Snub tetrahexagonal tiling

Uniform tiling 64-snub

Snub Tetrahexagonal Tiling is a unique and fascinating pattern in the realm of geometry and tiling. It is a type of uniform tiling that combines the properties of both snub and tetrahexagonal configurations, making it a subject of interest in the fields of mathematics, crystallography, and art.

Definition

The Snub Tetrahexagonal Tiling is characterized by its intricate pattern, which is formed by alternating between different types of polygons in a specific arrangement. This tiling is part of the Euclidean plane tilings, which means it extends infinitely in all directions on a flat plane. The unique aspect of this tiling is its combination of squares, hexagons, and triangles in a snub fashion, which means the triangles are irregular, and the pattern lacks mirror symmetry, distinguishing it from regular tilings.

Geometry

In the Snub Tetrahexagonal Tiling, each vertex is surrounded by two squares, one hexagon, and one triangle. This arrangement leads to a highly uniform but non-regular pattern due to the absence of reflective symmetry. The tiling can be constructed by a process known as snubification, applied to the tetrahexagonal tiling, which involves inserting triangles between the edges of the squares and hexagons and then twisting the figure to remove any mirror symmetries.

Mathematical Properties

The Snub Tetrahexagonal Tiling exhibits several interesting mathematical properties. It is a quasiregular tiling, meaning it has two types of vertices that alternate in a regular pattern. The tiling also has a high degree of symmetry, specifically belonging to the dihedral symmetry group, despite its lack of reflective symmetry. This makes it a point of study in the field of group theory and symmetry in mathematics.

Applications

While the Snub Tetrahexagonal Tiling is primarily of theoretical interest, it has applications in various fields. In crystallography, similar patterns can be observed in the arrangement of atoms in certain crystalline structures. In art and architecture, the aesthetic appeal of its intricate patterns has inspired designs in tiling floors, walls, and other surfaces. Additionally, the study of such tilings contributes to the understanding of periodic tilings and aperiodic tilings in both two-dimensional and three-dimensional spaces.

See Also

• Uniform tiling
• Euclidean plane
• Symmetry in mathematics
• Crystallography
• Art