Matpic
mathematics, in pictures
Concept

Regular polygon

A closed figure whose sides are all the same length and whose angles are all the same size. Which ones are constructible is decided by the Fermat primes, which is Gauss's theorem and the reason a seventeen-sided one can be drawn.

Named by 11 essays across 3 fields — each of them below, with the objects they name alongside it.

Which regular polygons a compass and straightedge can draw, up to 100. A grid of the integers with the constructible ones filled in, each verdict computed two independent ways.

Which polygons can be drawn

Three sides yes, seven no, seventeen yes. The list of constructible regular polygons is neither everything nor almost nothing, and the pattern in it is a fact about which numbers are one less than a power of two.

computation · Constructible numbers Every relabelling of a 4-gon's corners, and the 8 that are motions. All 24 permutations of the corners drawn one by one, with the 8 that preserve every distance marked; the rest deform the polygon and are not symmetries.

Eight ways to leave a square alone

A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not asserted here — it is what a search through all twenty-four relabellings of the corners comes back with.

algebra · Symmetry groups One perimeter of 300, spent five ways. Regular polygons all of the same perimeter, drawn to scale beside the circle of that perimeter, with the area each encloses and the ratio 4πA/L².

The most area a fence can hold

One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.

geometry · Isoperimetric The 7 7th roots of unity. 7 points spaced evenly around the unit circle, at the vertices of a regular 7-sided polygon.

The polygon an equation forces

The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.

algebra · Roots of unity Solids with every corner alike and more than one kind of face. truncated tetrahedron, cuboctahedron, truncated cube, icosidodecahedron, each cut from a Platonic solid and drawn in projection; every edge in each is the same length and every vertex is surrounded by the same faces.

Thirteen more when one word is dropped

The list of regular solids stops at five because the definition asks for two things at once. Ask for only the second — every corner alike — and thirteen more appear, each of them cut off a Platonic solid at a depth found rather than chosen.

geometry · Regular polyhedra The star polygon {5/2}. 5 equally spaced points joined every 2th, forming a closed path that winds 2 times about the centre with an interior angle of 36.0 degrees at each point.

The four that are allowed to cross themselves

Drop convexity from the definition of a regular solid and four more appear. Their faces are pentagrams, they pass through one another, and the alternating sum that gives two for every ordinary solid gives minus six for two of them.

geometry · Regular polyhedra Fences of 300 against a straight wall. Fences made of two, three and four straight pieces with both ends on a wall, beside a half-circle with the same length of fence, each labelled with the area it holds.

Half a circle against a wall

Lay a fence of fixed length with both ends against a straight wall and the best shape is a half-circle, holding exactly twice what a full circle of the same fence holds. The proof is a mirror: doubled in the wall, any fence becomes a closed curve with twice the length and twice the area, and the closed-curve answer carries over. In a corner the same mirrors give a slice of a circle — until the corner's angle stops dividing a half-turn.

geometry · Isoperimetric Böröczky's 12 points and their 6 ordinary lines. A disc standing for the projective plane: the 6 corners of a regular polygon inside, and 6 points at infinity marked in pairs on the rim. All 22 connecting lines are drawn, the 6 ordinary ones solid.

The fewest ordinary lines a polygon allows

Take the corners of a regular polygon and add the points at infinity where its parallel chords meet. Every chord then carries three points, the line at infinity carries all the new ones, and the only lines left with exactly two points are the tangents at the corners — half as many as there are points. Dirac guessed in 1951 that nothing does better, and Green and Tao proved it in 2013.

geometry · Ordinary lines Six thirtieths of a turn that add to nothing. On the left a unit circle with the roots of unity in a vanishing sum marked and coloured by origin; on the right the same unit vectors placed head to tail, returning to their start.

The sums of roots of unity that add to nothing

All n of the n-th roots of unity add to zero, and so does any regular polygon among them, turned. Those are not the only vanishing sums: six thirtieths of a turn close into a loop with no polygon in them. Which counts of roots can close at all is decided by the prime factors of n — no seven fifteenths ever add to zero — and the same question counts where the diagonals of a regular polygon cross.

algebra · Roots of unity Hippias's quadratrix, traced by two uniform motions. A unit square with a quarter circle, several positions of a turning radius and a falling horizontal line, their crossings, and the curve through them ending on the base at 2/π.

A curve that divides any angle

Let a radius turn at a steady rate while a horizontal line falls at a steady rate, both finishing together, and mark where they cross. The curve they trace turns heights into angles, so dividing a height — which a ruler and compass can always do — divides the angle in the same ratio. The same curve meets its base at 2/π of the side, a length from which a square with the area of a circle follows. It reaches what no marked ruler or conic can, and the ancient objection to it is exact.

computation · Neusis The lowest mode of the regular pentagon. Level lines of the first Dirichlet eigenfunction of a regular 5-gon; λ times area 18.9191.

Three sides and four are proved

Among all shapes of a given area, the disc has the smallest lowest eigenvalue. Among triangles it is the equilateral one, among quadrilaterals the square — both proved by sliding chords to an axis. For five sides the regular pentagon wins every computation, every nudge raises its value by the square of the nudge, and there is still no proof.

geometry · Isoperimetric

Named alongside it

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

Cyclic groupSymmetryAreaCircleConstructible numberConvexityCyclotomic polynomialDihedral groupDualityEuler characteristicOptimalityPlatonic solids

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