A team of physicists has created what they claim is the most difficult labyrinth ever, using a pattern from a chess game to create the structure. To the untrained eye, a maze looks like the most complex snowflake. But for the puzzle lovers among us, it probably looks like a challenge.
A maze is constructed from a Hamiltonian cycle, a graph cycle that visits each node in the graph only once. The same movement pattern is shown in the “Knight’s Path” in chess, where a chess piece can visit each square on the board once without repeating before returning to its starting piece.
The team maze works similarly; it is a set of Hamiltonian cycles in quasicrystals. Don’t worry, we’ll explain. A detailed explanation of the maze design was accepted for publication in Physical overview X.
“When we looked at the shapes of the lines we constructed, we noticed that they form incredibly complex mazes,” said Felix Flicker, a physicist at the University of Bristol and co-author of the paper in the university report. “The sizes of the subsequent mazes grow exponentially – and there are an infinite number of them.”
Quasicrystals are a rare type of matter. Ordinary crystals have a periodic structure, meaning that their building blocks repeat regularly. But the building blocks of a quasicrystal do not repeat themselves regularly; they have asymmetric, non-repeating structures that make them confusing in three dimensions and almost magical in others; in 2022, a team of physicists managed to keep a quantum system coherent longer by using lasers to direct a quasi-crystalline pattern onto individual atoms – in other words, a quasi-crystal in time. One example of a 3D quasicrystal is an icosahedron, a 20-sided shape similar in appearance to a standard soccer ball. As one physicist told Gizmodo in 2021:
“The moment you go from periodic to quasi-periodic, all bets are off on symmetry… All those 200-year-old rules are off—any symmetry is allowed, including the most famous forbidden symmetry for solids, which is icosahedron symmetry. With quasi-crystals, you suddenly have an infinity of possibilities.”
Quasi-crystals occur in nature in rare cases. Some have been found in lonsdaleite, a mineral harder than diamonds that does not occur naturally on Earth but rained down to us in meteorites. In 2021, physicists discovered that the quasi-crystals formed in trinitite, a bizarre material that formed after the Trinity bomb test in 1945 turned swathes of the New Mexico desert into glass.
A recent team presented an algorithm for creating cycles of Hamiltonian graphs on two-dimensional spaces called Ammann-Beenker tiles. The team says these two-dimensional mazes show Hamiltonian cycles mimicking the atomic patterns of the quasicrystal.
“We show that certain quasicrystals provide a special case in which the problem is unexpectedly simple,” Flicker said. “Therefore, in this setting, we make some seemingly impossible problems solvable.” This could include practical purposes involving different fields of science.
The formula actually has scientific implications. As stated in the university version, the Hamiltonian cycle provides microscopic imagers such as scanning tunneling microscopes with the fastest way to cover an object. There are also implications for the use of the quasicrystal in several physics problems, including one that can be used to model protein folding.
But if you’re not involved in any of these fields, you can step back and appreciate the way mathematics reveals some of the more exotic patterns in our physical universe.