Peaks, pits and passes

In 1870 James Clerk Maxwell published a short paper called “On Hills and Dales”. Arthur Cayley had made the observation in 1859. Think of the Earth’s surface, smooth enough that all of its heights can be described by contour lines, with peaks, pits and passes between them. Count them. Whatever the landscape looks like, however many mountains or valleys it has, the number of peaks plus pits minus passes is always 2. A new hill arrives together with a pass connecting it to what was there before, so a local change never changes the total.

That is the Euler characteristic of a sphere, read off a contour map. It is the first appearance of what later became Morse theory, in a paper about geography. Nearly a century later, in 1967, Thomas Banchoff found the version for polyhedra, and it comes with a proof that takes one line. And three years later he added something still more surprising. If the height is allowed to point in every direction, and the score of one vertex is averaged over all of those directions, the result is exactly the vertex’s angle defect. So curvature is the average of a count of comparisons.

The score of a vertex #

Take a closed polyhedral surface with triangular faces. Pick a direction, and call the coordinate along it the height, assuming that no two vertices have equal heights. Look at a vertex and go round the ring of its neighbours, marking each as higher or lower than the vertex. The marks form a cyclic pattern with some even number of changes.

If all neighbours are higher, it is a pit. If all are lower, it is a peak. Both have no changes. An ordinary point on a slope has two changes, with the higher neighbours in one arc and the lower ones in another. A pass has four: up, down, up, down. A monkey saddle has six, with three ways up and three ways down, and a polyhedron can have these as readily as passes. Define

$$i(v) = 1 - \frac{\text{number of changes round the ring}}{2}.$$

This gives $+1$ at peaks and pits, 0 at ordinary points, $-1$ at passes, and $-2$ at monkey saddles. Banchoff’s theorem says that the sum of $i(v)$ over all vertices is the Euler characteristic, in any generic direction. It has a very short proof. Each triangle has three vertices at three different heights, so that one of them is in the middle. Look at that middle vertex: its ring passes along the triangle’s opposite edge, from a higher neighbour to a lower one, and so this triangle contributes exactly one change to that vertex’s ring. It contributes no change to the rings of its top and bottom vertices. So the total number of changes over all vertices is exactly the number of faces $F$. Then

$$\sum_v i(v) = V - \frac F2 = V - E + F,$$

since each face has three edges and each edge two faces, so that $3F = 2E$ and $E = 3F/2$. That is the whole proof. It uses no continuity, and no notion of curvature, nothing but which of three numbers is the middle one. It is more elementary than the usual count of solids, and it works for any triangulated closed surface in space.

All directions at once #

The figures were tested as follows. Two lumpy surfaces were built with seeded noise, a sphere of 66 vertices and a torus of 240. The sum of indices was computed for 3,000 random directions, each direction checking the whole surface.

The sum never changes. It is 2 on the sphere in every case, and 0 on the torus. The number of critical points varies widely, from 2 to 12 on the sphere and from 4 to 40 on the torus, as lumps face different ways. This illustrates what the signs do. When a direction changes, the height order changes, one pair of adjacent vertices at a time, and critical points are created and destroyed in pairs with opposite signs, as with fixed points that come and go in pairs. It is actually the same theorem, because a critical point of a height is a zero of a tangent field, the gradient of the height, and the index is the Poincaré–Hopf index. A height function is a particular way of combing a surface. The result is that the Euler characteristic is what is left over after everything that can cancel has cancelled.

Averaging gives the curvature #

Now the second result. Fix a vertex and average its index over all directions of height, using the uniform measure on the sphere of directions. At a sharp peak, a vertex is often a local extreme, and it gets a large positive mean. On a saddle-shaped part, it is often a pass, and it gets a negative mean. On a flat part, it has two changes in every direction, for a mean of nought. That sounds like curvature, and it is.

The horizontal coordinate is the vertex’s angle defect, $2\pi$ minus the sum of the triangle angles there, divided by $2\pi$. That was computed from dot products of edge vectors, with no reference to any direction of height. The vertical coordinate is an average over 60,000 random directions, computed only from comparing heights. The two have nothing in common as procedures. And they agree to within 0.004 at all 306 vertices, which is about the sampling error of an average of that many values. Banchoff’s 1970 theorem says they are exactly equal. The proof is a small piece of spherical geometry. For each triangle at a vertex, the set of directions for which its two other corners lie on opposite sides is a pair of opposite lunes on the sphere, with total area set by the triangle’s angle at that vertex, and adding over the triangles gives the defect.

This gives the discrete Gauss–Bonnet theorem at once. Sum over vertices, and use the first theorem:

$$\sum_v \frac{\text{defect}(v)}{2\pi} = \sum_v \text{mean}\ i(v) = \text{mean} \sum_v i(v) = \chi.$$

So all defects total $2\pi\chi$, and this holds on any polyhedral surface, as it was tested to 12 digits on the five surfaces here. Descartes’s theorem on the corners of a solid is the case of a sphere. A very deep fact of geometry has been reduced to an average over simple comparisons. The right way to see it might be that curvature does not have to be thought of as something that lives at a point in any intrinsic way. It can be the density of critical points seen from all directions. There is also a real-world way of seeing it. The Gauss map of a smooth surface sends each point to its normal, and the points whose normal is a given direction are exactly the critical points of the height in that direction. So the count of how many times the Gauss map covers the sphere, with signs, is a count of critical points with signs, and it is this that the integral of Gauss curvature records.

Local rules with a global sum #

The argument has a family resemblance to several others. A triangulated surface’s vertices can be given another local score, one minus a sixth of the number of triangles there, and those scores also add to $\chi$. That is the same fact as the twelve pentagons a soccer ball must have, and it is a combinatorial curvature, in which a vertex with six triangles is flat. A third local score comes from angles, as above. So there are at least three different ways to spread $\chi$ over a surface’s vertices: by angles, by numbers of triangles, and by comparisons of height. They give different numbers at a given vertex, and they all have the same total. Only one of them depends on how the surface sits in space, in the sense that a change of height direction alters the local score. Yet averaging over all directions gives back a quantity that depends only on angles within the surface. That is a discrete version of Gauss’s remarkable theorem, the intrinsic nature of curvature, arriving by a surprising route: a thing defined using space, averaged over all ways of looking at it, does not depend on how space was involved at all.

Rising water #

Morse’s picture is a flood. Raise the water level across the surface and watch the submerged part. It is empty at first, and at a pit a new pool appears, which adds 1 to its Euler characteristic. At a pass, either two pools join or a pool closes up round a loop, and each subtracts 1. At a peak the last hole is capped, adding 1. On the torus here, the water starts with five pools in its lower reaches, joining as the level rises, and it is an annulus at around height nought with characteristic nought. It closes up through the upper half. The curve starts and ends at 0, as $\chi$ of the torus requires.

The sequence of changes contains more than the total. It is a recipe for building the surface one cell at a time: a cell of dimension 0 at each pit, of dimension 1 at each pass and of dimension 2 at each peak. This is the central idea of Morse theory, which Marston Morse developed from the 1920s. It yields the Morse inequalities: the number of critical points of index $k$ is at least the $k$-th Betti number. So a torus, with two independent loops, must have at least two passes, one pit and one peak, for four critical points altogether, in any direction and on any shape. In 1956 John Milnor used a refinement to prove that there are spheres with more than one smooth structure, so this flood picture leads directly to some of the deepest results in topology. Robin Forman’s discrete Morse theory of 1998 is another descendant. In practice it is used to simplify large meshes and data sets, and related techniques appear in counting targets from sensor networks.

The fewest critical points #

The Morse inequalities raise a natural question: can a shape reach the minimum in every direction at once? A convex solid can, with exactly one peak and one pit for every direction, and that is a good way of defining convexity for a surface. The cube and icosahedron have exactly 2 in all 4,000 directions tested. The regular torus drawn here has exactly 4 in all of them. Such shapes are called tight, and they are the most economical possible shapes for their topology: they have as little total curvature, in absolute value, as it is possible to have. Shiing-Shen Chern and Richard Lashof’s theorem of 1957 states the same thing. The average number of critical points over all directions is the total absolute curvature, up to a constant, and it is at least the sum of the Betti numbers. Equality holds exactly for tight surfaces.

The lumpy torus has between 4 and 38, with an average of well above the minimum. So curvature that is signed sums to a fixed value, and unsigned curvature is what measures lumpiness. There is a well-known application to knots: John Milnor and István Fáry showed in about 1950 that a knotted curve in space has total curvature of at least $4\pi$, in a theorem that has exactly this form, since a knotted curve’s height function must have at least two peaks in every direction. Nicolaas Kuiper studied tight surfaces at length, and he found that most closed surfaces can be made tight in ordinary space. The Klein bottle cannot, by a short argument, and the same goes for the projective plane. So there is no shape in space for which the Klein bottle attains its minimum critical points in every direction.

What these pictures miss #

A height function here must be generic: if two adjacent vertices have equal heights, the index is not defined. Random directions avoid this with probability one, but a surface placed specially relative to a direction needs special handling. Non-orientable surfaces and self-intersections need only a little care for the count, since the proof uses nothing about embedding, except for the Klein bottle remark above, which is about being tight. The picture of a flood is a sketch. The surface is not really a landscape, and the interpretation of each change as joining pools or closing a loop was read from the theory and not from the data.

Sampled averages are limited, too. A difference of 0.004 is at the level of sampling error for 60,000 directions, and this is not a proof of the equality. The proof is the short piece of spherical geometry mentioned, and it was not checked in formal detail here. The drawn torus’s 4 in all 4,000 directions is a sample, and the claim of tightness in every direction would need an argument. A regular polyhedral torus with 240 vertices might exhibit a slightly higher count in very rare directions.

What was computed and what was quoted #

The index sums were computed exactly as integer counts in 300 directions on each of five surfaces in a direct test, and in 3,000 more for the figure, and they never failed. Defects came from the triangles’ angles, and their sums were checked to 12 digits. The mean index was computed over 60,000 seeded directions. The sweep used one direction. The minimum and maximum critical counts used 4,000 directions on each surface, and the true extremes could be a little wider than those shown. The proof of the sum is given in full above.

Quoted: Cayley’s and Maxwell’s papers, Banchoff’s two theorems, Morse’s work, the theorems of Chern and Lashof, Milnor and Fáry, Kuiper, Milnor’s exotic spheres and Forman’s discrete theory.

Still open: from a polyhedron back to a smooth surface #

How close does discrete curvature come to smooth curvature, in general? Angle defects are a fine definition of curvature for a mesh, and for a good triangulation of a smooth surface, the defect over a suitable area does approach the true curvature. For a poorly chosen one it does not, even when the triangles are small, and the question of which triangulations are suitable has been studied at length. More active is the question for higher dimensions, in which the analogue of an index sum over vertices has been studied by Banchoff and others, and in which discrete curvatures of many kinds compete, none of them doing everything that smooth curvature does. Forman’s Ricci curvature and Ollivier’s are the best known, and how far they capture geometric facts is open.

For random surfaces there are sharper open questions, on which the essay about random discs gave one view. The number of critical points of a random smooth function on a sphere of high dimension is an active topic in the theory of spin glasses. Antonio Auffinger, Gérard Ben Arous and Jiří Černý computed its exponential growth rate by index, and showed a striking layered structure in the energy levels at which critical points of each index can live, in 2013. How to take this down to actual algorithms that seek out low-energy states is still open.

A count of who is higher #

On any closed surface of triangles, and for any generic height, each triangle has one middle vertex, so the numbers of changes round the rings total the number of faces, and that makes the sum of $1 - (\text{changes})/2$ the Euler characteristic. Averaged over all directions, a vertex’s index becomes its angle defect, and that proves Gauss–Bonnet.

The contrast between the two things being equated is the whole charm of it. On one side there is a calculation with angles, using lengths and inner products. On the other there is nothing but comparisons, which of two heights is greater, in direction after direction. Maxwell saw the first form of this looking at a map of hills and dales. That a whole body of geometry was waiting inside the observation that every new hill brings its own pass is the kind of thing that a count can do, and that a picture can show.