The same thing twice — page 13
Applied
The prices nobody can break away from
When houses are sold to buyers who value them differently, there is a whole range of prices at which nobody wants to walk away, and it has a remarkable shape — one corner best for every buyer at once, one best for every seller at once, and the buyers' corner pays each buyer exactly what the market would lose without them.
Computation
A rectangle made only of squares
A rectangle can be cut into finitely many squares — of any sizes, as many as wanted — exactly when its two sides are in whole-number proportion. Max Dehn proved it in 1903, and the proof that stuck, found by four Cambridge undergraduates in 1940, reads the squares as currents in an electrical circuit.
Computation
No odd number of equal triangles
A square can be cut into two triangles of equal area, or four, or any even number. It cannot be cut into three, or five, or any odd number — whatever shapes the triangles take. Paul Monsky's proof of 1970 has no geometry in its engine at all: it colours the points of the plane by how divisible their coordinates are by two.
Dynamics
Where the Collatz map is a coin
Extend the Collatz map from the whole numbers to the 2-adic integers — binary strings that run on for ever to the left — and it stops being mysterious. It becomes, after a change of coordinates, the simplest chaotic system there is: shifting a string of coin tosses one place. Everything about it is then known, and none of it says anything about the whole numbers, which is the most instructive failure in the whole story.
Geometry
The skeleton inside a shape
Take the Voronoi diagram not of a handful of points but of a shape's whole outline: the points inside the shape that have two nearest points on its edge. What comes out is a skeleton — a thin branching tree that describes the shape and, with one number attached to each point, rebuilds it exactly. It is also where fires lit all round the edge would meet, and it is violently unstable: a tooth too small to notice grows a branch several times its own size.
Geometry
Three mirrors make every solid
Every symmetry of a regular solid, reflections included, is produced by just three mirrors meeting at its centre, reflected in one another over and over — a kaleidoscope. Put a single point between the three mirrors and its reflections are the corners of a solid: the regular solid itself if the point sits in a corner, and every one of its truncated and expanded relatives if it sits anywhere else.
Logic
The program that prints itself
The diagonal argument has always been used to destroy — to show that a list misses something, that a sentence cannot be proved. Run the same move the other way and it builds. Kleene's recursion theorem says every program can be given its own text to work with, and the proof is a program that prints itself, thirty-two characters long, which can be run and checked.
Probability
An error bar for points that are not random
Evenly spread points integrate far better than random ones and give no error bar; random points give an error bar and integrate badly. Randomise the even points themselves — shift a lattice by a random vector, or scramble the digits of a Sobol' sequence — and both are kept: an unbiased estimate, a confidence interval from a handful of repeats, and an error that falls faster than any deterministic set's.
Topology
Counting targets by their holes
The Euler characteristic adds up like an area: glue two shapes together and it is the sum of theirs minus that of their overlap. So it can be used to measure, and measuring with it does something no area can. A field of sensors that each report only how many targets they detect — not which, not where — can have its readings added up, weighted by the Euler characteristics of the regions where the count is high, and the answer is exactly the number of targets.