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Things that cannot be done — page 4

Results that close a door rather than open one — and the peculiar difficulty of drawing a picture of something that does not exist.
The polynomial that squeezes π. On the left, xⁿ(π − x)ⁿ/n! drawn at several degrees, its largest value falling toward nothing; on the right, the derivatives of the same polynomial at zero, every one a whole number. Number

An integral that cannot be a whole number

Niven's proof that π is not a fraction is the same squeeze as the one for e, with a much harder multiplier. A polynomial supplies the whole number; its own smallness supplies the contradiction; and both halves are computable.

 How closely a fraction can come, and the barrier that says no closer. Two panels at very different scales: the approximations to √2, which stay above the barrier a degree-two number obeys, and the truncations of a constructed number, which fall below every barrier drawn. Number

Approached too fast to be algebraic

An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.

 Drop one condition, and something else satisfies the rest. A column for each of the four conditions, holding a sharing rule that breaks that one and keeps the other three, with the split each rule gives on a stated four-player game. Applied

None of the four conditions is spare

Four conditions pick out one sharing rule. The half that is usually shown is that they are enough; the other half is that each is needed — drop any one and a different rule satisfies the rest, so the list cannot be shortened.

 The affine plane of order 3, one parallel class at a time. The n² cells of a complete set of orthogonal Latin squares of order 3, with the rows, the columns and each square's symbol classes drawn as lines of a plane. Computation

The plane hiding in the squares

A complete family of orthogonal squares is not a collection of squares that happen to agree nowhere. It is a geometry — a plane with n² points in which every two points lie on exactly one line — and reading it that way is how the impossible orders were found.

 The 576 squares of order 4, sorted by whether they associate. Every Latin square of order 4, counted by whether it associates and by which group it is when it does. Computation

Sixteen of five hundred and seventy-six

A Latin square is a multiplication table in which every equation has exactly one solution. Ask it to be associative as well and almost every square drops out — sixteen of the five hundred and seventy-six of order four survive, and they are the two groups.

 Limit ordinals and the sequences that approach them. Several ordinals with the first terms of their fundamental sequences, and the successors marked as having a predecessor instead. Logic

Reached from below, or not at all

Every limit ordinal anybody meets is the end of an increasing sequence — ω, ω·2, ω^ω, all of them approached one step at a time. The first uncountable ordinal is not, and the reason it is not constrains the size of the continuum.

 A lattice of determinant 3, and the ellipse that must hold a point. A lattice with the parallelogram its basis spans, an ellipse centred at the origin, and the nearest non-zero lattice point it contains. Algebra

One point in every big enough shape

A determinant measures a lattice, not the basis that happened to describe it — and that measurement is an exchange rate. Any symmetric convex region with more than four times that area has to swallow a lattice point.

 A determinant of −5 and a permanent of 23 from the same six products. The six products of a three-by-three matrix listed once, added with signs to give the determinant and without signs to give the permanent, with a row operation applied to both. Algebra

The same sum without its minus signs

Delete the signs from the determinant's sum over permutations and what is left counts things directly rather than by cancellation. It is a better count and a far worse object — because the cancellation was what made the determinant computable.

 A room a trajectory cannot get out of, and one it can. A mushroom-shaped billiard table with two long trajectories: one confined to the cap by a conserved quantity, and one that enters the stem. Dynamics

A room that cannot be lit

Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.

 A path of 4 bounces that closes, in a triangle of 100°, 40°, 40°. A triangular billiard table with a periodic path found by an exhaustive sweep of starting positions and directions. Dynamics

The triangle nobody can settle

Does every triangular billiard table have a path that closes on itself? Acute triangles do, right triangles do, triangles with rational angles do — and for the rest the question has been open since it was asked.

 Stage 3 of a curve that has area. A square split into 64 smaller squares by removing crosses of decreasing width, the squares joined in Hilbert order; the kept area is 63.2 per cent and its limit is 0.5931. Topology

A curve that has area

The Jordan curve theorem assumes three things and nothing else — continuous, closed, no self-crossing. Everything else the eye supplies is false of some curve that satisfies all three, including the assumption that a curve is thin.

 Alexander's horned sphere at stage 3. A tree of clasped pairs of horns, 7 of them, each pair's two circles passing once through the other's disc; the horns shrink geometrically and their tips converge. Topology

A ball whose outside is not one

Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.

 A cycle showing every pair from 5 things exactly once. The complete graph on 5 points with an Eulerian circuit drawn, and the cyclic sequence of symbols it spells; every window of two consecutive symbols is a different pair. Computation

A cycle for every pair

A cyclic sequence in which every window of two consecutive symbols is a different pair of things. For five things it exists and for four it does not, and in both cases there are exactly as many pairs as there are places to put them.

 Two out of three, and never all three. A table of the five apportionment methods against three properties, each cell decided by a search over generated instances; no method has all three. Applied

Two out of three, and never all three

Stay inside every region's quota, never take a seat away when the house grows, never take one from a region that grew faster. Each pair is achievable. All three together are not, and the proof is that no rule anywhere manages it.

 16 ways to sort it. A small order with its 16 linear extensions counted, and for each incomparable pair the fraction of extensions putting one before the other. Discrete

How many ways to sort it

An order says some things come before others and leaves the rest open. Counting the orderings consistent with it measures how much is still unknown — and the counting is as hard as any counting problem gets.

 Where a congruence decides which primes a form represents, and where it does not. Rows of primes marked by whether each is represented by x squared plus n y squared, with the residue classes that decide it where such classes exist. Number

Which primes a form takes

A prime is the sum of two squares exactly when it is 1 modulo 4. Change the form slightly, to x² + 27y², and no congruence on p decides it at all — which is where the elementary subject ends and its successor begins.

 The four-square identity, and how few squares a number needs. A product of two whole quaternions with both sides of the four-square identity evaluated, above a strip colouring every number by the fewest squares that add to it. Algebra

The identity that multiplies sums of squares

A sum of two squares times a sum of two squares is a sum of two squares, and the same is true of four and of eight. It is true of three of nothing: 3 and 5 are each a sum of three squares, 15 is not, and one small pair of numbers rules the case out forever.

 The octonion multiplication table, drawn as seven lines. The Fano plane with its seven points labelled by the imaginary octonion units and its seven lines carrying an arrow each, giving the products of every pair. Algebra

What is lost at eight

Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.

 The classes, a selection from them, and the translates that cannot have a length. Points of several classes of the unit interval under translation by rationals, drawn one class per row, above rows showing rational translates of a selection that never overlap. Analysis

A set that has no size at all

Slide the unit interval along itself by every rational and the points fall into classes. Choose one point from each and the resulting set has no length — not zero, not positive, none: countably many disjoint copies of it would have total length nought or infinity, and the union needs something in between.

 Axioms, the conditions on the arrows they answer to, and the one that answers to none. A table of modal axioms with the property of the accessibility relation each corresponds to, every row decided by sweeping all relations on up to four worlds. Logic

The axiom with no property of the arrows

Each axiom of modal logic can be matched by hand to a condition on the arrows between worlds. There is a recipe that does it for a whole class of axioms, and there is an axiom the recipe cannot reach — not because nobody has looked, but because no condition on the arrows defines it at all.

 Four solids with the same counts and every volume. The Reeve tetrahedra at heights 1, 2, 3, 5, drawn in wireframe with a table of their lattice-point counts and volumes. All have four boundary points and none inside; their volumes run from 0.17 to 0.83. Discrete

The theorem that has no version in space

A lattice polygon's area is decided completely by two counts of dots. The obvious guess is that a lattice solid's volume is decided by the same two counts in three dimensions, and there is a family of tetrahedra with identical counts and every volume that says otherwise.

 Sliding one square past another. 4 stages of a slide in which two labelled squares inside a larger one exchange positions without ever overlapping. The larger square's boundary is the base point throughout, and the exchange is what makes the composition commutative. Topology

Why the second group commutes

Replace loops by spheres and the same construction gives a second homotopy group. It is always commutative, and the reason is not a fact about spheres or about any space — it is a two-line argument about any set carrying two compatible operations.

 A circle stays a circle, unless it meets the pole. 3 circles on a sphere beside their stereographic images in the plane, which are circles, together with one circle through the projection point whose image is a straight line. Topology

Angles survive and areas do not

Stereographic projection takes every circle on the sphere to a circle or a line, and every crossing angle to itself. It does both exactly, with no approximation anywhere, and it destroys area so thoroughly that a patch near the pole can be a thousand times its neighbour's size.

 The same banding at every magnification. The Hénon attractor drawn from 26000 points, followed by 2 magnifications of one part of it. Each magnification resolves what looked like a single curve into several parallel ones. Dynamics

Neither a surface nor a solid

The attractor has no volume, because the flow shrinks volumes at a rate that can be read off the equations. It is also not a surface, because a surface cannot carry chaotic dynamics. What is left is an object of dimension a little over two, and that number is measurable.

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