Snub pentahexagonal tiling

Uniform tiling 65-snub

Snub pentahexagonal tiling is a type of tiling of the Euclidean plane by irregular polygons. It is a unique and fascinating form of tiling that combines elements of both geometry and art, creating patterns that are both aesthetically pleasing and mathematically significant. The snub pentahexagonal tiling is characterized by its combination of pentagons, hexagons, and triangles in a repeating pattern that lacks mirror symmetry, making it a chiral pattern.

Definition

The snub pentahexagonal tiling is defined by its specific arrangement of polygons in a way that each tiling vertex is surrounded by two triangles, one pentagon, and one hexagon. This arrangement does not allow for a direct line of symmetry, contributing to its classification as a snub tiling. The term "snub" refers to the operation in geometry that involves extending the sides of a polygon and twisting the figure in a way that removes any mirror symmetry, resulting in a chiral pattern.

Geometry

In the snub pentahexagonal tiling, the geometric properties are determined by the specific angles and side lengths of the triangles, pentagons, and hexagons. The precise arrangement ensures that the tiling covers the plane without any gaps or overlaps, adhering to the principles of Euclidean geometry. The angles and lengths are typically defined in such a way that the tiling exhibits uniformity, meaning that each vertex configuration is identical throughout the entire tiling.

Classification

The snub pentahexagonal tiling belongs to the family of uniform tilings of the plane, which are tilings made up of regular polygons and exhibit a high degree of symmetry. However, due to its lack of mirror symmetry, it is classified as a snub form, distinguishing it from other types of uniform tilings that possess reflective symmetry.

Applications

The study and application of snub pentahexagonal tiling extend beyond pure mathematics into areas such as architecture, art, and design. Its unique pattern can be found in various cultural artifacts, decorative art, and architectural designs, serving both aesthetic and functional purposes. In mathematics, it serves as an example of the complexity and beauty of geometric patterns, often used in the study of tiling theory and symmetry.

See Also

• Tiling
• Euclidean geometry
• Uniform tiling
• Symmetry in mathematics