Angle trisection
Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.
Seven pieces and an equilateral middle
Morley's theorem has two proofs worth knowing, and they run in opposite directions. The trigonometric one starts from the triangle and computes each side of the inner one as 8R sin α sin β sin γ, symmetric in the three angles. Conway's starts from an equilateral triangle, builds six pieces round it from their angles alone, and shows they fit — so the triangle they make is whatever triangle was wanted, and its middle is equilateral because it was built that way.
Eighteen equilateral triangles
Every angle of a triangle has three trisectors, not one, once the angle and its outside are both counted. Choosing one at each corner gives twenty-seven ways to cut out a triangle, and eighteen of them give an equilateral one. The nine that fail are exactly the choices whose labels add to 2, 5 or 8 — and all eighteen equilateral triangles have their sides in the same three directions, fixed by a third of the difference between two angles.
The spiral that measures its own circle
Let a ray turn steadily while a point moves steadily out along it, and the point draws Archimedes' spiral. Distance from the centre is then proportional to angle, so dividing a length divides an angle in any ratio. The spiral's second power is stranger: the tangent at the end of the first turn cuts off, on a line through the centre, a straight length exactly equal to the circumference of the circle through that end — a curved length laid out straight, and with it the circle squared. The catch is the tangent itself.
A third reached only in the limit
No compass-and-straightedge construction trisects every angle. But a quarter, plus a quarter of a quarter, plus a quarter of that, and so on, adds up to a third — and each of those pieces is two bisections away. So an angle can be trisected by bisecting forever, every stage exact and the shortfall shrinking to a quarter each time. The construction never ends, and that is precisely what Wantzel's proof forbids: a construction is a finite thing, and a third of a general angle is only reached in the limit.
The trisectors on a sphere almost agree
Draw Morley's construction on a sphere and the triangle the trisectors cut out is not equilateral — but its sides agree to within a tenth of a percent even when the angles add to 222°. Every step of the rotation proof survives on the sphere except one, and that one is the angle sum.
Named alongside it
The objects these essays reach for when they reach for this one.
Morley theoremConstructible numberEquilateral triangleTranscendenceTrigonometryAngle sumArchimedean spiralBinary expansionCircumradiusExhaustive searchGeometric seriesHyperbolic geometry