Generator

wiring

A generator in the algebra library, called 8 times across 2 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.

all 24 permutations of 4 places are listed ×2
adding a detour changes the crossing count by an even number ×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
swapping two places flips the sign of every permutation ×1
swapping two tiles leaves the puzzle unsolvable ×1
the board is small enough to exhaust — at most six squares ×1
the detour is between 1 and 3 extra loops ×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 second decomposition is 1 to 3 swaps longer, in pairs ×1
the short decomposition really is this permutation ×1
the signed sum over permutations is the determinant ×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 view is one the family draws ×1
two decompositions of one permutation have the same parity ×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.

The whole library · What the figures prove