Theme

Things that cannot be done — page 3

Results that close a door rather than open one — and the peculiar difficulty of drawing a picture of something that does not exist.

a loop that dips through and back, and the punctures of the disc. A link drawn with a shaded disc spanning the first loop, seen at an angle, with every place the second loop passes through the disc marked with the direction it was travelling in.

Topology

Zero can mean two different things

The linking number counts how often one loop pierces a surface the other one bounds. Two punctures of opposite sign add to nothing, and a loop that never goes through adds to nothing as well — so the answer zero is two pictures wearing one number.

The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times. A permutation drawn as strings running from a row of numbered pegs to another, with every place two strings cross marked, and the crossing count checked against the number of pairs that are out of order.

Algebra

The crossings that will not come out even

Draw a rearrangement as strings from one row of pegs to another and count where they cross. The count depends on how the strings are drawn; whether it is odd or even does not, and that single bit is what makes determinants exist and a sliding puzzle unsolvable.

The subgroups of a polynomial's symmetries, against the fields they name. Two lattices side by side, one of the subgroups of the symmetry group of the cube roots of two and the other of the fields between the rationals and the splitting field, drawn so that one is the other turned upside down.

Algebra

The lattice that runs the other way

The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.

The quaternion multiplication table, from i² = j² = k² = ijk = −1. A four-by-four multiplication table of the quaternion units, with the row giving the left factor, every entry computed from Hamilton's rule, and the pair that differs between the two orders marked.

Algebra

A multiplication that remembers the order

Four characters — i² = j² = k² = ijk = −1 — define a multiplication in which ab and ba are different numbers. Everything follows from them, including the fact that a rotation of a solid body has two names, and that two full turns are needed to get one of them home.

The splits no group can beat, for three partners. The triangle of ways to split a fixed total between three players, with each coalition's demand drawn as a straight cut across it, and the region surviving every cut shaded.

Applied

A split nobody can walk away from

Every way of dividing what a group earns is a point of a triangle, and every coalition's threat to leave cuts a straight line across it. What survives all the cuts is the set of stable divisions — and for one three-player game there is nothing left.

3 consistent judges, and a majority that is not. A table of judges against three questions, every judge's row internally consistent, with the majority answer to each question underneath forming a combination no judge holds.

Applied

The court that contradicts itself

Three judges each answer three questions, and each answers them consistently. Take the majority on each question separately and the answers no longer hang together — the body as a whole asserts a combination no member of it holds, and no rearrangement of the procedure removes the problem.

A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

Computation

Equal area is enough, and equal volume is not

Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.

The midpoint of a segment, drawn with a compass and no straightedge. A segment with the arcs that step its length three times round one end to reach the point twice as far away, and the further arcs that send that point back to the midpoint, every one of them a circle.

Computation

The straightedge buys nothing

Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.

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