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mathematics, in pictures
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Things that cannot be done — page 1

Results that close a door rather than open one — and the peculiar difficulty of drawing a picture of something that does not exist.
The five Platonic solids. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, drawn at a common scale. Geometry

Why the list of perfect solids stops at five

There are infinitely many regular polygons and exactly five regular solids. The reason is not deep, but it is very sharp, and it can be checked on a single row of corners.

 Multiplying two complex numbers. In the complex plane, multiplying adds the two angles and multiplies the two lengths. Algebra

Multiplying is turning

Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.

 13 into 12. 13 items spread as evenly as 12 boxes allow. Even at their most even, some box holds 2, because 13 is more than 12 × 1. Discrete

More things than boxes

If there are more objects than containers, some container holds two. That is the entire principle, it is impossible to disagree with, and it settles questions that look nothing like it.

 A wheel of 5 rim regions needs 4 colours. A hub touching 5 rim regions arranged in a ring. The rim is odd, so the whole map needs 4 colours and no fewer. Discrete

Four colours, and a proof nobody can read

Every map on a plane can be coloured with four colours so that no two neighbours match. The statement is understandable by a child, it resisted a century of attempts, and the proof that settled it cannot be checked by a human being.

 A map of the interval must fix a point. a continuous map of the interval, drawn with the diagonal. Every continuous map of the interval into itself meets the diagonal somewhere; this one does so at x = 0.6944. Topology

Something always stays put

Stir a cup of coffee however violently and let it settle. Some molecule is exactly where it started. Crumple a map and drop it on the region it depicts, and one point lies over the place it names.

 The same terms, with the signs alternating. The partial sums of 1 - 1/2 + 1/3 - 1/4 + …, out to 24 terms. They close on 0.69315 from both sides at once, and the gap between consecutive sums is the next term, so the answer is trapped. Analysis

The same terms, in a different order, adding to whatever is asked

Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.

 Six people, and the trio that cannot be avoided. The fifteen pairs among six people, coloured at random. Whatever the colouring, three people are all mutual acquaintances or all mutual strangers — here 1, 2, 5. Discrete

Six people at a party

Among any six people, three are mutual acquaintances or three are mutual strangers. Five is not enough, and the arrangement that saves five is a pentagon. Beyond that the numbers become unknowable.

 the trefoil. the trefoil, drawn as a closed curve with 3 crossings. At each crossing the strand passing underneath is broken, which is the only information the flat picture carries that the curve alone does not. Topology

Three moves, and what they cannot undo

A knot is a closed loop of string, and two knots are the same if one can be wiggled into the other. Reidemeister reduced all possible wiggling to three local pictures — which is what makes it possible to prove that a knot is knotted.

 The field that cannot be combed. A tangent field on the sphere, flowing along the meridians. Every arrow is tangent to the surface, and at the two poles there is no direction for an arrow to take — the field is zero there, and no rearrangement removes both zeros. Topology

Nothing on a sphere can be combed flat

Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.

 8 multiples of φ in 7 boxes. The fractional parts of the first multiples of a number, dropped into equal boxes along the unit interval. Number

How close a fraction can get

Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.

 28 is perfect, because its divisors form this rectangle. Two rows of divisors: the powers of two, and the same powers multiplied by the Mersenne prime. Number

Numbers that are their own parts

Six is one plus two plus three. Twenty-eight is one plus two plus four plus seven plus fourteen. Euclid explained where such numbers come from; Euler proved there are no others of that kind; and whether an odd one exists has been open for two thousand years.

 Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked. Number

The square that cannot shrink

The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.

 Two orbits of the logistic map at 3.9, started 0.0001 apart. Two sequences from almost the same starting point, plotted together against the step number. Dynamics

A difference too small to draw

Two starting points a ten-thousandth apart, under the same rule, with nothing random anywhere. Within forty steps they have nothing in common — and the rule was not doing anything to them that it does not do to everything.

 Elementary cellular automaton, rule 110. A row of cells evolving downward, each cell decided by the three above it. Dynamics

The rule that computes

One of the 256 elementary rules can run any program. Not simulate one, not approximate one — a machine that can compute anything computable, built from a lookup table with eight rows and nothing else.

 Rotating by √2 − 1 of a turn, 40 times. Points on a circle produced by repeatedly turning through the same angle. Dynamics

The orbit that must come back

A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

 The Lorenz attractor at ρ = 28. A trajectory of the Lorenz equations, projected onto two of its three coordinates. Dynamics

Two lobes and no cycle

Three equations, three variables, and a trajectory that never crosses itself, never repeats, and never leaves a region of zero volume. The set it settles onto is not a point, not a loop, and not a surface.

 The Collatz orbit of 27. Halve an even number, triple an odd one and add one; the sequence plotted on a logarithmic scale. Dynamics

The question nobody can answer

Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.

 4 circles, and the 14 patterns they realise. Closed curves overlapping in the plane, with each region of the arrangement identified by which curves contain it. Logic

Four circles cannot do it

Three overlapping circles cut the plane into exactly the eight regions three sets need. Four circles cut it into fourteen, and sixteen are required — so the diagram everyone draws stops working at four, and the reason is a count.

 A line, a point, and many parallels. A disc whose lines are arcs meeting the boundary at right angles, showing several lines through one point that never meet a given line. Logic

Two worlds that both obey the rules

A statement is independent of a list of axioms when there is a structure satisfying the axioms where it holds and another where it fails. That is not a claim about what nobody has managed to prove — it is a proof that nobody can.

 The diagonal, and the row built to be off the list. A table of rows of ones and zeros with the diagonal marked, and beneath it the row obtained by flipping every diagonal entry. Logic

The row that is not on the list

Write down a list of infinite sequences, any list at all, and there is a rule that builds a sequence missing from it. The rule reads one entry from each row, and it is the single most reused argument in this field.

 Does this set contain that one — and the row that is missing. A membership table with the diagonal marked, and beneath it the complement of the diagonal, which is not among the rows. Logic

A list that cannot contain itself

The set of all sets that do not contain themselves is not a set. The argument is the diagonal again, applied to a table whose rows and columns are the same objects, and it destroyed the foundations of mathematics in a postcard.

 The diagonal, and the row built to be off the list. A table of rows of ones and zeros with the diagonal marked, and beneath it the row obtained by flipping every diagonal entry. Logic

The sentence that says it has no proof

Number every sentence and every proof, and a formal system can talk about itself. Then the diagonal is available one more time, and what it builds is a sentence that is true exactly when it is unprovable.

 Every rational number that could be a root of x³ − 2. A table of the candidate rational roots allowed by the rational root theorem, with the polynomial's exact value at each. Computation

The cube that will not double

Doubling a cube needs an edge in the ratio of the cube root of two. That number satisfies an equation of degree three, three does not divide any power of two, and the oldest open problem in geometry closes in a line.

 Which angles with a rational cosine can be cut in three. A dial of angles marked trisectable or not, beside the cubic whose rational roots decided each one. Computation

The angle that will not divide by three

Halving an angle costs one circle. Cutting it in three means solving a cubic, and for sixty degrees that cubic has no rational root — but plenty of angles do trisect, and which ones is a question with a countable answer.

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