Matpic
mathematics, in pictures
Concept

Equilateral triangle

A triangle with three equal sides, and therefore three angles of 60 degrees. It is the shape that appears whenever a construction is forced to be symmetric under a third of a turn, which is why it answers questions about angle trisectors, about tilings and about the densest packing of circles.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Trisect every angle, and an equilateral triangle appears. A triangle with angles 78°, 54°, 48°, its six angle trisectors, and the triangle whose corners are where the trisectors nearest each side meet. That inner triangle is equilateral, which is Morley's theorem.

Three trisectors and a triangle nobody expected

Cut every angle of a triangle into three. The trisectors nearest each side meet in three points, and those three points are always the corners of an equilateral triangle — for every triangle there is, with no exceptions and no reason anybody finds obvious.

geometry · Morley Conway's seven pieces, pulled apart and measured. A triangle with angles 78, 54, 48 degrees divided along its angle trisectors into an equilateral centre, three pieces touching it along its sides and three along the outer sides, spread apart, with each corner's angle labelled as a third of an outer angle plus a multiple of sixty degrees.

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.

geometry · Morley Twenty-seven choices of trisector, and the eighteen equilateral triangles. Twenty-seven small panels, one for each choice of trisecting line at each corner of a triangle, each drawing the triangle and the triangle the chosen lines cut out; the eighteen equilateral ones are marked.

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.

geometry · Morley The point nearest in total to three corners, and the lines that find it. A triangle with equilateral triangles on its sides, the three lines from each corner to the opposite apex meeting at the Fermat point, where the sides subtend 120°; least total distance 7.7090.

The point nearest in total to three corners

Three towns want one depot with the least total road to all of them. The answer is not the centroid or any of the classical centres: it is the point from which every side of the triangle is seen at 120°, found by building equilateral triangles on the sides. Three straight lines through it have the same length, and that length is the least total. When an angle of the triangle reaches 120°, the answer jumps to that corner — and the same 120° rule shapes soap films and the shortest networks joining many points.

geometry · Triangle centres Seven towns joined with junctions and without. Seven random points: spanning tree 1.65407, Steiner tree 1.54997 with 2 junctions, ratio 0.93707.

What a junction can save

Join a set of towns by roads that may meet only at towns, and the shortest such network is easy to find. Allow junctions anywhere and the network gets shorter — but never, it is conjectured, by more than 13.4%, the saving the Fermat point makes on an equilateral triangle. Computed exactly on 1,304 random sets of up to seven towns, the average saving is about 3%. An adversarial search drives it down to the bound and no further, and a hexagon with its centre attains it exactly. The proof of the bound that was accepted for twenty years has a gap.

geometry · Triangle centres

Named alongside it

The objects these essays reach for when they reach for this one.

Angle trisectionExhaustive searchMorley theoremOptimisationSteiner treeTriangle centresTrigonometryAngleCircumradiusConjectureConstructionConvexity

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