How many times, not whether

This ladder has three rungs and every one of them answers a question with yes or no. No map of the whole sphere is continuous and one-to-one. North cannot be up everywhere. Every continuous map of the sphere to a page glues an antipodal pair. Each is a theorem with no tolerance in it, and that absence is the anchor’s whole identity: the sibling anchor measures how far off a map is, and this one establishes that certain things are not off by an amount at all.

There is one quantity in this subject that counts without measuring, and this rung is it. A continuous map of the sphere to the sphere covers it a whole number of times, that number is the degree, and it survives any continuous deformation of the map. It is not a distance and it has no units. It is also not a yes or no: it is 1, or 2, or −3, and the difference between 1 and 2 is the difference between a map that could be one-to-one and one that cannot.

A map of the sphere onto itself, built out of projections #

Degree needs a map from a closed surface to a closed surface, and every map in this collection so far goes to a page, which is not closed. So the object has to be built, and it can be built entirely from things this site already draws.

Project stereographically to the plane. Raise the complex coordinate to the nth power. Project back.

That is three steps, two of which are a projection and its inverse, and the middle one is the simplest non-trivial thing that can be done to a plane. The result is a genuine map of the whole world onto the whole world. It is conformal everywhere except at two places — because an analytic function of the stereographic coordinate is a conformal map, which is what the families ladder found — and its effect is to draw the world n times.

In coordinates it is as plain as it sounds. The stereographic radius is tan(π/4 + φ/2), so the map takes that to its nth power and multiplies the longitude by n. A place at 30° east goes to 60° east under the square, and so does a place at 210° east.

Counting the preimages #

The first way to get the degree is to pick a place and ask what lands on it. For the nth power map the answer is exact rather than searched: the n preimages of a place are its own stereographic radius to the power 1/n, at longitudes spaced 2π/n apart. Each one is verified by pushing it forward again and checking it arrives, to 1.5 × 10⁻⁸ radians at worst.

Every generic place has exactly n preimages. Not approximately n, not on average n — exactly, for every place except two.

The two exceptions are the poles, and they are what makes this a topological statement rather than an arithmetic one.

Integrating instead of counting #

A count is a fragile measurement, because it depends on finding all of them. The second route knows nothing about the formula at all.

At every point of the sphere, the map takes a small patch of ground to a small patch of ground, and the ratio of the two areas — signed, so that a map which reverses orientation contributes negatively — is the Jacobian. Integrate it over the sphere and divide by the sphere’s own area, and the result is the degree, because a map that covers the target n times sweeps n times its area.

Sampled on a Fibonacci lattice of forty thousand points so no latitude is over-weighted:

The residual is the sampler’s own discretisation and it grows in proportion to the degree, which is what a discretisation error does when the thing being integrated has n times as much structure in it.

Two routes agreeing is the discipline this collection applies everywhere — curvature is computed from the embedding and from the metric and required to match, and the same reason applies here. A count could be short by one and nothing would say so. An integral cannot be short by one; it can only be inaccurate, and inaccuracy shows up in a digit rather than in an integer.

The two places where the count fails #

The poles are branch points. Every other place has n preimages; the north pole has exactly one, and so does the south. The n sheets of the covering come together there.

That is not a defect and it is not avoidable. The Riemann–Hurwitz relation says how much branching a covering of a given degree must have:

$$\chi = n\,\chi - \sum (e - 1)$$

For the sphere, χ is 2 on both sides, and the nth power map has two branch points each of multiplicity n:

It balances at every degree, and it balances by the branch points paying for exactly the excess the covering created. A degree-5 map would like to have five times the Euler characteristic; the two branch points remove eight; and 10 − 8 is 2, which is what the sphere has and what the map’s image must therefore have.

That identity is the same accounting Gauss–Bonnet does with curvature, done with counting instead. It is the second time this anchor has met the number 2 from two directions — the indices of the north field’s zeros sum to it too — and it is why the Euler characteristic keeps arriving in arguments that appear to have nothing to do with each other.

A third power, for the shape of it #

Drawing the cube rather than the square makes the mechanism visible in a way the algebra does not. Each meridian of the original sphere is drawn onto a meridian of the target, and three of them arrive at the same place. The parallels are still parallels — the map moves latitude but does not mix it with longitude — so what the picture shows is a graticule whose longitudinal symmetry has been multiplied by three while its latitudinal structure is merely stretched.

Two things are worth noticing in it. The equator does not stay the equator: tan(π/4 + φ/2) is 1 at the equator and 1 to any power is 1, so the equator is fixed, and it is the only parallel that is. And the crowding near the poles is not a distortion of the drawing — it is the branching, visible as the place where the three sheets have to come together.

The branch point counts the degree by itself #

The preimages have to go somewhere as the target approaches a branch point. They coalesce, and the rate at which they coalesce is a third route to the degree.

Measured over three decades of approach:

The gap closes as the nth root of the distance from the branch point. So the degree is readable from a neighbourhood of one place, without counting anything and without knowing what happens on the rest of the sphere — the local behaviour at a branch point knows the global covering number.

That is the kind of statement this anchor exists for. The degree is not assembled from local information by adding it up; it is a global fact that shows through at every branch point, in the same way the total curvature shows through in the angle sum of one small triangle.

The two degrees a power cannot reach #

The powers produce 1, 2, 3 and upward and nothing else, which leaves out the two values that say what kind of quantity this is.

A reflection has degree −1. Send every place to its mirror image in the prime meridian. The map is continuous, it is one-to-one, and it covers the sphere exactly once — the other way round. The integral returns −1.0000. The sign is orientation: a map of degree −1 is a homeomorphism that turns the world inside out, which is exactly what happens to a map read through the back of the paper, and is why an axis swap is a real defect that does nothing at all along the diagonal.

A map that misses anything has degree 0. Squash the whole sphere into the northern hemisphere. The map is continuous, it is onto a large part of the sphere, it is nowhere constant — and its degree is 0.00002, which is zero within the sampler’s resolution. Every place in the southern half has no preimage at all, and a covering number of zero is what “some places are not reached” means when it is turned into an integer.

The second of those is the one worth keeping. Degree is not a measure of how much of the sphere is covered; the squash covers half of it and scores zero, while a degree-2 map covers all of it twice and scores two. What is being counted is signed coverings of a generic place, and a place with no preimages contributes nothing.

Why this is the bridge to the other anchor #

The curvature anchor’s arguments all have a size and a tolerance. Its results are of the form this map is wrong by 11.84 degrees over this region, and every one of them supports a sentence beginning if that much is acceptable.

The three rungs before this one have no such form. A map is one-to-one or it is not, north is a continuous choice or it is not, and no amount of accepting anything moves the answer.

The degree is between the two, and deliberately so:

It is a number, which the earlier rungs are not. Two is more than one, five is more than two, and a map of degree five is further from injective than a map of degree two in a sense that can be stated and measured.

It is an integer, which the curvature results are not. There is no map of degree 1.5, so there is no continuous path from a covering to an injection, and there is nothing to make small. A tolerance argument is available against every result in the curvature anchor and against none in this one — and the degree shows why: the quantity moves in steps, and a step is not something a tolerance can be set inside.

Injectivity forces it to ±1. A one-to-one continuous map of the sphere onto itself is a homeomorphism, and a homeomorphism has degree ±1. So the degree is an obstruction counter: it says not only that a map fails to be one-to-one but by how many sheets.

What the degree does not do for a projection #

It is worth being clear about the limit of this, because the obvious next sentence is false.

A projection does not have a degree. Its target is a page, the page is contractible, and every map into a contractible space is continuously deformable to a constant — so any degree defined that way is zero for every projection and says nothing about any of them. The degree needs both ends closed.

What a projection has instead is a local version: how many places land on one page point, which varies from region to region of the page. That is what the orthographic’s 47 per cent is a measurement of, and what makes Craig’s retroazimuthal projection impossible to read backwards — two latitudes on one meridian drawn at the same point, which is a local degree of two on part of the page and one on the rest.

So the rung’s finding is not that projections have degrees. It is that the covering number is the quantity the first rung was missing. No map of the whole sphere is continuous and one-to-one is a statement with no size in it; this map covers the world twice, and here are the two places it sends to every place has a size, it is an integer, and it is the smallest amount of quantitative structure the impossibility permits.

What a cartographer would do with this #

The construction above is not a projection and nobody would print it, so the fair question is what a map-maker gets from an integer that counts coverings.

Three things, and the first is the one this ladder cares about.

It names what a cut is for. Every world map has an edge, and the first rung established that the edge is forced rather than chosen. The degree says what the cut is buying: an uncut sphere admits no injective map into a page, and cutting it turns a closed surface into one with a boundary, for which no degree is defined and no obstruction exists. The cut does not reduce the covering number — it removes the question.

It separates two things a reader would call the same defect. The orthographic draws 47 per cent of the world twice and Craig’s projection folds two latitudes onto one point. Both are failures of injectivity, and they are different kinds: the orthographic’s is a whole region covered twice, which is a local degree of two over half the page, while Craig’s is a fold along a curve, which is a local degree that changes across it. The word fold covers both and the count does not.

It explains why the interrupted maps look the way they do. A projection that gives up continuity to keep something else — and giving up continuity is a decision the choosing ladder prices — is trading one obstruction for another. What the lobes buy is a page on which the map is injective; what they cost is the tears. That trade is available precisely because the obstruction is topological, and it would not be available against a curvature obstruction, which no amount of cutting removes.

Why no amount of design changes the count #

There is a reason the third item above is the only escape, and it is worth making explicit because it says something about the whole design space this collection works in.

The degree is a homotopy invariant. Two continuous maps that can be deformed into one another have the same covering count, and the deformation is allowed to be as violent as it likes so long as it never tears.

Now look at what a projection designer actually has to work with. Moving a standard parallel, tilting the axis, changing an aspect, blending two projections with a weight that runs from zero to one, fitting a polynomial to minimise a distortion functional — every one of those is a continuous deformation, because the parameter can be turned smoothly from the old value to the new one and the map moves smoothly with it. The whole toolkit is homotopies.

So the covering count is not a quantity a designer can push on. It is fixed for an entire connected family of projections at once, and the only operations that move it are the ones that break continuity: cutting the sphere, restricting the domain to less than the whole of it, or accepting a fold.

That is the precise sense in which an interrupted map is a different kind of object rather than a further refinement. It is not further along the same axis; it has left the space in which the axis is defined.

What this rung establishes #

A continuous map of the sphere to itself has a whole-number covering count, obtainable three ways that share no arithmetic: exact preimage counting, integration of swept area agreeing to 10⁻³, and the exponent at a branch point recovering 1/n to four decimal places.

Branching is forced, and Riemann–Hurwitz says how much. Two branch points of multiplicity n pay for exactly the excess a degree-n covering creates, at every degree, and the identity closes on the same Euler characteristic that the curvature integral and the north field’s indices both produce.

Injectivity forces degree ±1, which turns the first rung’s yes-or-no into a count — and the count is an integer, so the impossibility this anchor is about still admits no tolerance. It admits an amount, which is a different thing, and this is the rung that says so.

The field’s other anchor prices the failure of distance in degrees and per cent. This one now prices the failure of injectivity in sheets. Neither number can be made small; the second one cannot even be made continuous.