Mathematicians from the U.K. and Switzerland have designed the world’s most challenging maze based on geometric principles found in quasicrystals.
These materials have an atomic structure that repeats in a pattern, similar to the Penrose tiling. This research aims to explore the potential of quasicrystals, particularly in adsorption—a process where atoms or molecules stick to a surface, often used in carbon capture technology.
Hamiltonian Cycles & Ammann-Beenker Tilings
A new paper in Physical Review X details the creation of what could be the world’s most difficult maze, using Hamiltonian cycles and Ammann-Beenker tilings.
Scientists expanded this to irregular structures like Ammann-Beenker tilings, mimicking quasicrystals.(ref)
Hamiltonian Cycles
- Origin: The concept originates from the “Knight’s Tour” puzzle, where a knight on a chessboard must visit every square only once.(ref)
- Definition: In graph theory (the study of networks), a Hamiltonian cycle is a path that visits every vertex (node or point) exactly once and returns to the starting vertex.
- Significance: Finding Hamiltonian cycles is a classic problem in mathematics and computer science, with applications in routing, scheduling, and DNA sequencing.
Ammann-Beenker Tilings
- Nature: These are aperiodic tilings, meaning they lack translational symmetry (they don’t repeat in a regular pattern).
- Structure: Ammann-Beenker tilings are made of two types of rhombic tiles arranged in a specific pattern. They exhibit self-similarity, meaning they look similar at different scales.
- Connection to Quasicrystals: The structure of Ammann-Beenker tilings closely resembles quasicrystals, which are materials with ordered but non-repeating atomic structures.
Hamiltonian Cycles on Ammann-Beenker Tilings
- Challenge: Constructing Hamiltonian cycles on aperiodic tilings is a complex task due to their irregular nature.
- Breakthrough: The researchers in the Physical Review X paper developed a method to create Hamiltonian cycles on Ammann-Beenker tilings.
- Implication: This finding provides insights into the properties of quasicrystals and opens up new possibilities for their applications.
The Maze Connection
The Hamiltonian cycles on Ammann-Beenker tilings essentially form intricate mazes. The path of the cycle becomes the maze’s solution, winding through the tiling’s structure. The larger the tiling, the more complex the maze becomes, leading to the claim of the “world’s most difficult maze.”
Quasicrystals Are Slices of Six-Dimensional Crystals
Quasicrystals are materials that defy the conventional rules of crystallography. Unlike regular crystals, which have a repeating, periodic atomic structure, quasicrystals exhibit order but lack translational symmetry.(ref) This means that their patterns do not repeat at regular intervals, yet they maintain a long-range order.
Key Characteristics
- Aperiodic Structure: The most defining feature of quasicrystals is their aperiodic nature. Their atomic arrangement follows mathematical rules, but it never repeats exactly.
- Forbidden Symmetries: Quasicrystals can exhibit rotational symmetries that are impossible in traditional crystals. For example, they can have five-fold, eight-fold, or ten-fold rotational symmetry.
- Unique Properties: Due to their unusual structure, quasicrystals possess unique properties, such as low friction, high hardness, and low thermal conductivity. They are also promising candidates for various applications, including coatings, catalysts, and energy storage.
Infinite Mazes from Quasicrystal Cycles
Applying Hamiltonian cycles to quasicrystals creates incredibly intricate mazes called fractals. “The sizes of subsequent mazes grow exponentially—and there are an infinite number of them,” says lead author Felix Flicker.(ref)
Rare Quasicrystals & Carbon Capture
Quasicrystals are rare, found in a Siberian meteorite and created by the Trinity nuclear test.(ref) When Hamiltonian cycles are applied, they form mazes that could improve adsorption that are vital for carbon capture.
Quasicrystals vs. Crystals for Adsorption
Adsorption is when molecules adhere to a surface, unlike absorption (dissolving). Co-author Shobhna Singh explains, “Quasicrystals may be better than crystals for some adsorption applications.”(ref) Their irregular structure and brittleness create more surface area, ideal for capturing molecules like carbon dioxide.
The special traits of quasicrystals, along with the complex patterns found in mazes created from them, might lead to new ways to capture carbon, store energy, and make other important discoveries. By studying the link between shapes and materials, we could find solutions for a better future in unexpected places.
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Nancy Maffia
Nancy received a bachelor’s in biology from Elmira College and a master’s degree in horticulture and communications from the University of Kentucky. Worked in plant taxonomy at the University of Florida and the L. H. Bailey Hortorium at Cornell University, and wrote and edited gardening books at Rodale Press in Emmaus, PA. Her interests are plant identification, gardening, hiking, and reading.
