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Symmetry: Culture and Science
Volume 35, Number 3, pages 377-380 (2024)
https://doi.org/10.26830/symmetry_2024_3_377

HIGHLY SYMMETRIC (a, b, c, d) TILINGS WITH CUBICAL SYMMETRIES

Mark D. Tomenes¹, Ma. Louise Antonette N. De Las Peñas²

¹ Ateneo de Manila University, Quezon City, Metro Manila 1108, Philippines.
Email: mtomenes@ateneo.edu

² Ateneo de Manila University, Quezon City, Metro Manila 1108, Philippines.
Email: mdelaspenas@ateneo.edu

Abstract: Understanding how molecules arrange themselves in crystals is important in the construction of new materials. One way to model molecular and crystal structures is through the use of tilings. In particular, we consider (a,b,c,d) tilings, tilings that have a orbits of vertices, b orbits of edges, c orbits of faces and d orbits of tiles under the action of their respective symmetry groups, for small values of a, b, c and d. (a,b,c,d) tilings are important in crystal chemistry, as the structures most desirable as prime targets for synthesis are the ones that have high symmetry structure. The method presented in this paper is an extension of the work done by the authors on (a,b,c) tilings of the Euclidean plane E^2, the two-dimensional analogue of (a,b,c,d) tilings (Tomenes & De Las Peñas, n.d.a, n.d.b). We apply the method to obtain highly symmetric (a,b,c,d) tilings with cubical symmetries.

Keywords: (a, b, c, d) tilings, cubical symmetries, transitivity properties

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