In an innovative study, physicists have used the complexity of chess to design intricate mazes that might help solve some of the world’s most pressing issues. These unique labyrinths, inspired by the Knight’s moves on a chessboard, could assist in addressing challenges such as industrial processes, carbon capture, and fertilizer production.
Lead researcher Dr. Felix Flicker, Associate Professor of Physics at the University of Bristol, explained, We noticed that the shapes of the lines we constructed formed incredibly intricate mazes. The sizes of these mazes grow exponentially, and there are an infinite number of them.
A Knight’s tour involves the chess piece moving in an L-shape, visiting each square on the board exactly once before returning to its starting point. This tour exemplifies a ‘Hamiltonian cycle’, a path that visits each point once. Theoretical physicists at the University of Bristol have constructed infinitely large Hamiltonian cycles in irregular structures representing an exotic form of matter called quasicrystals.
Quasicrystals differ from regular crystals like salt or quartz because their atomic patterns do not repeat at regular intervals. Instead, they can be described mathematically as slices through six-dimensional crystals. Natural quasicrystals have only been found in a single Siberian meteorite, while the first artificial quasicrystal was accidentally created during the 1945 Trinity Test, the atomic bomb explosion depicted in Oppenheimer.
The group’s Hamiltonian cycles visit each atom on the surface of certain quasicrystals exactly once, creating uniquely complex mazes described by ‘fractals.’ These paths allow an atomically sharp pencil to draw straight lines connecting all neighboring atoms without lifting or crossing over. This is beneficial for ‘scanning tunneling microscopy’, where the pencil represents a microscope tip capturing images of individual atoms. Hamiltonian cycles provide the fastest possible routes for the microscope, crucial since producing a top-tier image can take a month.
Solving Hamiltonian cycles in general contexts is immensely challenging and could resolve many significant unsolved problems in mathematical sciences. Dr. Flicker noted, Certain quasicrystals provide a special case where the problem is unexpectedly simple. This makes seemingly impossible problems manageable, potentially serving practical purposes across various scientific areas.
For example, ‘adsorption’ involves molecules adhering to crystal surfaces and is vital in industrial processes. Currently, only crystals are used for industrial-scale adsorption. If surface atoms admit a Hamiltonian cycle, appropriately sized flexible molecules can pack efficiently along these atomic mazes. The research shows quasicrystals can be highly efficient adsorbers, which is important for carbon capture and storage to prevent CO2 emissions.
Co-author Shobhna Singh, a doctoral researcher in Physics at Cardiff University, stated, Quasicrystals can be better than crystals for some adsorption applications. Flexible molecules will find more ways to land on the irregularly arranged atoms of quasicrystals. They are also fragile, meaning they break into tiny grains, maximizing their surface area for adsorption.
Efficient adsorption also suggests quasicrystals could be effective catalysts, enhancing industrial efficiency by reducing the energy required for chemical reactions. For instance, adsorption is crucial in the Haber process, used to produce ammonia fertilizer for agriculture.
SOURCES
Shobhna Singh, Jerome Lloyd y Felix Flicker, Hamiltonian cycles on Ammann-Beenker Tilings. Physical Review X
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