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Things that cannot be done — page 6

Results that close a door rather than open one — and the peculiar difficulty of drawing a picture of something that does not exist.
sin(2πkx) for k up to 10, and the largest gap between every pair. Members of the sequence sin 2πkx drawn on one pair of axes, beside a table of the largest vertical gap between every two members, none of which is less than one. Analysis

The subsequence that has to exist

Every bounded list of numbers has a part that settles down. A bounded list of functions need not: the waves sin 2πkx never come within 1.76 of one another. One extra condition — that no member may change faster than a bound they all share — restores the guarantee, and it is the reason a differential equation with a continuous rule has a solution at all.

 Biproportional seats against their fair shares. A table of votes for 3 districts and 4 parties beside the seats the biproportional method gives, each with the fair share from the continuous fit, and the cell whose seats fall outside its quota marked. Applied

The table inside every quota

Give seats to districts and parties at once, and every cell of the table has a fair share it ought to round from. A table rounding every cell to its floor or its ceiling, with every total exact, always exists. The biproportional method does not always choose one: here it gives a party 2 seats where its fair share is 3.088.

 Sixteen halves in a three-by-three-by-three table of seats. A three-way table of fair shares drawn as three slices, one per group, with sixteen cells holding a half and every line total, along districts, parties and groups, equal to zero or one. Applied

Where the rounding runs out

In two dimensions a table of seats inside every fair share always exists. Add a third family of totals — every district and party split between groups — and it need not. Sixteen halves in a three-by-three-by-three table meet every total, and no whole table does it without a seat where the fair share is nothing, because the halves close a loop of seven.

 Five certificates against any two of three decide. A table of every minimal balanced family on three players, what each demands of the game, and whether the grand coalition's value covers it — the complete test for whether a stable split exists. Applied

Five weighings and the question is closed

Searching the triangle of splits can only ever fail to find a stable one, which is not the same as there being none. Weighing five families of coalitions against the whole settles the question outright — and the family that fails is the proof that nothing survives.

 The share of a ball that is its own edge. A plot of the proportion of each ball formed by its outermost shell against the radius, one line per group — falling towards nothing for the lattice groups and holding steady for the free group. Algebra

The edge that is as big as the ball

In a lattice the boundary of a large ball is a negligible fraction of it. In a tree it is two thirds of it at every size — and that single ratio, not the group's size, is what decides whether a set can be cut into pieces and reassembled into two copies of itself.

 Which polygons two instrument sets reach, up to 24. A strip of the polygons from 3 to 24 sides, each marked according to whether compass and straightedge reach it and whether a conic or a trisector does, with the degree of its cosine beneath. Computation

Two instruments with one reach

Allow every conic to be drawn at will, or allow an angle to be cut in three. The two permissions look nothing alike and reach exactly the same numbers — because what an operation buys is a degree, and both of these buy three.

 The polynomial the 11-sided polygon needs. The minimal polynomial of twice the cosine of the central angle of an 11-sided polygon, with its degree, the rational root test applied to it, and whether that degree is reachable by cubic steps. Computation

A quintic a sliding mark reaches

The eleven-sided polygon needs a number of degree five, and five is not a product of twos and threes — so no conic and no angle trisector reaches it. A ruler with two scratches does, which places the marked ruler strictly above the conics and leaves its exact reach unknown.

 A graph whose matching misses 2 of its 10 vertices. A graph with its largest matching drawn thick, a set of vertices ringed, and the pieces left when that set is deleted marked by whether they hold an odd number of vertices. Discrete

The piece that cannot pair off

Take the sides away and the obstruction to a matching changes character completely. It is no longer a shortage of partners; it is a parity, and the quantity that measures it counts pieces of odd size rather than vertices of any size.

 Six lists of cycle shapes, and how many coverings each has. A table of lists of cycle shapes over a sphere, each with the Euler characteristic the Riemann–Hurwitz count gives, the number of lists of permutations with that product, and the number of those that connect all the sheets. Topology

A count that can say zero

The branched count ends on a list of cycle shapes that passes every test and describes no covering. There is an exact formula for how many coverings a list has — a sum over the character table of a symmetric group — and it returns nought without giving any reason why.

 A rule that meets the diagonal, and the same rule with a hole in it. Two panels, each the graph of a rule assigning a set of values to every point of the unit interval, drawn against the diagonal: the first meets the diagonal at a filled-in jump, the second has the jump left open and misses it. Topology

A map that offers a choice

Brouwer's theorem needs a function, and the object it was most wanted for is not one — a best reply is a whole set whenever a chooser is indifferent. Allow a point to be sent to a set and the fixed point survives, provided the sets are convex, and the convexity is the entire hypothesis.

 An area of starting points from which z³ − 2z + 2 is never solved. The complex plane coloured by which root of z³ − 2z + 2 Newton's method reaches from each starting point, with the points that reach no root left uncoloured. Dynamics

An area that never finishes

Newton's method's famous failure is a boundary, and a boundary has no area — a random start misses it with probability one. The real failure is different in kind: a polynomial with small whole-number coefficients whose method has a region of starting points, with area, from which it provably never terminates.

 67 starting points that find all 5 roots. The roots of z⁵ − 1 with a ring of starting points around them, each start marked by which root the method reaches from it, and every root reached by at least one. Dynamics

Covering rather than avoiding

Two arguments say no starting guess is safe: the boundary is fractal and some regions are permanently trapped. The repair is not a better guess. It is a fixed list of starting points, computed from the degree alone, from which every root of every polynomial of that degree is found.

 Two graphs every count agrees on, and one question that does not. Two sixteen-point graphs drawn on a four-by-four grid, with one point marked and its six neighbours highlighted in each, the neighbours forming two triangles in one and a six-cycle in the other. Logic

One gadget defeats every refinement

Colour refinement fails on two triangles against a hexagon; its two-dimensional version fixes that and fails on a pair of strongly regular graphs. For every k there are two graphs the k-dimensional version cannot separate, and they are built from one local piece whose only symmetry is a parity.

 Where the aliquot sequence of every number up to 1000 goes. A grid of the starting values 2 to 1000 coloured by the fate of each aliquot sequence: 964 reach 1, 19 reach a perfect number, 3 enter a cycle, and 13 pass 10²² undecided. Number

The sum of the parts, taken again

Replace a number by the sum of its proper divisors and do it again. Most numbers fall to 1, a few land on a perfect number or a cycle, and some climb for hundreds of steps. Below a thousand there are twelve whose fate nobody knows — and the thing that keeps them climbing is a perfect number hiding in their factorisation.

 The Seifert circles of the trefoil. The trefoil with an orientation, cut at each of its 3 crossings and reconnected the way the orientation allows. The 6 segments form 2 circles, and the surface built from them has genus 1. Topology

The surface a knot bounds

Every knot is the edge of a surface with two sides, and Seifert found a way to build one from any diagram: smooth the crossings, fill the circles that result with discs, and join them with twisted bands. Counting the handles gives an upper bound on how complicated the knot is, the Alexander polynomial gives a lower one, and for the simplest knots the two meet.

 The Petersen graph, which a five-edge count rules off the plane. The Petersen graph drawn as an outer pentagon, an inner pentagram and five spokes: ten points, fifteen edges, girth five, and more edges than the 13.33 a flat drawing with five-edge faces allows. Discrete

Five spokes squeezed into K5

The Petersen graph has no point with four neighbours, so no stretched copy of K5 can sit inside it. Contract its five spokes and K5 appears anyway. Kuratowski's theorem forbids stretched copies and Wagner's forbids squeezed ones, the two notions disagree on this graph — and they still name exactly the same planar graphs.

 Area out to infinity, for three powers. Left: the curves 1/√x, 1/x, 1/x² from x = 0 to 10, with the region beyond x = 1 shaded under the lowest. Right: the area from 1 to T for each, on logarithmic scales, for T up to 10^6. 1/√x keeps growing, 1/x keeps growing, 1/x² levels off at 1. Analysis

An endless region with a finite area

A region that runs off to infinity can still have a finite area, and for the curves 1/xᵖ the exponent that makes the far end finite is exactly the one that makes the end at zero infinite. 1/x fails at both, no power succeeds at both, and a horn can hold less than π while needing infinite paint.

 The odd ring that forbids a stable pairing. The ranked lists of 6 people and a stable partition of them drawn on a circle: pairs as plain chords, a ring of three or more as arrows from each person to the one they hold. An odd ring is present, as it is in every stable partition of this instance. Applied

A ring that no pairing can break

Put everybody in one pool and a stable pairing may not exist. Allow rings as well as pairs and something stable always exists — and the pairs-only answer fails exactly when that stable arrangement contains a ring of odd length. Two sides make every ring even, which is the whole reason the two-sided theorem holds.

 A local rule taking a vote. A space-time diagram of the GKL rule on 149 cells from a random row with 69 ones. Black and white regions grow and meet along slanting boundaries, and after 69 steps the whole ring is 0. Dynamics

No local rule can count the votes

A ring of cells, each holding 0 or 1, has to agree on whichever value is in the majority — every cell seeing only its neighbours. The best-known rule gets it right most of the time and wrong near a tie; no rule of any radius gets it right always. Yet two rules run one after the other do, on every ring, and the first of them is the traffic rule.

 An additive function that is nowhere a line. A square window with 1157 points of the graph of an additive function that sends √2 to 0, scattered across the window and meeting every part of it, beside a dashed diagonal marking the straight line y = x. Logic

A function that adds and is nowhere a line

Every continuous function with f(x + y) = f(x) + f(y) is a straight line through the origin. Drop continuity and, given the axiom of choice, there are others — functions that add perfectly and whose graphs are scattered densely over the whole plane. A finite piece of the construction can be drawn exactly; the whole of it needs a basis of the real numbers that no one can write down.

 Infinitely many guessers, finitely many wrong. Three rows over the first 40 places: the hats worn, the chosen representative of their class, and a row of marks showing each guess right or wrong. The 5 wrong guesses all fall within the first 14 places, up to a marked place; every later guess is right. Logic

Infinitely many guessers, finitely many wrong

An infinite line of people each wears a black or white hat, sees every hat in front and none of their own, and must guess their own colour. With a finite line, each guesser is right half the time whatever they agree in advance. With an infinite line and the axiom of choice, they can agree a strategy under which all but finitely many are right — and nobody can carry it out.

 The tangent built from a continued fraction. The curve tan x on (−1.55, 1.55) with 4 of Lambert's convergents: a straight line, then rational curves that bend ever closer to the tangent and follow it towards its poles. Number

The fraction Lambert built for the tangent

The first proof that π is not a fraction, from 1761, does not look at π at all. It writes the tangent as an endless continued fraction, shows that the fraction's value at any rational point other than zero cannot be rational — because its tails are trapped between nothing and one — and then notes that tan(π/4) = 1.

 The powers of 2 in 1 to 12: one number, 8, stands alone. The whole numbers from 1 to 12, each with a bar whose height is the power of 2 dividing it. A single number has the tallest bar, which is why the harmonic number H(12) has an even denominator and an odd numerator. Analysis

The sum that steps over every whole number

The harmonic sum 1 + 1/2 + 1/3 + … passes 2 at the fourth term, 3 at the eleventh, 4 at the thirty-first, and eventually every whole number there is. It never lands on one. The proof is a single number in the list 1, 2, …, n that carries more factors of two than any other — and the same arithmetic makes the numerators divisible by squares of primes they have no business knowing about.

 Where discs of radius one cover the lattice, and where they leave holes. Six lattices of algebraic integers drawn as points in the plane with a unit disc around each; for five of them the discs cover the whole plane and for the sixth, the integers of the field of the square root of minus nineteen, uncovered holes remain. Number

Factoring uniquely with no way to divide

Unique factorisation is proved by dividing with a small remainder, and in the Gaussian integers that works because discs of radius one cover the plane. In the integers of ℚ(√−19) the discs leave holes, no division algorithm of any kind can be made to work — and factorisation is unique anyway. The same field is why n² + n + 41 is prime forty times running.

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