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       Physicists Have Created The World's Most Fiendishly Difficult Maze
        
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       Daedalus could have learned 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 have created what they say is the most fiendishly
       difficult maze ever devised.
        
       Led by physicist Felix Flicker of the University of Bristol in the UK,
       the group has generated routes called Hamiltonian cycles in patterns
       known as Ammann-Beenker tilings, producing complex fractal mazes that,
       they say, describe an exotic form of matter known as quasicrystals.
        
       And it was all inspired by the movement of a Knight around a chess
       board.
        
       "When we looked at the shapes of the lines we constructed, we noticed
       they formed incredibly intricate mazes. The sizes of subsequent mazes
       grow exponentially - and there are an infinite number of them,"
       Flicker explains.
        
       "In a Knight's tour, the chess piece (which jumps two squares forwards
       and one to the right) visits every square of the chessboard just once
       before returning to its starting square. This is an example of a
       'Hamiltonian cycle' - a loop through a map visiting all stopping
       points only once."
        
       An example of an open Knight's tour of a chessboard, with visited
       squares shaded. For the tour to be 'closed', the Knight needs to end
       on a square one Knight's move from where it started (so it can return
       to the starting square and go around the board again). (Ilmari
       Karonen/CC0/Wikimedia Commons)
        
       Quasicrystals are a form of matter only found very extremely rarely in
       nature. They're sort of a strange hybrid of ordered and disordered
       crystals in solids.
        
       In an ordered crystal - salt, or diamonds, or quartz - the atoms are
       arranged in a very neat pattern that repeats in three dimensions. You
       can take a section of this lattice and superimpose it on another, and
       they'll match up perfectly.
        
       A disordered, or amorphous, solid, is one in which the atoms are just
       all higgledy-piggledy. These include glass and some forms of ice
       usually not found on Earth.
        
       A maze generated by finding a Hamiltonian cycle on an Ammann-Beenker
       tiling. Don't worry. They can get much, much larger and harder. A
       solution can be seen down below. (University of Bristol)
        
       A quasicrystal is a material in which the atoms form a pattern, but
       the pattern does not repeat perfectly. It might seem pretty self-
       similar, but superimposed sections of the pattern will not match up.
        
       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 identically repeat.
        
       The famous Penrose tiling is one of these. The Ammann-Beenker tiling
       is another.
        
       Using a set of two-dimensional Ammann-Beenker tilings, Flicker and his
       colleagues, physicists Shobhna Singh of Cardiff University in the UK
       and Jerome Lloyd of the University of Geneva in Switzerland, generated
       Hamiltonian cycles that they say describe the atomic pattern of a
       quasicrystal.
        
       An Ammann-Beenker tiling with a thicker black line tracing out the
       Hamiltonian path by visiting each vertex. The purple lines are not
       part of the tiling. (Singh et al., _Phys. Rev. X_ , 2024)
        
       Their generated cycles visit each atom in the quasicrystal only once,
       connecting all the atoms in a single line that never crosses itself,
       but cleanly continues from beginning to end. And this can be scaled
       infinitely, generating a type of mathematical pattern known as a
       fractal, in which the smallest parts resemble the largest.
        
       This line then naturally produces a maze, with a start point and an
       exit. But the research has far greater implications beyond
       entertaining antsy children in diners.
        
       For one, finding Hamiltonian cycles is extremely difficult. A solution
       that would allow for Hamiltonians to be identified has the potential
       to solve many other tricky mathematical problems, from complex route
       finding systems to protein folding.
        
       And, interestingly, there are implications for carbon capture via
       adsorption, an industrial process that involves hoovering up molecules
       in a fluid by sticking them to crystals. If we could use quasicrystals
       for this process instead, flexible molecules could pack themselves
       more tightly by lying along the Hamiltonian cycle therein.
        
       One possible solution to the maze above. (University of Bristol)
        
       "Our work also shows quasicrystals may be better than crystals for
       some adsorption applications," Singh says.
        
       "For example, bendy molecules will find more ways to land on the
       irregularly arranged atoms of quasicrystals. Quasicrystals are also
       brittle, meaning they readily break into tiny grains. This maximizes
       their surface area for adsorption."
        
       And if you happen to have a minotaur you need to stash away somewhere,
       we think we know someone who can help.
        
       The research has been published in _Physical Review X_.
        
        
        
        
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