Daedalus could learn something from a team of physicists in the UK and Switzerland.
Based on the principles of fractal geometry and the strategic game of chess, they have created what they say is the most fiendishly difficult maze ever devised.
The group, led by physicist Felix Flicker of the University of Bristol in the United Kingdom, created paths called Hamiltonian cycles in patterns known as Ammann-Beenker tilings, creating complex fractal mazes that they say describe an exotic form of matter known as quasicrystals.
And everything was inspired by the movement of the knight around the chessboard.
“When we looked at the shapes of the lines we built, we noticed that they formed incredibly complex mazes. The sizes of the subsequent mazes grow exponentially – and there are an infinite number of them,” explains Flicker.
“In a knight’s tour, a chess piece (which jumps two squares forward and one to the right) visits each square of the board only once before returning to its starting square. This is an example of a ‘Hamiltonian cycle’ – a map loop showing all stopping points only once.”
Quasi-crystals are a form of matter that occurs very rarely in nature. They are a kind of strange hybrid of ordered and disordered crystals in solids.
In an ordered crystal—salt, diamonds, or quartz—the atoms are arranged in a very neat pattern that repeats itself in three dimensions. You can take part of this grid and overlay it on another and they will match perfectly.
A disordered or amorphous solid is one in which all the atoms are just jumbled up. These include glass and some forms of ice not normally found on Earth.
A quasicrystal is a material in which the atoms form a pattern, but the pattern does not repeat itself perfectly. It may seem quite similar, but the overlapping parts of the pattern will not match.
These similar-looking but non-identical patterns are very similar to a mathematical concept called aperiodic tilings, which involve patterns of shapes that do not repeat themselves in the same way.
The famous Penrose tiling is one of them. Another is the Ammann-Beenker dressing.
Using a set of two-dimensional tilings, Ammann-Beenker Flicker and his colleagues, physicists Shobhna Singh of Cardiff University in the UK and Jerome Lloyd of the University of Geneva in Switzerland, created Hamiltonian cycles that they say describe the atomic pattern of the quasicrystal. .
The cycles they generate visit each atom in the quasi-crystal only once, joining all the atoms in a single line that never crosses but continues cleanly from start to finish. And this can be scaled infinitely, creating a type of mathematical pattern known as a fractal, in which the smallest parts resemble the largest.
This line then naturally creates a maze with a starting point and an exit. But the research has far bigger implications than just keeping nervous kids busy in cafeterias.
First, finding Hamiltonian cycles is extremely difficult. A solution that would allow the identification of Hamiltonians has the potential to solve many other tricky mathematical problems, from complex path-finding systems to protein folding.
And interestingly, there are implications for capturing carbon through adsorption, an industrial process that involves vacuuming up molecules in a liquid by sticking them to crystals. If we could instead use quasicrystals for this process, the flexible molecules could pack more tightly by lying within them along a Hamilton cycle.
“Our work also shows that quasicrystals may be better than crystals for some adsorption applications,” says Singh.
“For example, bending molecules find more ways to land on the irregularly arranged atoms of quasicrystals. Quasicrystals are also brittle, meaning they break up easily into tiny grains. This maximizes their surface area for adsorption.”
And if you happen to have a minotaur you need somewhere to hide, we think we know someone who can help.
The research was published in Physical overview X.