Daedalus could learn a thing or two from a team of physicists in the UK and Switzerland.
Taking principles from fractal geometry and the strategic game of chess, they created what they say is the most fiendishly difficult maze ever created.
Led by physicist Felix Flicker of the University of Bristol in the UK, the group has generated pathways called Hamiltonian cycles in patterns known as Amann-Beenker pads, creating complex fractal mazes that they believe describe an exotic form of matter known as quasicrystals.
And all this was inspired by the movement of a knight around a chessboard.
“When we looked at the shapes of the lines we constructed, we noticed that they formed incredibly complex labyrinths. The sizes of the subsequent mazes grow exponentially – and there are an infinite number of them,” explains Flicker.
“In the knight’s tour, the chess piece (which jumps two squares forward and one to the right) visits each square on the chessboard only once before returning to the starting square. This is an example of a ‘Hamiltonian loop’ – a loop through a map visiting all stopping points only once.”
Quasicrystals are a form of matter that is extremely rare in nature. They are sort of a 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 top of another and they will match perfectly.
A disordered or amorphous solid is one in which all the atoms are simply heterogeneous. 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 perfectly. It may look quite similar to itself, but the overlapping areas of the pattern will not match.
These similar-looking but non-identical patterns are very similar to a mathematical concept called aperiodic tiling, which involves patterns of shapes that do not repeat identically.
The famous Penrose tiling is one of them. The Ammann-Beenker liner is another.
Using a set of two-dimensional Ammann-Beenker pads, Flicker and his colleagues, physicists Shobna Singh of Cardiff University in the United Kingdom and Jerome Lloyd of the University of Geneva in Switzerland, generated Hamiltonian cycles that they believe describe the atomic model of a quasicrystal.
Their generated loops visit each atom in the quasicrystal only once, connecting all the atoms in a single line that never intersects but continues cleanly from start to finish. And this can be scaled infinitely, generating a kind 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 much bigger implications beyond entertaining nervous kids in cafeterias.
On the one hand, finding Hamiltonian cycles is extremely difficult. A solution that would allow the identification of Hamiltonians has the potential to solve many other difficult mathematical problems, from complex route-finding systems to protein folding.
And, interestingly, it has implications for carbon capture through adsorption, an industrial process that involves transferring molecules into 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 along the Hamiltonian cycle within them.
“Our work also shows that quasicrystals may be better than crystals for some adsorption applications,” says Singh.
“For example, bending molecules will find more ways to land on the misaligned atoms of quasicrystals. Quasicrystals are also brittle, meaning they break up into small grains easily. This maximizes their surface area for adsorption.”
And if you happen to have a minotaur you need to hide somewhere, we think we know someone who can help.
The study was published in Physical examination X.
