Measuring curvature from inside

Gaussian curvature is intrinsic: computable from measurements made entirely inside a surface, with no reference to any surrounding space. That is the fact that makes a faithful map impossible.

It is usually left there, as a statement about a formula. It has a much more concrete form, and the concrete form is a measurement anybody confined to a surface could actually carry out.

Draw a triangle and add the angles #

On a plane the angles of a triangle sum to exactly $\pi$. On a curved surface they do not, and the discrepancy is the curvature inside:

$$\alpha + \beta + \gamma - \pi = \int\int_\Omega K\,\mathrm{d}A$$

That is the Gauss–Bonnet theorem for a geodesic triangle. On a unit sphere $K = 1$, so the excess is the area — a spherical triangle with an angle sum of 200° encloses $20° = 0.349$ steradians, and that is not an approximation.

The extreme case is worth drawing because it is so unlike a plane triangle. Take a vertex at the north pole and two on the equator, ninety degrees of longitude apart. Every angle is a right angle. The angle sum is 270°, the excess is 90° $= \pi/2$, and the triangle covers one octant of the sphere, which has area $4\pi/8 = \pi/2$.

The site computes both routes for that triangle and for two arbitrary ones. The excess comes from the three vertex angles; the area comes from integrating the surface element over ninety thousand interior samples. They agree to 3×10⁻⁶.

Why this is a measurement rather than a formula #

The reason the theorem matters here is what it licenses.

A two-dimensional creature on the surface cannot see the surface bending, because bending is a statement about a third dimension it has no access to. What it can do is walk in straight lines — geodesics, which it can identify as locally shortest paths without any external reference — and measure the angles between them.

So it draws a large triangle, adds the angles, and gets 200°. It concludes that its world has positive curvature and that the total curvature enclosed by its triangle is 20°. It has measured a property of its universe with a ruler and a protractor.

That is the operational content of “intrinsic”, and it is why the impossibility of a faithful map is not a statement about cartographers. The obstruction is a quantity the surface has, measurable from within it, and a map that preserved distances would have to preserve it.

The scale of the effect #

The excess is proportional to area, which is why nobody notices it at human scale and why geodesists have always had to.

A triangle with sides of one kilometre on the Earth encloses about 0.43 km², which is $1.1\times10^{-8}$ steradians — an angle excess of 0.0022 arcseconds. No survey instrument has ever resolved that.

A triangle with sides of a hundred kilometres encloses about 4,300 km², an excess of 22 arcseconds. That is well within the resolution of a nineteenth-century theodolite, and it had to be accounted for in every large triangulation network. Legendre’s theorem — subtract a third of the excess from each angle and solve the resulting plane triangle — was the standard working method.

So the curvature of the Earth appears in survey arithmetic as a correction of a few tens of arcseconds, and the correction is the Gauss–Bonnet excess. Gauss knew this from the field before he wrote the theorem down: he had spent years measuring triangles across Hanover, and the Theorema Egregium came out of that work rather than preceding it.

The version with no geometry in it at all #

Gauss–Bonnet has a second form, and it is the more startling of the two.

For a closed surface — one with no boundary — the theorem gives

$$\int\int_S K\,\mathrm{d}A = 2\pi\chi$$

where $\chi$ is the Euler characteristic: vertices minus edges plus faces of any polyhedral decomposition of the surface. That is a counting number. It contains no distances, no angles, and no geometry whatsoever.

A sphere has $\chi = 2$, so its total curvature is $4\pi$, whatever its radius and whatever shape it is deformed into. A torus has $\chi = 0$, so its total curvature is exactly zero.

The torus is the case worth dwelling on. Its curvature is nowhere zero except on two circles. Integrating the positive part alone gives 12.566 — the same as the whole sphere’s — and integrating the negative part gives −12.566. The two cancel to eleven decimal places, and they do so because the surface has one hole.

Getting zero out of an integral of a function that is almost nowhere zero is not something a tolerance can be tuned to produce. It is the theorem, and it is the reason the site computes the positive part separately and asserts that it is large: a cancellation to zero is only evidence if there was something to cancel.

What the two forms have in common #

Both say the same thing in different registers: local geometry adds up to something that is not geometric.

In the triangle form, the curvature inside a region is determined by the angles on its boundary — so the interior is fixed by the edge. In the closed form, the curvature over the whole surface is determined by its topology — so the geometry is fixed by the shape’s connectivity.

Deform a sphere however violently: push in a dent, pull out a spike, stretch it into an ellipsoid. The curvature redistributes, and the integral stays at $4\pi$ because $\chi$ stays at 2.

That is exactly the structure of the projection problem, and stating it that way makes the impossibility feel less like an obstruction and more like a conservation law. A projection redistributes distortion; it cannot reduce the total, because the total answers to something a map cannot reach.

The flatlander’s other experiment #

The triangle is not the only measurement available from inside, and the alternative is worth knowing because it is more practical.

Walk out a distance $r$ from a point in every direction and measure the circumference of the resulting circle. On a plane it is $2\pi r$. On a positively curved surface it is less; on a negatively curved one it is more. Expanding:

$$C(r) = 2\pi r\left(1 - \frac{K r^2}{6} + \ldots\right)$$

so $K$ can be recovered from the deficit. The same works for the area of a disc.

That is a genuinely different experiment from the triangle — it uses distances rather than angles — and it returns the same $K$, which is another instance of the pattern this site keeps applying: where two independent routes to a number exist, compute both and require agreement.

The site’s own machinery does exactly this at the level of the formula. Curvature is computed extrinsically, from how the surface sits in space, and intrinsically, from the first fundamental form alone. The two agree to 1.7×10⁻⁵ across six surfaces, and the agreement is the theorem exercised rather than cited.

Curvature on a shape with no curvature #

There is a discrete version of the closed form, and it is the cleanest demonstration in the subject because it can be checked by hand in a minute.

Take a cube. Its faces are flat and its edges are flat — a strip across an edge unrolls without stretching, exactly as a cylinder does. All of the curvature is concentrated at the eight vertices, and at a vertex it appears as an angle defect: three square faces meet, contributing 90° each, so the angles round the vertex sum to 270° rather than the 360° a flat point would have. The defect is 90°, or $\pi/2$.

Eight vertices at $\pi/2$ each is $4\pi$.

A tetrahedron: four vertices, three equilateral faces at each, angles summing to 180°, defect $\pi$ each. Four times $\pi$ is $4\pi$.

An icosahedron: twelve vertices, five triangles at each, angles summing to 300°, defect $\pi/3$. Twelve times $\pi/3$ is $4\pi$.

Every convex polyhedron gives $4\pi$, because every one of them is topologically a sphere and $\chi = 2$. The result is Descartes’ theorem on the total angular defect, from about 1630, and it predates Euler’s formula by a century and Gauss–Bonnet by two.

That is worth knowing for what it says about where curvature lives. A cube has no curvature anywhere except at eight points where it is infinite, and the integral is the same as a smooth sphere’s. Curvature is not a property of how something looks; it is a property of how the surface closes, and the two are only loosely connected.

The boundary term Bonnet added #

Gauss’s version needs the triangle’s sides to be geodesics. Most regions are not bounded by geodesics, and Bonnet’s 1848 generalisation says what happens when they are not:

$$\int\int_\Omega K\,\mathrm{d}A + \oint_{\partial\Omega} k_g\,\mathrm{d}s = 2\pi\chi(\Omega)$$

where $k_g$ is the geodesic curvature of the boundary — how much the boundary curve turns relative to a geodesic. For a geodesic boundary $k_g = 0$ and the second term vanishes, which recovers the triangle case with the corner angles absorbed into $\chi$.

The circle experiment above is an instance. A circle of radius $r$ drawn on a sphere is not a geodesic, its geodesic curvature is $\cot r$, and the theorem then gives the circumference deficit directly. So the two measurements a flatlander could make — angles of a triangle, circumference of a circle — are the same theorem with the boundary term switched on and off.

That unification is the reason Bonnet’s name is on it. Gauss’s version is a statement about a special shape; Bonnet’s is a statement about any region, and the price is one line integral.

Where the theorem reaches #

Naming a few consequences outside cartography makes the result feel less like a technicality.

The hairy ball theorem — that a continuous tangent vector field on a sphere must vanish somewhere — is a consequence of $\chi = 2$, and it is why there is always a point of zero horizontal wind on the Earth’s surface.

Every closed surface of genus $g$ has total curvature $2\pi(2-2g)$, so a two-holed surface has total curvature $-4\pi$ and cannot be given a metric of everywhere-positive curvature. Topology forbids geometries.

General relativity is written in the machinery this generalises to. The reason spacetime curvature can be a physical quantity rather than a description of an embedding is exactly the intrinsic property Gauss found: there is no higher-dimensional space for spacetime to curve into, and the curvature is measurable from within.

The line from a survey of Hanover to the field equations runs directly through this theorem, which is a fair claim for a result whose statement is that the angles of a triangle do not add up.

Why this is the theorem the site rests on #

Worth saying explicitly, because it is easy to read Gauss–Bonnet as an elegant aside.

The impossibility argument rests on curvature being intrinsic, and Theorema Egregium establishes that at a point. That is enough to prove no isometry exists, and it says nothing about how much distortion is forced or where.

Gauss–Bonnet is what turns the prohibition into a budget. It says the curvature over a region is a fixed quantity determined by the boundary, and over a closed surface a fixed quantity determined by counting — so a projection is redistributing something conserved rather than fighting something vague. The scale rule is that budget expressed in parts per million, and the exact bound for a cap is what a perfectly chosen projection does with it.

The two theorems therefore answer different questions. One says a faithful map is impossible; the other says how impossible, for this region, in units a surveyor can use. Almost every treatment of projections cites the first and none of them cites the second, which leaves the impossibility as a fact with no consequences — true, unarguable, and no help at all in deciding anything.

A surveyor confined to the surface can in principle detect not only that the Earth is curved but that it is curved by different amounts in different places.

What was computed here #

The angle excess is computed from the three vertices by projecting each edge direction into the tangent plane at the vertex and taking the angle between them. The area is computed by integrating the surface element over the interior, parameterised barycentrically and mapped radially onto the sphere.

The two share no arithmetic, which is the point. They agree to 3×10⁻⁶, and the octant case — three right angles — comes out at exactly $\pi/2$ to machine precision.

The interior integration was wrong the first time by 0.17%, which read as a 0.3% failure of Gauss–Bonnet. The parameter domain is a triangle and it was being sampled by midpoints of sub-triangles, which silently omits the down-pointing cells and overcounts the total weight by $1/2n$. Substituting the unit square onto the triangle with the Jacobian $(1-s)$ fixed it. A theorem failing by a fifth of a per cent is almost always the quadrature.

The total curvature of the closed surfaces is integrated over the full parameter domain with the area element from the first fundamental form. Three things are asserted: the sphere must give $4\pi$ to half a per cent, the torus must give zero to the same tolerance relative to its own positive part, and that positive part must exceed one — because a cancellation to zero is not evidence unless there was something to cancel.

What the pictures cannot show #

The intrinsic route, which is the whole subject. Its defining feature is that it uses no information from outside the surface, and every drawing of a surface is a view from outside. The triangle figure shows a sphere seen from a distance; the flatlander doing the measurement has no such view and does not need one.

The figures also cannot show the integral being taken. The curvature-integral figure shows two bars and the numbers they represent; what happened between the surface and the bar is ninety thousand samples of a quantity with no appearance.

Who found it, and when #

Gauss proved the local form for a geodesic triangle in the Disquisitiones generales circa superficies curvas of 1827, as a corollary of the Theorema Egregium. He had derived the spherical excess relation for surveying much earlier.

Pierre Ossian Bonnet generalised it in 1848 to regions whose boundaries are not geodesics, adding the geodesic-curvature term along the boundary. The global form — the connection to the Euler characteristic — came later still, with Dyck and then with the twentieth-century development of algebraic topology.

The generalisation to arbitrary dimension, the Chern–Gauss–Bonnet theorem, was proved by Chern in 1944, and it is one of the results that established the modern relationship between geometry and topology. What began as a correction to survey triangles ended as a statement about what shapes can carry what geometries.

What the flatlander still does not know #

The measurement is complete and the knowledge it produces is not, and the gap is worth stating precisely because it is the boundary between two of this collection’s subjects.

The local measurement gives the metric. A flatlander who measures the excess of triangles everywhere knows the Gaussian curvature at every point, and by Gauss’s theorem that is exactly what is determined by distances measured inside the surface — no more and no less.

The integral gives only the topology. Totalling that curvature over the whole closed surface returns $2\pi\chi$, and $\chi$ is an integer that counts handles. It is the same $4\pi$ for a round sphere, an egg, a pear and a lumpy potato, because all four are topological spheres. The integral has thrown away everything except the shape’s genus.

Neither gives the embedding. A flatlander on a plane and a flatlander on a cylinder measure identical zero curvature and can perform no interior experiment that separates them — which is the same fact, read from inside, as a cylinder unrolling without distortion.

So the ladder’s intrinsic route buys the geometry and refuses the picture. That refusal is not a limitation of the method; it is the content of the theorem, and it is why a map of the Earth can be judged without ever leaving it.

Where this goes next #

The theorem this one extends is no map is faithful. What its integral says about how much distortion a map must impose is total curvature and the scale rule. And the surfaces where the integral vanishes are what can be unrolled.