North cannot be up everywhere

The previous rung is about position: no map of the whole sphere can put every place somewhere, once, continuously. This one is about direction, and the obstruction turns out to be the same integer.

Ground north is not an opinion. At every place on the Earth except two there is a well-defined direction along the meridian towards the pole, and it is what every compass, every grid convergence calculation and every rotated map is a statement about. It is a tangent vector field: an arrow attached to each point of the sphere, varying smoothly, and vanishing nowhere except at the two places where the meridians meet.

A tangent field on a sphere that vanishes nowhere at all does not exist. That is the hairy ball theorem, and Poincaré–Hopf is the sharper version of it: the zeros of such a field have integer indices, and the indices sum to the Euler characteristic of the surface, which for a sphere is two. Not approximately two. Two.

What is being counted, and what is not #

The index is a winding number. Walk a small loop round a place, watch the direction of the arrow, and count how many net turns it makes by the time the loop closes. Nothing else enters: no length, no scale factor, no second derivative, no notion of how far apart two arrows are. Two arrows are either the same direction or they are not, and the count is of how many times the answer came back to itself.

That is worth stating plainly because this site spends most of its time doing the opposite. The whole distortion apparatus is a machine for turning a projection into numbers with units, and every one of those numbers is a how much. A winding number is a how many, it is an integer, and an integer has no error bar and cannot be improved on by a better projection.

The same integer in seven pictures #

An index is computed in a chart, and a chart is a choice. If the answer depended on the choice it would be a property of the picture rather than of the field, and there would be nothing to say.

So it is checked rather than assumed. The ring two degrees from the north pole was drawn in seven charts: four azimuthal projections centred on the pole, one centred forty degrees away so that the ring is not a circle and the arrows are not symmetric, and two oblique pseudocylindricals which are not azimuthal at all and which draw the ring as a lopsided oval.

One chart was tried and rejected, and the rejection is worth keeping. A transverse Mercator ought to be the ideal non-azimuthal witness — the pole is an ordinary interior point on it rather than an edge. It is not a chart there at all: the rotation that lays the cylinder on its side puts the projection’s own seam exactly through both poles, so the ring comes back broken rather than wrong. A chart has to be verified at the point it is used and not at the point it looks reasonable, and this is the second time this site has been caught by a seam that moved when the aspect did.

The map on which the problem has been solved #

Now the objection. On a plate carrée, ground north is page up at every single point. Draw any loop anywhere on that map and the arrow makes no net turn whatever. The field appears to have no zeros, and the theorem says it must have zeros summing to two.

The map is right and the theorem is right, and the two are compatible because the plate carrée is not a map of the sphere. It is a map of the sphere with two points removed, and a sphere with two points removed is a cylinder. A cylinder has Euler characteristic zero, a field of parallel arrows on a cylinder is perfectly consistent, and every world map on which north is reliably up has made that trade without saying so.

This is the previous rung’s finding arriving from the other side. There it was measured as a page length: the pole is 6.28 units of paper for a place with no extent. Here it is the reason a field can look combed. The two statements are the same fact, and neither is visible to any instrument that measures distortion — the indicatrix at 89.9° north is a well-behaved ellipse on a plate carrée and says nothing at all about what happens at 90.

And the convergence angle is the working version of this. Grid north is not north prices the difference between the direction a grid calls north and the direction the meridian does, and finds it running to several degrees at the edge of a zone. What the present rung adds is that the difference cannot be defined away by a better grid: the field being corrected has a zero of index +1 at each pole, and any grid that claims a consistent north over the whole sphere has either omitted those two places or torn.

The same two, from a field that has nothing to do with north #

Poincaré–Hopf is a statement about the surface, not about the field. Any tangent field with isolated zeros gives the same total, so the theorem can be exercised on something with no cartographic content at all.

The natural second witness is already here. The gradient ladder works on analytic scalar surfaces built from spherical harmonics — elevations, in effect, with exact closed-form derivatives — and the gradient of such a surface is a tangent field whose zeros are the peaks, the pits and the passes. A peak counts +1, a pit counts +1, a pass counts −1, and the theorem says the total is two. Which is the mountaineer’s version: a landscape on a sphere has two more summits and hollows than it has passes, and cannot be all summits.

The hypothesis is a measurement too #

Two of the six fields fail the theorem’s own precondition, and they fail it for two different and instructive reasons.

The zonal field depends on latitude alone. Its gradient therefore vanishes along a whole parallel wherever its latitude profile turns over, so its critical set is a circle rather than a set of points. Poincaré–Hopf counts isolated zeros and this field has none; the search returns 149 “critical points” which are 149 samples of two rings.

The terrain field is built from local caps on a background that is exactly zero. Outside the caps the Gaussian tails underflow, the surface is flat to the last bit of a double, and the gradient is zero over most of the sphere. Same failure, opposite cause: one field is too symmetric and the other is too local.

Neither is a counterexample. Both are what it looks like when a theorem’s hypothesis is treated as something to be checked rather than recited, which is the habit this whole site runs on: an assertion that has never rejected anything proves nothing, and a hypothesis that has never excluded anything is decoration.

The place the second derivative gets it wrong #

There is a way to compute an index that does not involve a winding number, and it is the way anyone would reach for first. At a critical point of a scalar field, form the Hessian; if its determinant is positive the point is a peak or a pit and counts +1, and if negative it is a pass and counts −1. That is standard, it is fast, and on most of these fields it agrees with the winding number exactly.

It is also wrong, and the sectoral field shows where.

The sectoral field has three peaks, three pits, and a monkey saddle at each pole. Counted by winding it is 3 + 3 − 2 − 2 = 2. Counted by determinant it is 6, because the Hessian at a monkey saddle is not merely small but identically zero, so what the determinant returns there is the sign of arithmetic noise: −1 at the north pole and +1 at the south, from the same field, at points related by a symmetry. A classifier that gives two different answers to one question is not one short of the truth. It is not answering.

This is the site’s own habit landing on itself. The rule everywhere else is that a quantity should be computed two independent ways and the two required to agree — curvature from the embedding and from the metric, the indicatrix analytically and by walking the circle of directions. Here the two routes disagree, and the disagreement is the finding: one of them differentiates twice and the other does not, and the theorem is about the one that does not.

The third route to the same number #

There are now three ways to reach the number two on this site and they share no machinery at all.

From the metric. Gauss–Bonnet computed rather than cited integrates the Gaussian curvature of the whole sphere and gets 4π, which is 2πχ, which gives χ = 2. That calculation needs a curvature, so it needs a first fundamental form, so it needs distances. It also needs the surface to be smooth, and it gives a different answer on a body with a different shape only if that body has a different topology — two surfaces with the same curvature are the standing demonstration that the metric knows more than the total does.

From counting faces. The cut has to go somewhere counts vertices, edges and faces of a polyhedron and gets V − E + F = 2, which is why a hexagonal tiling of the sphere is forced to carry exactly twelve pentagons however finely it is subdivided. That calculation needs no curvature and no smoothness, only a tiling.

From winding a direction. This essay. No curvature, no tiling, no distance — only a field of arrows, a loop, and a count of turns.

Three calculations with nothing in common, one integer. That is not a coincidence to be admired; it is what it means for a quantity to belong to the surface rather than to any of the ways of describing it, and it is the same discipline this site applies to a projection’s invariants at a much smaller scale. Scale factors along the meridian and along the parallel depend on how the sphere was parameterised. The principal scales do not. Neither, by an enormous margin, does two.

The nearest relative of the winding number here is a direction carried round a loop, which parallel-transports a vector round a closed circuit and finds it returns rotated by the enclosed curvature. That is the metric measurement of the same phenomenon: the holonomy is a real number and depends on the loop, while the index is an integer and does not depend on the loop at all — only on what the loop encloses.

Where the model stops #

The index is computed on a ring of finite radius. Two degrees for the north field, half a degree for the critical points. It is an integer and it is stable — the accumulated turning comes back at 1.000000 rather than at 0.98 — but a field with two zeros closer together than the ring would report their sum rather than each, and the code has no way to notice. Every field examined here has well-separated zeros and that has been checked by eye rather than by machinery.

The critical-point search is a search. Nine hundred Newton starts on a Fibonacci sphere, deduplicated at one and a half degrees. It found every zero of every field it was pointed at, and it can only report zeros it converged to; a field with a very narrow basin could hide one. The check that it has not is the total, which is exactly the theorem — so the count and its test are not independent, and this is the one place in the essay where the argument is circular if read carelessly. What is genuinely established is the consistency: four fields, four different counts of critical points, one total.

Orientation matters and is handled explicitly. The sign of a winding number depends on which way the ring is walked and on whether the chart preserves the surface’s orientation. Both are pinned down in the code — the ring is traversed anticlockwise about the outward normal, and the chart’s own Jacobian determinant supplies the sign — because an index computed in an orientation-reversing chart comes back negated and looks like a discovery.

Who found it, and when #

Poincaré proved the two-dimensional case in 1885, working on the singular points of differential equations rather than on surfaces; Heinz Hopf generalised it to all dimensions in 1926. Brouwer had proved the hairy ball theorem — the corollary that there is no nowhere-zero tangent field on an even-dimensional sphere — in 1912, in the same period as the invariance of domain result that closes the previous rung.

The cartographic reading is older than any of it and was never stated as a theorem. Every projection with a pole line, every azimuthal map that puts a place on its rim, every atlas that changes its aspect between plates is an accommodation to the fact that a sheet cannot carry a consistent north over the whole world. What Poincaré supplies is that the accommodation is forced and that its size is fixed: not “north is hard to keep”, but two, exactly, always, and distributed among however many singular places the projection chooses to have.

The same two, in hardware and on a screen #

The theorem is about a tangent field on a sphere, and a projection is only one thing that produces one. Two others are worth naming because they are met daily and they pay the same bill.

A device compass. The quantity a magnetometer reports is the horizontal component of the geomagnetic field, which is a tangent vector field on the Earth’s surface — so the count applies to it unchanged. It vanishes exactly where the field is vertical, which is at the magnetic dip poles, and there are two of them. A compass at the north magnetic dip pole is not inaccurate; it has no defined heading, because the vector it is measuring the direction of has length zero. The instrument’s manual calls that a region of unreliable readings and the theorem calls it an index.

A north-up control on a globe view. A rendered globe has no singularity, because nothing has been flattened and no field of arrows is being drawn. Offer the reader a keep north up control, though, and one has been created: the control is a rule assigning a screen direction to north at every place, which is a tangent field, and it must vanish at two. On the common implementation those two are the geographic poles, which is why every such viewer becomes uncontrollable at high latitude and why they nearly all clamp the tilt before the reader gets there.

Neither is a bug and neither can be engineered away, which is the practical content of an index. A better magnetometer moves nothing; a better renderer moves nothing. The only thing available is placement — where the two go — and the whole design question is whether they can be put where nobody is looking. The magnetic field does not offer that choice and a projection does, which is the difference between an instrument and a map.

Where this goes next #

The two rungs so far have shown what a map must give up, and both failures have been about a place or a direction being lost. The third is about something being doubled, and it is the sharpest of the three: a projection may be continuous everywhere or one to one everywhere, and when it takes continuity, the pair of places it glues together is always a pair of opposite places. That is not a tendency of the projections in the library. It is a theorem about every continuous pair of quantities on a sphere, and the search for the glued pair either finds one to fourteen decimal places or refuses.