical does not really need one. And it says why a compromise projection cannot be made symmetric by any amount of tuning: the seven broken ones are all compromises, and a compromise is a projection built by asking for something at every place, which is precisely a construction that treats places differently.

What the taxonomy was already doing #

It is worth being clear that the classification is not wrong. It has organised the subject for four centuries and it groups projections that really do behave alike. What this rung changes is the account of why.

Three things the old account struggled with come out for free.

Why a projection can be in a family without a surface. The developable surface was never necessary finds five named maps that no perspective construction reproduces, and four of the five are still, unarguably, cylindrical or azimuthal. Under the symmetry account there is nothing to explain: the Lambert cylindrical is cylindrical because a rotation slides its page, and whether any cone or cylinder was ever wrapped round anything is irrelevant.

Why the conic family is continuous and the taxonomy is not. A cone constant runs over an interval and the classes are three labels, so the taxonomy has always had an awkward join at its ends. The ratio measured here runs over the same interval and reaches both ends exactly, which makes the classes the two limits rather than three boxes.

And why the aspect is free. Because the axis is the only thing a symmetry has, and where it points is not part of the symmetry.

The one measurement that could have refuted this #

A classification derived from the pictures is only worth having if it could have come out differently, and it could have, in two ways.

A projection could have been symmetric about an axis its construction does not name. The search covers the whole sphere, so a hidden symmetry would have shown as a second dark patch in the axis figure. None of the twenty-one has one — the fourteen have exactly one axis and its antipode, and the seven have none anywhere.

Or a pseudocylindrical could have turned out symmetric. Nothing about the arithmetic prevents it: a projection whose parallels are straight and whose meridians are curved could in principle still slide under a rotation. The measurement says the seven do not, at a rate four orders of magnitude above the floor, and says so at every angle tried.

That last figure is worth reading against the cone-constant one. Within a class, changing the standard parallel changes the map a great deal and changes the symmetry not at all; between classes, changing the cone constant changes the symmetry continuously. The taxonomy separates the two kinds of parameter and the shape account does not — a cylinder with a different standard parallel is a different cylinder, and a cone with a different apex angle is a different cone, and the shape account has no way to say that only one of those is a change of kind.

What a reader can do with this #

The test is short enough to be worth stating as a procedure, because it needs no formula and can be run on a map whose projection is unknown.

Take two copies of the map, one of the world rotated a little about some axis. If the second is the first slid sideways, the projection is cylindrical and the axis is its aspect. If it is the first turned about a point, the projection is azimuthal and the point is its centre. If it is the first turned about a point through a smaller angle than the sphere was rotated, the projection is conic and the ratio is its cone constant. And if the second is not a rigid motion of the first at all, the projection is in none of the three classes, whatever its parallels look like.

That is a classification anybody can apply to a finished map, and it is the one the identification ladder would reach for first if it were asking about family rather than about identity.

Where the model stops #

The residual floor is the search’s, not the arithmetic’s. The fourteen symmetric projections come back at between 3 × 10⁻⁷ and 2 × 10⁻⁶ rather than at 10⁻¹⁶, because the axis is found by a polish that stops at about a millionth of a radian and a slightly wrong axis leaves a proportional residual. Given the exact axis the residual is machine noise. The four orders of magnitude to the broken group is the number that matters, and it would only grow with a better polish.

The classification is of the library. Twenty-one projections is a sample, and the argument that there can be no fourth kind is a statement about the rotation group rather than a measurement. What the table establishes is that this collection’s own classification survives being re-derived from the pictures.

Only continuous symmetries are looked for. Every projection here has discrete symmetries too — reflections in the equator and in the central meridian, which most of the library has and Robinson has approximately — and those are a different classification that this rung does not touch. A projection with no continuous symmetry can have a great deal of discrete structure, which is why a Mollweide still looks orderly.

And the test is of the map, not of the formula. Two projections that differ by a similarity of the page are the same map to this measurement, which is deliberate and is the same convention the identification ladder uses for the same reason: a projection scaled or rotated on the page is the same projection.

Who found it, and when #

The group-theoretic fact — that one-parameter subgroups of the rotation group are all rotations about an axis and all conjugate — is elementary Lie theory and is nineteenth-century. Its application to map projections is standard in the mathematical treatments: any modern differential-geometry account of cartography defines a cylindrical projection as one equivariant under rotations about the polar axis, and derives the classical formulae from that.

What is unusual here is the direction. The mathematical treatments define the classes by symmetry and derive the shapes; the cartographic tradition defines them by shape and finds the symmetry incidental. This rung starts from a library of finished maps, measures the symmetry, and finds it recovers the taxonomy exactly — including the aspect, the cone constant and the pseudo- prefix, none of which was put in.

A conjecture about the seven, offered as one #

The ladder has no account of why the projections without a family are the ones people choose, and this rung’s own machinery suggests one. It is a conjecture rather than a finding, and it is stated that way because nothing here tests it.

A continuous symmetry is a strong constraint on where the distortion can go. If a projection is unchanged by rotation about an axis, then every quantity derived from it — the scale factors, the angular deformation, the areal factor — is constant along each orbit of that rotation. On a cylindrical projection that means the distortion is a function of latitude alone: fixed along every parallel, whatever is underneath.

A world map’s requirements are not distributed symmetrically. The land is in particular places, the oceans are in others, and the regions a general-purpose sheet most wants to treat gently — the populated mid-latitudes, the continental interiors — form a set with no rotational symmetry whatever. A projection constrained to spend its distortion in bands cannot put the good behaviour where the land is, because the land is not a band.

So the freedom the symmetric families give up is exactly the freedom a world map wants. A projection with no continuous symmetry can shape its distortion two-dimensionally, and the seven — Robinson, Winkel tripel, Eckert IV, Mollweide, sinusoidal, Hammer, the polyconic — are precisely the ones that were designed by fitting a compromise rather than by imposing a construction.

What would test it is measuring, for each of the seven, how much of its advantage over the best symmetric member survives when the criterion is made rotationally symmetric — a distortion score integrated over the sphere with no regard for where the land is. If the seven’s advantage largely disappears under that criterion, the conjecture holds: their value is in placing distortion, not in reducing it. If the advantage survives, they are simply better maps and the symmetry has nothing to do with it.

Stating it as a conjecture with a test attached is the most this rung can honestly do with it, and it is more than leaving the seven unexplained.

That measurement is available to this collection and has not been made, which is why the paragraph above is a conjecture and the ladder’s honest position is still that it has no account of the seven.

Where the ladder goes next #

The ladder has now taken the families apart three ways: as surfaces, which rung seven shows is a story about discovery; as a one-parameter perspective construction, which holds six of eleven; and as a symmetry, which holds fourteen of twenty-one and explains the other seven.

What none of the three explains is the seven. A projection with no continuous symmetry is not a member of a family in any sense, and yet the seven are not a random collection — they are Robinson, Winkel tripel, Eckert IV, Mollweide, sinusoidal, Hammer and the polyconic, which is to say almost exactly the set of projections anybody would choose for a world map. The projections with no family are the ones people use, and the ladder has no account of why.