The trisectors on a sphere almost agree

Three trisectors and a triangle nobody expected stated Morley’s theorem: cut each angle of a triangle into three, and the trisectors nearest each side meet in the three corners of an equilateral triangle. Seven pieces and an equilateral middle gave three proofs and ended by observing that the theorem fails on a sphere and in the hyperbolic plane, where the angles of a triangle do not add to a half turn — so any explanation has to use flatness somewhere, and every known proof uses it only through the equation $\alpha + \beta + \gamma = 60°$ for the thirds of the angles.

This essay draws the failure. On a sphere the construction still makes sense: the sides of the triangle are arcs of great circles, the angles between them can be measured and cut into three, and the trisectors are arcs of great circles that meet. The question is what the triangle in the middle looks like when the theorem no longer applies. The answer is surprising in a direction the statement does not prepare for. The middle triangle is not equilateral, and it is very nearly so.

The construction on a sphere #

Every step of the construction is exact on the sphere, and each is checked. A point is a unit vector. A great circle is the set of unit vectors perpendicular to some fixed vector, its pole. The angle of a triangle at a corner is the angle between the directions in which the two sides leave it, measured in the plane that touches the sphere there. A trisector is the great circle that leaves the corner at a third of that angle from one side, and two great circles meet at the point whose direction is perpendicular to both poles. For every corner of every inner triangle drawn here, the angle it makes at the vertex is checked to be a third of the vertex’s angle to nine decimal places.

The triangle in the first figure is large. Its angles are 76.1°, 97.3° and 48.8°, which add to 222.2° — the sphere’s triangles add to more than 180°, by an amount proportional to their area, and this one covers about a ninth of a hemisphere. Its trisectors cut out a triangle with sides of 0.23902, 0.23881 and 0.23902 radians. Two happen to agree to five places; the third is shorter by about nine parts in ten thousand. The middle triangle is not equilateral. A drawing could never show it.

The same in the hyperbolic plane #

The hyperbolic plane is the surface on which triangles add to less than 180°, by an amount proportional to their area. It can be computed with the same linear algebra as the sphere, using vectors on a hyperboloid instead of a sphere and a form with one minus sign in place of the dot product, and drawn in the Poincaré disc, where the whole infinite plane is shrunk into a disc and its straight lines appear as arcs meeting the rim at right angles.

The disc is the hyperbolic plane’s counterpart of the stereographic map of the sphere, and it shares that map’s property of keeping every angle exactly while distorting every length: the trisectors in the drawing really do cut the drawn angles into thirds, although sides that look short near the rim are long in the plane itself. That is why the inner triangle looks small and lopsided in the picture while its true sides, computed on the hyperboloid, are nearly equal.

The triangle in the figure has angles of 39.3°, 75.1° and 18.3°, together 47.4° short of a half turn. The trisectors cut out a triangle whose sides differ by seven parts in ten thousand. The theorem fails in the hyperbolic plane exactly as it does on the sphere: in the opposite direction, and by about as little.

How far, against how curved #

The two examples suggest a rule, and the next figure measures it. Three shapes of triangle — one nearly equilateral, one scalene, one obtuse isosceles — are each drawn at twelve sizes on the sphere and twelve in the hyperbolic plane, and for each the spread of the inner triangle’s sides is plotted against how far the angle sum is from 180°.

Every curve passes through nought at the middle line, where the angle sum is exactly 180°, and leaves it in proportion to the excess or the shortfall. For small triangles the spread is a fixed multiple of the angle excess, and the multiple depends on the shape: about four ten-thousandths per radian of excess for the nearly equilateral triangle, a little over one thousandth for the scalene one, and a few thousandths for the obtuse isosceles triangle, whose inner triangle, by symmetry, has two equal sides and a third that drifts. Even the most extreme triangles drawn, with angles adding to more than 300°, have inner triangles within about half a percent of equilateral.

That makes Morley’s theorem an unusual kind of flat-geometry fact. Many statements of plane geometry fail badly on a sphere: the angle sum itself, Pythagoras’s theorem, the existence of similar triangles of different sizes. Morley’s statement fails, but the quantity it describes changes only in its third or fourth significant figure. The failure is real and it is never large.

Where flatness enters the rotation proof #

The second proof in seven pieces and an equilateral middle is Alain Connes’s, and it is built from rotations. Let $g_A$ be the rotation about the corner $A$ through two-thirds of the angle at $A$, and likewise $g_B$ and $g_C$. Two facts about them drive the proof.

The first is that their cubes compose to nothing: $g_A^3 g_B^3 g_C^3$ is the identity. The cube $g_A^3$ is the rotation about $A$ through twice its angle, which is a reflection in one side through $A$ followed by a reflection in the other; doing the same at $B$ and $C$, the reflections cancel in pairs. The second is that each product of two rotations fixes one corner of the inner triangle: $g_A g_B$ holds the corner nearest the side $AB$ still, because $g_A$ is a reflection in side $AB$ followed by a reflection in the trisector at $A$, $g_B$ is the same with the order reversed at $B$, and the two reflections in $AB$ cancel, leaving the product of reflections in the two trisectors — a rotation about the point where they meet.

Neither fact uses flatness, and the figure checks both on the sphere with rotation matrices about the three corners: the composed cubes differ from the identity by less than $10^{-15}$, and each product moves its corner by less than $10^{-15}$. Both arguments are about reflections in lines through points, and reflections in great circles through points of the sphere behave exactly the same way.

What does not survive is the last step. In the plane, rotations are maps $z \mapsto a z + b$ of the complex numbers, multiplication by a number of length one followed by a shift, and the “turning part” $a$ of a composite is the product of the turning parts. Connes’s lemma says that if three such maps have cubes composing to the identity and the product of their turning parts is a primitive cube root of unity, then the fixed points of the three pairs form an equilateral triangle. The product of the turning parts of $g_A$, $g_B$ and $g_C$ is a turn through two-thirds of the angle sum, and that is a third of a turn — a primitive cube root of unity — exactly when the angle sum is 180°. On the sphere there is no turning part to multiply: rotations about different points do not split into a turn and a shift, and the lemma has nothing to apply to. The figure’s third column records the one number that differs, and the last column the consequence.

Conway’s pieces cannot be assembled on a sphere #

The other proofs fail at other places, and seeing where is a second way of locating flatness. John Conway’s proof builds the answer instead of computing it. Start with an equilateral triangle, attach to its sides three triangles with angles chosen from the thirds $\alpha$, $\beta$, $\gamma$ and sixty degrees, and three more outside those, seven pieces in all; check that the angles around every interior point add to a full turn and that the pieces fit along their shared edges; and the assembled figure is a triangle with angles $3\alpha$, $3\beta$ and $3\gamma$ whose inner triangle is equilateral by construction. Since every triangle with those angles is a scaled copy of this one, every triangle has an equilateral Morley triangle.

The last sentence is where the sphere objects. In the plane a triangle’s angles fix its shape and leave its size free, so each of Conway’s pieces can be built at whatever scale makes its edges match its neighbours’. On a sphere a triangle’s angles fix it completely, size included — there are no similar triangles of different sizes, because the angle excess is the area — and the seven pieces, each determined by its angles, will not in general have matching edges. The assembly does not close up. The pieces exist and their angles are right; they are simply the wrong sizes to fit.

The trigonometric proof fails more quietly. It computes each side of the inner triangle from the circumradius and the sines of the angle thirds, using the plane’s sine rule and the identity for $\sin 3\theta$, and the three sides come out as the same symmetric product. The sphere’s sine rule relates the sines of the sides rather than the sides, and the computation carried out with it produces three expressions that are close to each other — the figures above show how close — and not equal.

Reflections, which behave the same everywhere #

The step of Connes’s proof that survives deserves one more look, because it is a fact about mirrors that holds on every surface. A reflection in a line, followed by a reflection in a second line meeting the first at a point, is a rotation about that point through twice the angle between the lines. That is true in the plane, on the sphere with great circles as the lines, and in the hyperbolic plane with its own straight lines, and it is the whole content of both of Connes’s structural facts.

It is also the principle behind three mirrors make every solid, where three planes through the centre of a regular solid, meeting at angles of $\pi/2$, $\pi/3$ and $\pi/n$, generate every symmetry of the solid by repeated reflection: on the sphere around the centre the three mirrors cut out a triangle, and the products of reflections in its sides are rotations about its corners, exactly as here. In that setting the angles are chosen so that the rotations generate a finite group. Morley’s construction uses the same mirrors with arbitrary angles, and asks a different question of them; the mirrors answer it the same way on every surface, and the difference between surfaces shows up only when the rotations are added up.

Why so close #

The proof explains why the theorem fails off the plane but not why it fails by so little. The figures suggest an answer in terms of size. A small triangle on a sphere is nearly flat; its angle excess is its area on the unit sphere, and every quantity in the construction differs from its flat value by a correction proportional to that area. The spread of the inner triangle’s sides is one such correction, and the measurements show that it is a small multiple of the excess even for large triangles — a few thousandths per radian. The flat construction is, in that sense, unusually stable: the equal sides are not a coincidence that a small perturbation destroys, but a property that a perturbation of the geometry bends only slightly.

That fits the observation that every proof uses flatness only through the single equation for the angle sum. The construction itself — trisecting each angle, intersecting adjacent trisectors — is the same on every surface, and the two structural facts that make the inner triangle special survive intact. What changes is one number, the total turn of the three rotations, and the inner triangle responds to that number smoothly. A theorem whose hypotheses are local and whose conclusion depends on one global quantity can be expected to fail gently when that quantity moves.

The flat theorem, for comparison #

In the plane every triangle, however lopsided, has an exactly equilateral Morley triangle, and the trigonometric proof gives its side as $8R \sin\alpha \sin\beta \sin\gamma$, symmetric in the three thirds of the angles, where $R$ is the radius of the circle through the corners. On a sphere there is no circumradius formula of that shape — the sine rule changes, the law of cosines changes — and the symmetric expression has no counterpart that could force the three sides equal.

The eighteen equilateral triangles of eighteen equilateral triangles, which appear when the outside trisectors are included, depend on the cube roots of unity in the same way. Each of the three choices of trisector at a corner is a different cube root of the rotation through twice the angle, and the eighteen equilateral triangles are the choices whose turning parts multiply to a primitive cube root of unity. On the sphere the lemma has nothing to apply to for any of the eighteen at once, since the turning parts it multiplies do not exist there.

What the computations do not show #

The figures measure particular triangles, and the claim that the spread is always small is a description of what was found rather than a theorem. A triangle with one very small angle and a large excess could in principle behave differently, and the curves for the obtuse isosceles shape, which rise fastest, suggest that shapes with an angle close to 180° are the ones to watch. The hyperbolic curves are also less regular than the spherical ones, because large hyperbolic triangles have small angles, and trisecting a small angle gives trisectors that meet far from the corner.

The computations are exact in their construction and limited only by floating-point arithmetic, which here means errors around $10^{-15}$, far below the spreads of $10^{-4}$ to $10^{-3}$ being measured. The trisector check at every corner and the two rotation identities guard the construction; the spread is what is left.

Still open: what the curved triangle is #

On the sphere and in the hyperbolic plane the inner triangle exists and is nearly equilateral, and nobody has found a description of it as clean as the flat one. Is there a natural quantity — a weighted combination of its sides, or its angles, or some function of its area — that the construction makes exactly equal at the three corners on every surface of constant curvature, reducing to equal sides in the plane? Some curved generalisations of Morley’s theorem have been proposed, with trisectors replaced by other lines chosen so that an equilateral triangle results, but they change the construction rather than describing what the original one does.

And the size of the failure invites a sharper question. The spread grows in proportion to the angle excess with a coefficient that depends on the shape; what that coefficient is, as a function of the three angles, and which shape maximises it, could in principle be worked out from the first-order correction to the construction, and the measurements here only sample it. A formula for it would say precisely how close to equilateral a curved Morley triangle must be — a quantitative version of the flat theorem, measuring the price of curvature in the one place the proof says it is paid.