Seven hundred and twenty degrees of gap

Take a corner of a cube and unfold the three squares meeting there onto a flat page. Three right angles is $270°$, and a full turn is $360°$, so the unfolded corner leaves a wedge of $90°$ open. The gap is what makes it a corner: a point where the faces close up exactly is not a corner at all, but a flat patch with a crease drawn on it.

Eight corners at $90°$ each is $720°$: two full turns. That is not a fact about cubes.

Four corners of $180°$, eight of $90°$, twenty of $36°$. Twelve corners of $60°$ on the icosahedron, six of $120°$ on the octahedron. Every one of them totals two turns, and so does every irregular solid, every squashed one, every one with a hundred faces of no particular shape.

The quantity being added up #

The gap at a corner has a name — the angle defect — and a definition that needs no regularity: at a vertex $v$, take the angles of all the faces meeting there and subtract their sum from a full turn.

$$\delta(v) = 360° - \sum_{\text{faces at } v} \text{angle}.$$

For a convex solid every defect is positive, because the faces at a corner must leave room to fold up. Descartes’ theorem says

$$\sum_{v} \delta(v) = 720°,$$

whatever the solid is.

That total is fixed even though nothing else is. Squashing a cube changes every one of its eight defects and leaves the sum alone. Slicing a corner off replaces one vertex by three, redistributing its defect among them — and the arithmetic works out exactly, which is worth checking on an example rather than believing.

Cut the corner off a cube with a plane through the midpoints of the three edges meeting there. The vertex worth $90°$ is gone. In its place are three new vertices, and at each of them two of the cut squares — now pentagons — meet the new triangular face. A square cut across its corner has an angle of $135°$ at each cut end, and the triangle contributes $60°$, so the faces at a new vertex total $135° + 135° + 60° = 330°$ and its defect is $30°$.

Three new vertices at $30°$ is $90°$, which is exactly what the removed vertex was worth. The other seven corners of the cube were not touched. The total is $720°$ before the cut and $720°$ after it, and no arrangement was made to bring that about.

Why it is the same theorem as $V - E + F = 2$ #

The two look unrelated. One is a count of combinatorial pieces; the other is a sum of angles measured in degrees. They are the same statement, and the translation is four lines of arithmetic.

Add up every face angle of the solid, in two ways.

First, corner by corner. At each vertex the angles sum to $360° - \delta(v)$, so over the whole solid

$$\text{total angle} = 360°V - \sum_v \delta(v).$$

Second, face by face. A face with $k$ sides has interior angles summing to $(k-2)\cdot180°$, so

$$\text{total angle} = 180°\sum_{f}(k_f - 2) = 180°\left(\sum_f k_f\right) - 360°F = 180°(2E) - 360°F,$$

using the fact that every edge is shared by two faces, so the side counts add to $2E$.

Setting the two equal:

$$360°V - \sum_v \delta(v) = 360°E - 360°F \quad\Longrightarrow\quad \sum_v \delta(v) = 360°(V - E + F).$$

So the total defect is $360°$ times the Euler characteristic, and $720°$ is what that comes to when the characteristic is $2$. On a solid with a hole, where the characteristic is $0$, the defects cancel to nothing — the inner corners of the hole have negative defect, more than $360°$ of face crowding into them, and there is exactly enough of that to cancel the outside.

The proof above is a counting the same thing two ways argument, which is the most reliable trick in the subject, and its two ways are the two halves of the alternating sum. The vertex tally produces $V$; the face tally produces $E$ and $F$. The characteristic falls out because the same pile of angles was sorted into different heaps.

Why the list of solids stops at five, again #

Descartes’ theorem gives a one-line proof of a fact this collection has already met from the other side.

A regular solid with $V$ corners has the same defect $\delta$ at each, so $V\delta = 720°$, so $\delta = 720°/V$ — and since $\delta$ must be positive and is determined by the face polygon and the number meeting at a corner, the whole enumeration follows from the few positive values available. The tetrahedron’s $180°$ gives $V = 4$; the cube’s $90°$ gives $8$; the icosahedron’s $60°$ gives $12$; the dodecahedron’s $36°$ gives $20$.

That is the same argument as the angle budget with the conclusion read in the other direction. The budget argument says which corners can exist; Descartes’ theorem says how many of them a solid must have, which is $720°$ divided by the gap each one leaves. A solid with a $1°$ gap at every corner needs seven hundred and twenty corners.

The dual, and where the gap goes #

A cube has eight corners of $90°$; an octahedron has six of $120°$. Both come to $720°$, and the two solids are duals of each other — swap vertices for faces and one becomes the other.

Duality swaps $V$ and $F$ and leaves $E$ alone, so it leaves $V - E + F$ alone, so it leaves the total defect alone — while redistributing it completely. The dodecahedron has twenty corners of $36°$ and its dual the icosahedron has twelve of $60°$, and $20 \times 36 = 12 \times 60$.

Nothing above is a coincidence of the regular solids. Duality is a general operation on polyhedra, it preserves the characteristic, and the total defect is the characteristic in disguise, so any dual pair distributes the same two turns differently. That is a fair test of whether a quantity is combinatorial or geometric: the individual gaps are geometric and move; their sum is combinatorial and does not.

Curvature, concentrated #

The most useful reading of the defect is not as a leftover angle but as curvature that has been squeezed into a point.

A sphere is curved everywhere and a polyhedron is flat everywhere except at its corners — along an edge it is flat too, since an edge unrolls onto the page without stretching. All the bending is at the vertices, and the defect measures how much bending is there.

Round off the corners of a cube smoothly and the total curvature over each rounded region is exactly the $90°$ that used to be a gap, in the sense the smooth theory measures curvature. Round them all off and keep going until the cube is a sphere, and the total is still $720°$ — which is $4\pi$ steradians, the total curvature of any sphere of any size.

That is Gauss–Bonnet, and Descartes’ theorem is its polyhedral case:

$$\int_S K \, dA = 2\pi\chi(S).$$

The total curvature of a closed surface does not depend on its shape at all, only on its characteristic. Squash it, dent it, stretch it into a pear: the curvature moves around and the total does not change, because the total is counting handles rather than measuring shape.

The theorem that was lost for two centuries #

Descartes wrote this down around 1630, in a short manuscript called De solidorum elementis, and it was not published in his lifetime. The manuscript passed to Leibniz, who copied it out in Paris in 1676; the original was subsequently lost — reportedly in a shipwreck while Descartes’ papers were being moved — and Leibniz’s copy sat unnoticed in Hanover until 1860, when Foucher de Careil found and published it.

By then Euler’s formula had been known for a century, and von Staudt had proved it properly thirteen years earlier. So the earlier theorem was published second, and arrived as a curiosity attached to a subject that had already grown past it.

There is a question worth asking about that gap, and the honest answer is nobody knows: whether Descartes saw that his angle sum was equivalent to a count of vertices, edges and faces. The manuscript contains a table of $V$, $E$ and $F$ for several solids, and the numbers in it satisfy $V - E + F = 2$, and Descartes did not write the relation down. He was two lines of arithmetic away from Euler’s formula and either did not take them or did not think them worth recording.

The moral usually drawn is about publication, and there is a better one about statement. The reason Euler’s version became the theorem and Descartes’ did not is that the alternating sum names a quantity — the characteristic — that turned out to belong to the surface, generalise to every dimension, and survive into homology. The angle sum is the same information tied to a particular geometric realisation, and a quantity is more useful the fewer things it is attached to.

Carrying a direction around a corner #

There is a second way to feel the defect, and it is the one that makes the word curvature deserved.

Walk a small loop on the surface of a polyhedron, carrying an arrow and keeping it pointing in a fixed direction as far as the surface allows — never turning it deliberately, only sliding it flat across each face and over each edge. Around a loop that stays on one face, or crosses edges but encircles no corner, the arrow comes back exactly as it left.

Around a loop that encircles a corner, it does not. It comes back rotated, and the angle it has turned through is precisely the defect at that corner. A cube’s corner turns the arrow by $90°$; a tetrahedron’s by a half turn.

This is holonomy, it is a measurement made entirely on the surface without any reference to the space the solid sits in, and it means a creature living on the surface could find every corner and measure every defect without ever seeing the solid. Adding the results up, that creature obtains $720°$ and therefore knows it is living on something sphere-like rather than on something with a handle — a global fact about the shape of its world, assembled from local measurements it can make with a stick and some patience.

It is also the reason the total is what it is. Walking a loop that gradually expands until it has swept over the whole surface returns to where it started having turned by the total defect, and a loop that has swallowed the entire closed surface must have turned by a whole number of full turns. That the whole number is two, rather than one or three, is the same $\chi = 2$ arriving by a third road.

Where the theorem needs a condition #

Descartes’ theorem as stated above needs the solid to be convex — or at least to have all its defects positive — for the sum to be a sum of positive things. The theorem itself does not: the identity $\sum \delta = 360°\chi$ holds for any polyhedral surface, with negative defects allowed at vertices where more than a full turn of face crowds in.

Those vertices are easy to build and unsettling to look at. Take six equilateral triangles and glue them into a fan at a point: they close up flat, defect zero. Take seven and the surface will not lie flat — it ruffles, like a lettuce leaf, and its defect is $-60°$. A surface made entirely of such vertices is a discrete model of the hyperbolic plane, and it is why a crocheted hyperbolic plane frills.

So the condition is not really convexity. It is that the count be finite and the surface be closed. Drop closed and the theorem fails in the obvious way: a flat sheet has defect zero at every interior vertex and a great deal of missing boundary, and no total is meaningful until the boundary is accounted for — which is what the general Gauss–Bonnet theorem does with an extra integral round the edge.

What the picture cannot show #

The unfolded corner in every figure here is a lie of a specific and unavoidable kind. The faces are drawn lying flat on the page with a gap between the first and the last, and on the actual solid there is no gap: the first and last faces are glued along an edge, and the picture has cut that edge open to lay them out.

So the gap in the drawing is not a hole in the solid. It is the amount by which the solid refuses to be drawn flat, which is a property of the folding rather than of the object, and the figure represents it by an absence — which is the one thing a picture is worst at.

Nor can any figure show the sum. Each shows one corner of one solid; the theorem is about all the corners at once, and the total appears only as a number in a table. The site’s usual claim — that the picture is the proof — is weaker here than elsewhere: the picture shows what a defect is, and the argument that the defects add to two turns is the arithmetic in the section above, which no drawing contains.

The ladder from here #

Below: the alternating sum itself and the two-tree proof of it. Above: the solids where the total is not $720°$, and beyond the ladder, Gauss–Bonnet in its smooth form, where the same statement is an integral.

Sideways: the defect is the discrete curvature that a triangle on a sphere measures continuously, it is what makes the list of regular solids finite, and it is the quantity that decides whether a cone made from a sector of paper is a cone at all.

Two turns, and nothing else #

The lasting point is what the number $720°$ is doing.

It is not a fact about angles. It is $360°$ multiplied by a count of combinatorial pieces, dressed in degrees, and the reason a sum of measured quantities comes out at a fixed number is that the measurement was never independent of the counting. Every angle in the solid appears once in the vertex tally and once in the face tally, and the difference between the two tallies is forced.

This is the mechanism behind a whole class of results that look like coincidences: a geometric quantity that cannot be moved, because some combinatorial quantity underneath it cannot be moved. The winding number of a curve, the parity that decides whether a puzzle is solvable, and the total curvature of a surface are all of this shape — a measurement that turns out to be a count, and is therefore stuck at a whole number times a constant.