A direction carried round a loop

A flatlander with a ruler can find out its world is round by drawing a triangle and adding the angles. A flatlander with a compass can find out faster, and without drawing anything: walk a closed circuit holding one direction as steady as the ground permits, and see whether the direction comes back the way it left.

The operation, written without a symbol in it #

Parallel transport has a reputation for machinery — Christoffel symbols, covariant derivatives, an index or two — and none of it is needed to compute one.

Take the vector. Take a small step along the curve. The vector wants to remain the same vector in the surrounding space; the surface will not have it, because it has tilted; so remove whatever component has left the new tangent plane and renormalise. Repeat. In the limit of small steps that is parallel transport, because it is exactly the statement that the vector’s rate of change has no component inside the surface.

Written that way it needs a surface with an embedding and four lines of arithmetic, and it applies unchanged to the plane, the cylinder, the cone, the sphere, the torus and the pseudosphere.

Round a loop that bounds a disc, the turning is the curvature #

The first measurement is on a rectangle in parameter space — a loop whose inside is an ordinary patch of surface with nothing missing from it. Against it, the curvature integrated over that patch:

The two columns agree to a few parts in ten thousand, which is the discretisation of both. That is the Gauss–Bonnet theorem in the form that has no triangle in it, and it is a second independent route to the statement measuring curvature from inside reaches through the angle sum of a geodesic triangle.

Both flat surfaces return zero, and the cone returns zero, which matters: the cone is curved in space and flat within itself, and a routine that measured the bending rather than the curvature would report something for it. That is the refusal built into the check, and it is the same distinction what can be unrolled is about.

Round a parallel, the turning is something else #

The obvious next loop is a parallel of latitude, and it does not obey the rule above. At 45° the transported direction comes back turned by 4.443 radians while the cap it encircles carries 1.840 of curvature. The numbers are not close and the discrepancy is not numerical.

They sum to a full revolution, everywhere, and that is the whole explanation. The turning was measured relative to the local north, and the loop goes round the pole, where north does not exist. Following the north direction round the parallel winds it through one complete turn whatever the latitude, and the transported vector’s angle against a winding frame is the enclosed curvature plus the winding.

$$2\pi\sin\varphi \;+\; 2\pi(1-\sin\varphi) \;=\; 2\pi$$

So the honest statement of the theorem has two clauses. On a loop that bounds a disc inside the surface, the turning is the curvature enclosed. On a loop that does not, the turning is the curvature enclosed plus a whole number of revolutions, and the whole number is a fact about the hole rather than about the geometry.

Why the answer does not depend on the loop’s shape #

Two circuits enclosing the same region turn a direction by the same angle, whatever route they take, and that is worth pausing on because nothing about the operation suggests it. The transport is performed step by step along a particular curve; a different curve is a different sequence of steps; and the answers agree exactly.

They agree because the difference between two loops enclosing the same region is a loop enclosing nothing, and a loop enclosing nothing returns a direction unchanged. What the turning depends on is the region, through the integral of its curvature — a single number that cannot tell one boundary from another.

This is the same structural fact as total curvature and the scale rule: a region’s total curvature is a property of the region and is what any flat map of it has to absorb. Transport is that quantity made observable without a map, by an instrument that only ever knows where it is pointing.

The quantity 2π sin φ has three other names #

A Foucault pendulum at latitude $\varphi$ precesses through $2\pi\sin\varphi$ in a sidereal day, which is the same transport round the same circle performed by a swinging weight rather than by arithmetic.

A gyrocompass carried round a parallel accumulates the same angle, which is why an inertial system integrating its own heading has to be told the latitude.

And a conic projection standing on that parallel has cone constant $n = \sin\varphi$. That is the one this site owes an explanation for, and it is not an analogy.

A tangent cone touching the sphere along a parallel unrolls to a sector of angle $2\pi\sin\varphi_0$ rather than a full disc, because the parallel’s circumference is $2\pi R\cos\varphi_0$ and its distance from the apex along the cone is $R\cot\varphi_0$ — and a circle of that circumference at that radius subtends $\sin\varphi_0$ of a turn. The missing sector is what a paper cone leaves open when it is cut and laid flat.

Transporting a direction round the parallel is transport on that cone, which is flat, so in the unrolled picture the vector does not turn at all and the frame does — through the sector’s angle. The cone constant, the pendulum’s precession and the holonomy of the parallel are one number arrived at three ways, and the site’s machinery reaches two of them independently and is required to agree.

The cone is the case that decides what the theorem says #

A loop drawn round the apex of a cone is the sharpest example there is, because the cone’s curvature is exactly zero everywhere it is defined.

The apex is a puncture. A loop around it cannot be shrunk to a point without leaving the surface, so there is no disc for Gauss–Bonnet to integrate over, and the turning is pure winding with no curvature anywhere in it.

Anybody told “holonomy measures curvature” gets this case wrong, and it is not an exotic case: it is the surface every conic projection in the library is built on.

The cone’s turning is the sphere’s formula with the latitude replaced #

The cone measurement above is quoted as 172.6 degrees for a half-angle of 0.5 radians, and the two facts are one fact: 2π sin(0.5) is 0.4794 of a revolution, which is 172.6 degrees. A loop round a cone’s apex turns a direction through 2π sin α, with α the cone’s half-angle — the same expression the sphere’s parallel obeys with the latitude in place of the half-angle.

That is not a resemblance. The cone tangent to the sphere along the parallel at φ has its apex on the polar axis, and the angle between its axis and its own surface works out to exactly φ: the generator from the apex to a point of the parallel makes an angle whose tangent is cos φ sin φ / cos²φ, which is tan φ. So the tangent cone’s half-angle is the latitude, and the two formulae are the same formula evaluated on the same object.

Which settles the relation between the essay’s two hardest cases. The parallel’s turning of 2π sin φ was explained above as the winding of the north frame round a hole; the cone’s turning of 2π sin α was explained as pure winding round a puncture with no curvature anywhere. They are the same measurement, because transport along a curve depends only on a neighbourhood of that curve, and the sphere and its tangent cone agree on a neighbourhood of the parallel to first order. The sphere’s version happens to have curvature inside the loop and the cone’s does not, and transport along the loop cannot tell — which is precisely why the cap term had to be subtracted to recover the curvature.

So the flatlander’s two-circuit test has an exact form. On a sphere, a loop of radius r about a point turns a direction through 2π(1 − cos(r/R)), which is πr²/R² for small r and therefore falls as the square of the radius: halve the loop and the turning falls by four. Round a cone’s apex the turning is 2π sin α whatever the loop’s size, because there is no area contributing anything. Two loops, one twice the other, and a ratio of four against a ratio of one — which is a decisive experiment, needs no third dimension, and is the sharpest thing a two-dimensional instrument can say about whether the space it is in is curved or merely punctured.

What this adds to the impossibility #

The site’s founding argument is that a sphere and a plane have different Gaussian curvature and therefore no isometry between them exists. That argument is made pointwise: the curvature at a point is intrinsic, and two surfaces with different values cannot be laid on each other.

Transport gives the same conclusion in an integrated form, and the integrated form is the one a cartographer meets. A map is a promise that directions can be compared across it. On a plane, carrying a direction round any closed circuit returns it unchanged — that is what makes a single north possible at all. On a sphere it does not, and the discrepancy for a circuit enclosing a region of area $A$ is exactly $A/R^2$, with no dependence on the circuit’s shape.

So a map that preserved directions everywhere would be a map on which every closed circuit returned a direction unchanged, which is a map of a surface with no curvature. The grid on a projected map has exactly one north by construction, and the price is grid north is not north: the angle between the grid’s north and the ground’s varies from place to place, and its accumulated variation round a circuit is this same number.

The size of it on a real sheet #

For a survey sheet 100 kilometres square at mid-latitude, the enclosed area is $10^{10}$ square metres and the Earth’s radius squared is $4.06\times10^{13}$, so the holonomy is $2.5\times10^{-4}$ radians — 51 arcseconds.

That is not negligible. It is four times the tolerance a first-order traverse works to, and it is the reason the two ways to spread a misclosure has an angular misclosure to spread in the first place: a closed traverse’s bearings, carried round the loop and compared with the bearing they started from, do not agree, and the discrepancy that is not observational is this.

The correction has a name in survey practice — the spherical excess, distributed across the angles of the figure — and it is the same quantity as the triangle’s angle sum, the transported direction’s turn and the cone’s missing sector.

The size scales with area, so it is the sheet rather than the instrument that decides whether it matters. A site 1 kilometre square carries 5 milliarcseconds of it and nothing on Earth can observe that; a first-order triangulation figure 50 kilometres on a side carries 13 arcseconds and every angle in it has to be corrected before the figure will close. That is how small is flat enough asked about angles instead of about distances, and it gives the same quadratic law with a different constant.

The projection that makes the winding visible #

Every conic projection in the library is a cone unrolled, and the sector it leaves open is the winding term measured above. That gives a picture of the abstraction: the missing wedge is where the extra revolution went.

The cylindrical limit is the case worth stating separately. A cylinder is flat and a loop round it turns nothing — measured above as exactly zero — and that is why cylinders, cones and planes can present the family as a taxonomy at all: the three shapes are three values of one winding number, and the taxonomy is a coarse reading of a continuous parameter.

What the transport had to be checked against #

Four things, and the second is the one that was wrong first.

The rectangle against the integrated curvature, on surfaces of positive, mixed and negative curvature. A routine returning the loop’s length, or its area, or anything monotone would pass a single-surface check and fails this one.

The angle had to be accumulated, not read off at the end. A single arctangent at the finish returns a value between −π and π, so a circuit turning a direction through more than half a revolution comes back reporting the wrong number with a straight face. The parallel at 15° north encloses a cap of area 4.657 and was measured at −1.627, which is $4.657 - 2\pi$ and looks like a 65 per cent error rather than a wrap. Following the angle round the loop and unwrapping each step keeps the revolutions.

The flat surfaces must return zero, which is the refusal. Without it every other clause would be satisfied by a routine measuring how far the surface bends in space, which is an extrinsic quantity and is precisely what Gauss’s theorem says is not the answer.

The cone constant had to be measured off the projection, not computed from $\sin\varphi_0$ and compared with itself. The angle is read from where the forward map actually puts a full circuit of longitude — which needed the cone’s apex located from two points on the central meridian rather than from the pole, because a shallow conformal conic’s radius at the pole is a seventieth of its radius at the equator rather than zero, and using it put the sweep 1.3 per cent out.

What is left of the flatlander #

The opening image was a two-dimensional creature discovering its world is curved. Transport sharpens what such a creature can and cannot find out.

It can measure the curvature of any region it can walk round, exactly, with a compass and no instruments pointed at the sky. It cannot distinguish a curved region from a flat one with a puncture in it by any single circuit, because both return a direction turned — that takes two circuits of different sizes around the same place, and the curved one’s turning shrinks with the loop while the puncture’s does not.

Nothing in either measurement requires the creature to know that a third dimension exists, which is the content of no map is faithful restated as an experiment rather than as a theorem. The impossibility of a faithful map is not a fact about drawing; it is a fact two-dimensional instruments can establish about their own world.

That is a topological measurement made with a compass, and it is the same distinction the cone forced above.

One consequence of that identity is worth stating for the projections rather than for the flatlander. Because the tangent cone and the sphere agree on the turning round the parallel they touch along, a conic projection standing on that parallel inherits the sphere’s holonomy exactly at its standard parallel and nowhere else — which is the angular counterpart of the statement that its scale is true there and nowhere else. The cone constant is not merely a parameter of the family; it is the one place the flat sheet and the curved ground agree about how a direction comes home.

Where this ladder goes #

The curvature ladder has now taken the impossibility from a statement about a point (a sphere has curvature, a plane has none) to a statement about a region (the total is fixed by counting) to a statement about a circuit (a direction does not come back).

What it has not done is connect the last of those to what a projection does to a bearing, which is where the practice field takes it: the convergence of the meridians on a grid, the arc-to-chord correction between an observed direction and a plotted one, and the angular misclosure a closed traverse is required to distribute. All three are this quantity, arriving in a specification rather than in a theorem.