Generator

The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times

A generator in the algebra library, called 28 times across 6 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

wiring is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

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.

`show: "determinant"`

A 3×3 determinant as six signed products, totalling −43. The six permutations of three places, each drawn as the three matrix entries it selects, with the sign of the permutation attached and the signed total checked against a cofactor expansion.

`show: "moves"`

One permutation routed two ways: 3 crossings and 5. The same permutation drawn twice, once with straight strings and once with one string pushed across another and back, so the two diagrams have different crossing counts and the same parity.

`show: "swaps"`

(1 3 4 2) as 3 swaps and as 5. One permutation taken apart into swaps of neighbouring places in two different ways, drawn as stacks of layers, with the two lengths differing and their parity the same.

`show: "table"`

Every permutation of 4 places, by sign. All 24 permutations of 4 places listed in cycle notation with the number of pairs each puts out of order, coloured by whether that number is even or odd, and the two halves counted.

`show: "puzzle"`

A 2×3 sliding puzzle: 360 arrangements of 720 can be reached. Two arrangements of a small sliding puzzle side by side, the solved one and the one with two tiles exchanged, with the count of positions reachable by sliding found by walking every move.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the homomorphisms to the cyclic group of order 2 number 2 ×5
the class of shape 3+1 has 8 members by the formula too ×4
the class of shape 2+1+1 has 6 members by the formula too ×3
all 24 permutations are listed ×2
all 24 permutations of 4 places are listed ×2
the class of shape 1+1+1+1 has 1 members by the formula too ×2
the class of shape 4 has 6 members by the formula too ×2
a single closed tour moves the tiles in one cycle ×1
a tour of 4 squares cycles 3 tiles ×1
adding a detour changes the crossing count by an even number ×1
and at least one target admits the sign ×1
and exactly two of the squares are unit squares ×1
and its sign is what a cycle of that length has ×1
and no face turn changes the joint parity of the two permutations ×1
and one of the two is the sign ×1
and the half they generate is the even permutations ×1
and the other is the odd side ×1
and the sign is constant on it, which is why the sign is a function of the shape ×1
and the squares of their dimensions add to the size of the group ×1
and they are the same size ×1
because a single exchange is odd ×1
between five and two hundred scrambles ×1
between twenty and two thousand scrambles ×1
each scramble is between five and two hundred turns ×1
each scramble is between ten and four hundred moves ×1
each step of the tour moves the blank one square ×1
every board one legal move from a reachable one has the same parity ×1
every one of the eighteen turns was tried at every position ×1
every reachable arrangement has the same sign-and-blank-distance parity ×1
exactly half of the 720 arrangements can be reached ×1
exactly half the permutations are even ×1
exactly two of them are one-dimensional ×1
exchanging two tiles puts the board on the other value of the invariant ×1
no face turn flips the edges as a whole ×1
no face turn twists the corners as a whole ×1
no legal move leaves the component it starts in ×1
one component is the even side ×1
one corner twisted breaks its own invariant ×1
one corner twisted breaks only its own invariant ×1
one edge flipped breaks its own invariant ×1
one edge flipped breaks only its own invariant ×1
swapping two places flips the sign of every permutation ×1
swapping two tiles leaves the puzzle unsolvable ×1
the blank comes back to where it started ×1
the board is between nine and sixteen squares ×1
the board is between two and twelve squares ×1
the board is small enough to exhaust — at most six squares ×1
the characters are orthonormal under the group's own average ×1
the class of shape 1+1+1+1+1 has 1 members by the formula too ×1
the classes account for every permutation exactly once ×1
the commutators generate exactly half the group ×1
the commutators themselves are inside the subgroup they generate ×1
the cyclic groups tested have orders between two and eight ×1
the detour is between 1 and 3 extra loops ×1
the even classes together are exactly half the group ×1
the group is on three, four or five places ×1
the identity's column holds each representation's dimension ×1
the invariant is the same at both ends of the tour ×1
the long decomposition is the same permutation ×1
the matrix is 3×3 with whole entries no larger than 20 ×1
the number of swaps has the parity of the crossing count ×1
the permutation sends each of its 4 places somewhere different ×1
the reachable side has twelve arrangements ×1
the route is between three and nine squares of the board ×1
the second decomposition is 1 to 3 swaps longer, in pairs ×1
the short decomposition really is this permutation ×1
the sign of the arrangement alone does change, so it is not by itself the invariant ×1
the signed sum over permutations is the determinant ×1
the solved board has parity zero ×1
the squares are drawn for three, four or five places ×1
the squares of the dimensions add to the size of the group ×1
the strings cross exactly as often as the pairs are out of order ×1
the table is between 3 and 8 columns wide ×1
the table is drawn for 2, 3 or 4 places ×1
the table is drawn for three, four or five places ×1
the tour starts from the blank's home square ×1
the two components account for every arrangement ×1
the view is one the family draws ×1
the whole graph is drawn only for the two-by-two board ×1
there are exactly as many irreducible characters as conjugacy classes ×1
two decompositions of one permutation have the same parity ×1
two pieces exchanged breaks its own invariant ×1
two pieces exchanged breaks only its own invariant ×1
with nothing odd among them ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Algebra

A shared root, found without finding it

Two polynomials have a root in common exactly when one determinant built from their coefficients is zero. No root is computed, nothing is approximated, and the same construction turns two equations in two unknowns into one equation in one.

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.

Algebra

The only bit that survives

A shuffle can be called even or odd, and the label behaves under composition. Ask whether some cleverer label — a number out of three, or out of four — could behave the same way, and the answer is that nothing else can — one bit is exactly what a permutation gives up.

Algebra

The puzzle that is exactly half solvable

A sliding puzzle sold with two tiles swapped is not a hard puzzle; it is an impossible one, and the proof is a quantity that no slide can change. The same argument, run three times at once, says that one arrangement of a scrambled cube in twelve is reachable.

Number

The symbol is the sign of a shuffle

Multiplying every residue modulo p by a fixed number rearranges them. That rearrangement is a permutation, permutations have a sign, and the sign is exactly the Legendre symbol — so a question about squares becomes a question about crossings.

Algebra

When the label may be a matrix

A permutation carries exactly one bit into any commutative target, and commutativity is the restriction doing all the work. Drop it — let the label be a matrix — and what survives is a short finite table, computed here from traces and checked for orthogonality over every pair of rows.

The whole library · What the figures prove