A code on the 3-cube, and the balls around its words
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 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.
- the tour of 6 choose 2 steps only along edges ×11
- the tour of 6 choose 2 visits every vertex once ×11
- step 1 changes exactly one place ×7
- graphs with smallest degree 3 that have a tour ×3
- the dimension of the cube is a whole number between 2 and 4 ×3
- closed tours of the 3 × 10 board ×2
- 100 to 1,000 processes a size ×1
- 8, 10, 12 or 14 points ×1
- a board from 5 to 12 squares each way ×1
- a board that carries a closed tour ×1
- a closed tour is found ×1
- a code is more than one word and fewer than all of them ×1
- a knight on an outer row always moves to an inner row ×1
- a path through every point with unjoined ends exists ×1
- a process whose graph at τ₂ has a tour is found ×1
- a tour visits every point once ×1
- a tour visits every square once ×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
- an open tour of the 5 × 5 board exists ×1
- and its shortest cycle has seven edges ×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 the same number of them whichever vertex is removed, by symmetry ×1
- and visits none of them twice ×1
- but it has an open one ×1
- closed tours of the 4 × 6 board, by exhausting every path ×1
- counting up in the ordinary way changes more places than that ×1
- Coxeter's graph has no closed tour, by exhaustive search ×1
- Coxeter's graph has twenty-eight vertices ×1
- each cycle was met once in each direction ×1
- each step changes one place ×1
- each step of the tour is a knight's move ×1
- each step of the tour is an edge ×1
- each step, the last back to the first included, swaps one element out and one in ×1
- each with three neighbours ×1
- every board the theorem allows has a tour found and checked ×1
- every codeword has the length the cube has dimensions ×1
- every corner meets one edge per coordinate ×1
- every graph meeting Dirac's condition has a tour ×1
- every graph meeting Ore's condition has a tour ×1
- every knight's move changes colour ×1
- every other Kneser graph tried has a closed tour ×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 vertex's removal leaves a closed tour of the other nine ×1
- every walk is enumerated for at most the 4-cube ×1
- every word within 1 of a codeword decoded back to it ×1
- labelled graphs on six points ×1
- neither graph has a closed tour ×1
- Petersen's graph has exactly six perfect matchings ×1
- Petersen's graph has no closed tour ×1
- removing a perfect matching leaves two pentagons, never one ten-cycle ×1
- seed is read only by the path and tour views ×1
- ten to eighteen points ×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 board size is read only by the tour view ×1
- the code is a word list or one of repetition, parity, none, even2 ×1
- the counting finds a crossing pair ×1
- the cycle visits every word of the two levels once ×1
- the drawn walks are the first of the ones counted ×1
- the graph has a closed tour ×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 measured waiting time matches the Poisson prediction at each size ×1
- the number of ones is a whole number between 2 and 4 ×1
- the number of places is a whole number between 3 and 11 ×1
- the one exception is Petersen's graph ×1
- the prediction lies above γ and is coming down towards it ×1
- the radius of the balls drawn is a whole number between 0 and 2 ×1
- the rule usually succeeds and sometimes fails ×1
- the search closes a tour of the odd graph ×1
- the search found a cycle through the middle two levels ×1
- the share of processes with a tour at τ₂ rises with the number of points ×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 smallest degree of the complete bipartite graph ×1
- the smallest degree of the two cliques ×1
- the three-row exceptions have no closed tour, by search ×1
- the tour is one of reflected, balanced ×1
- the tour of the full graph of thirty-five steps only along edges ×1
- the tour of the full graph of thirty-five visits every vertex once ×1
- the tour steps only along edges ×1
- the tour visits every corner once ×1
- the tour visits every vertex once ×1
- the vertex removed is a whole number between 0 and 9 ×1
- the view is one the family draws ×1
- the walk visits all 8 corners ×1
- three-row boards of length 4, 6 and 8 have no closed tour, by exhausting every path ×1
- turning the places one step sorts the twenty-eight words into four rings of seven ×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.
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.
GeometryCircles 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.
ComputationDistance 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.
DiscreteEvery 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.
ComputationFinding 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.
DiscreteHalf the neighbours forces a tour
Deciding whether a graph has a closed tour through every point is one of the hardest problems computers are asked to solve. Gabriel Dirac found a condition that settles it in one glance: if every point is joined to at least half the others, a tour exists. The proof is a counting argument on a single path, it runs as an algorithm, and two small graphs show that the half cannot be lowered — while random graphs show how much stronger than necessary it is.
ComputationSixteen 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.
DiscreteThe boards a knight can tour
A knight can visit every square of a chessboard once and land a move from where it began. On a 5 × 5 board it cannot, on a 4 × 100 board it cannot, and on a 3 × 8 board it cannot, though on 3 × 10 it can in sixteen ways. Allen Schwenk found in 1991 the complete list of rectangles without a closed tour, and the reasons on it are of two kinds — a colouring that counts squares, and a second colouring that a tour would have to make agree with the first.
DiscreteThe symmetric graphs no tour can close
Take the words of five places with two ones and join two when they share no one. The ten words look exactly alike to the graph, and no closed tour runs through them — because every way of leaving out one edge at each word leaves two pentagons. Only four connected graphs this symmetric are known to fail like that, and nobody knows whether a fifth exists.
DiscreteThe tour arrives with the last second neighbour
Add the edges between n points one at a time in a random order. A tour through every point cannot exist while some point has fewer than two neighbours, and in a large random graph that is the only thing stopping it: the tour appears at the very edge that gives the last point its second neighbour. Watched in thousands of small graphs, the coincidence is already the rule, and the exceptions are tiny traps around the points of lowest degree.
DiscreteThe 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.