Matpic
mathematics, in pictures
Concept

Impossibility

A theorem stating that no object of a described kind exists, as against a report that none has been found. What makes one is an argument covering every candidate at once, usually a quantity a candidate would have to change and provably cannot.

Named by 15 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

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 endorses a combination no member of it holds, and no rearrangement of the procedure removes the problem.

applied · Judgement aggregation 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.

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.

computation · Scissors congruence 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.

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.

computation · Latin squares 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.

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.

dynamics · Billiards 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.

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.

applied · Apportionment Seats to districts and to parties at once. A 4 by 3 table of seats, with every row total and every column total prescribed. The entries come from scaling the votes by one factor per row and one per column and rounding, and all the totals come out exactly right.

Seats to parties and places at once

Seats can be given to regions in proportion to one list of populations, and no rule does it perfectly. Ask for seats to regions and to parties simultaneously and the object stops being a list — and the impossibility that closed the subject does not apply.

applied · Apportionment 16 rules, and none that survives. A table of every systematic anonymous aggregation rule for 3 judges: one row per rule, showing the verdict it gives at each count of yes-votes, whether it decides every proposition, and whether it is consistent. No row has both.

No rule escapes the doctrinal paradox

A court whose members each hold a consistent position can reach an inconsistent verdict by majority. One such case is easy to build, which invites the hope that a better rule would avoid it — and every rule that responds to the votes at all fails somewhere.

applied · Judgement aggregation The quantity a cut cannot change and a turn can. 4 polygons, each with the spikes of its translation invariant drawn round a dial: the length of the edges facing each direction, less the length of those facing the opposite way. It vanishes everywhere for 3 of them.

Slid, but never turned

The classical dissections all turn their pieces. Forbid the turn — allow the pieces to be slid and nothing else — and equal area stops being enough, for a reason that is a single number attached to each direction and that a cut cannot change.

computation · Scissors congruence What the chain costs on a 6-gon: 39 pieces. A regular 6-gon fanned into 4 triangles, each with the three cuts that turn it into a rectangle, beside the running count of the pieces the whole chain produces — 39 of them.

Finitely many, and nobody says how many

The theorem promises a dissection exists and the proof produces one. Running the proof on a hexagon produces thirty-nine pieces, ingenuity produces five, and there is no method for proving that five cannot be four.

computation · Scissors congruence Volume and one more number decide what a solid can be cut into. A table of a cube, a prism, a sixth of a cube, a regular tetrahedron, a regular octahedron and a collection of two tetrahedra with one octahedron, giving each one's volume, its Dehn invariant computed from its measured dihedral angles, and whether it can be cut into a box of equal volume.

The obstruction that was the only one

Dehn showed in 1901 that a cube cannot be cut into a regular tetrahedron of the same volume, because a number built from edges and angles disagrees. For sixty-four years nobody knew whether that number was the whole story. Sydler proved in 1965 that it is: volume and Dehn's number together decide every case.

computation · Scissors congruence 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.

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.

applied · Apportionment 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.

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.

applied · Apportionment 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.

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.

dynamics · Cellular automata The plane of order 3 as a table, and the table times its transpose. The 13 × 13 incidence table of the projective plane of order 3 and its product with its transpose, which has 4 on the diagonal and 1 in every other cell.

The orders a plane cannot have

Every counting condition allows a projective plane of order six, and there is none. The proof that rules it out looks at one matrix identity — each point on seven lines, each two points on one — and turns it, by way of Lagrange's four squares, into the statement that six would have to be a sum of two squares. Run on the planes that do exist, the same argument hands back their orders as sums of two squares; run on six, it asks for something no arithmetic can supply.

computation · Finite geometry No pair of weights modulo ten catches everything. Two 10 × 10 grids over pairs of alternating weights modulo ten, shaded by the share of single errors and of swaps each catches. No pair is full in both; weights 3 and 1, ringed, catch every single error and 88.9% of swaps.

Ten digits need a symmetry that does not commute

A check digit should catch one wrong digit and two neighbours swapped. Over eleven symbols a weighted sum does both; over the ten decimal digits no weighted sum can, no scheme of any shape built on adding modulo ten can, and the reason is the same as the reason Euler's thirty-six officers cannot be paraded. What works is the ten symmetries of a pentagon, which do not commute.

computation · Error-correcting codes

Named alongside it

The objects these essays reach for when they reach for this one.

Exhaustive searchCounterexampleInvariantApportionmentDissectionOperation setAreaDivisor methodLatin squareMajorityMatrixPolygon

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