Abstract

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Symmetry: Culture and Science
Volume 36, Number 1, pages 059-066 (2025)
https://doi.org/10.26830/symmetry_2025_1_059

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

Mark D. Tomenes^1*, 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

^* corresponding author

Abstract: This paper presents a method for constructing tilings of the Euclidean space . Here, an tiling is defined as a tiling with , , and transitivity classes of vertices, edges, faces, and tiles, respectively, under the action of its symmetry group. By employing the process of dualization and subgroup structures of symmetry groups, tilings with varying transitivity properties can be derived from a known tiling. Using the method, one can use the subgroup structure of the symmetry group of a known tiling to construct new tilings. In particular, application of this method to the regular cubical tiling gives rise a new (1,1,1,2) tiling or a quasiregular tiling.

Keywords: tilings, cubical symmetries, transitivity properties.

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