Generator

A code on the 3-cube, and the balls around its words

A generator in the computation library, called 26 times across 7 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

cube-code 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.

With nothing chosen

A code on the 3-cube, and the balls around its words. The corners of a hypercube with the chosen codewords marked and the words within one error of each shaded.

A closed walk on the 3-cube changing one place at a time

Counting to 15 two ways

All 6 closed walks on the 3-cube

The 1,344 tours of the 4-cube, by how often each place changes

The reflected code on 4 places: changes 2, 2, 4, 8

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.

step 1 changes exactly one place ×7
the dimension of the cube is a whole number between 2 and 4 ×3
a code is more than one word and fewer than all of them ×1
a tour with changes 4, 4, 4, 4 exists and the search found it ×1
a tour with changes 8, 6, 6, 6, 6 exists and the search found it ×1
and its worst single step changes all 4 places at once ×1
and none twice ×1
and the changes add up to the number of steps ×1
and the last corner is one step from the first, so the walk closes ×1
and visits none of them twice ×1
counting up in the ordinary way changes more places than that ×1
each cycle was met once in each direction ×1
each step changes one place ×1
each step, the last back to the first included, swaps one element out and one in ×1
every codeword has the length the cube has dimensions ×1
every corner meets one edge per coordinate ×1
every place changes an even number of times ×1
every step of the code changes exactly one place ×1
every subset appears ×1
every walk is enumerated for at most the 4-cube ×1
every word within 1 of a codeword decoded back to it ×1
the 3-cube carries 6 ×1
the 3-cube carries exactly 6 such walks ×1
the 4-cube carries 1,344 tours ×1
the 4-cube carries exactly 1,344 ×1
the balls partition the cube exactly, so the code is perfect ×1
the code is a word list or one of repetition, parity, none, even2 ×1
the cycle visits every word of the two levels once ×1
the drawn walks are the first of the ones counted ×1
the half-length k is a whole number between 1 and 4 ×1
the length of the words is a whole number between 2 and 4 ×1
the levels hold every word ×1
the number of places is a whole number between 3 and 11 ×1
the radius of the balls drawn is a whole number between 0 and 2 ×1
the search found a cycle through the middle two levels ×1
the size of the set is a whole number between 3 and 7 ×1
the size of the subsets is a whole number between 1 and 5 ×1
the tour is one of reflected, balanced ×1
the tour visits every corner once ×1
the view is one the family draws ×1
the walk visits all 8 corners ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Discrete

A walk that changes one thing at a time

Counting from nothing to fifteen in binary changes four digits at once somewhere in the middle. There is another order through the same sixteen words in which every step changes exactly one — and it is a closed walk on a four-dimensional cube.

Geometry

Circles that are diamonds and squares

The theorem hands over a formula for distance. Take the formula as a definition, change the exponent in it, and the set of points one unit from the origin stops being round — while remaining, in every sense that matters, a circle.

Computation

Distance is a picture

A message is a corner of a cube and an error is a step along an edge. Everything a code can do is decided by how far apart the corners it uses are — and that is a fact about a drawing.

Discrete

Every place changes back

A closed walk through every corner of a cube changes one place at each step, and each place, having changed, must change back before the walk returns home. So every place changes an even number of times — which is why no walk on three places can share the work evenly, why perfect sharing is possible only when the number of places is a power of two, and what sorts the 1,344 walks on the 4-cube into exactly four kinds.

Computation

Finding the error without reading the message

Three parity checks on a seven-bit word produce three bits. If they are all zero nothing is wrong; otherwise they are the number of the position that broke. The message is never consulted, because the answer does not depend on it.

Computation

Sixteen spheres that fill a cube

A hundred and twenty-eight seven-bit words, sixteen of them chosen, and a ball of eight around each. Sixteen times eight is a hundred and twenty-eight exactly — so the balls tile the space with nothing left over, and the code wastes nothing at all.

Discrete

The walk through the middle levels

On seven places, the words with three ones and the words with four number thirty-five each. Is there a closed walk through all seventy, changing one place at a time and never leaving those two levels? On five places the answer is 24 walks, on seven and nine a search finds one in moments — and whether one exists for every odd length was open for thirty years, until Torsten Mütze proved in 2016 that it always does.

The whole library · What the figures prove