Snub tetraheptagonal tiling

Uniform tiling 74-snub

Snub Tetraheptagonal Tiling is a unique and complex form of tiling that exists within the realm of geometry, specifically within the study of tilings of the hyperbolic plane. This tiling method is characterized by its use of both polygons and snub transformations to create a pattern that is non-periodic and fills the hyperbolic plane without gaps or overlaps. The snub tetraheptagonal tiling is a part of the broader category of uniform tilings that exhibit a high degree of symmetry and regularity.

Definition

The snub tetraheptagonal tiling is defined by its composition of tetraheptagons, a polygon with 47 sides, arranged in a snub pattern. In the context of hyperbolic geometry, a snub pattern refers to a configuration where polygons are arranged in a skewed, non-regular manner, but still follow a uniform rule of arrangement. This tiling is a hyperbolic tiling, meaning it cannot be realized in the Euclidean plane but only in the hyperbolic plane, where the rules of Euclidean geometry do not apply, and parallel lines may diverge.

Properties

One of the most notable properties of the snub tetraheptagonal tiling is its ability to cover the hyperbolic plane completely without any overlaps or gaps, a property known as tessellation. This tiling is also uniform, meaning it has the same pattern of polygons meeting at each vertex. However, due to the high number of sides on the tetraheptagons and the complex nature of the snub transformation, visualizing this tiling can be challenging without the aid of computer graphics.

Mathematical Background

The study of snub tetraheptagonal tiling falls under the broader category of hyperbolic geometry, a non-Euclidean geometry that describes the geometry of saddle-shaped surfaces, unlike the flat surfaces of Euclidean geometry. The angles of a polygon in a hyperbolic tiling add up to less than the sum of the angles in a similar Euclidean polygon, which allows for the existence of tilings that would not be possible in a Euclidean setting.

Applications and Significance

While the snub tetraheptagonal tiling may seem purely theoretical, it has applications and significance in various fields. In mathematics, it serves as an example of the rich diversity of geometric forms possible in hyperbolic space, contributing to the understanding of hyperbolic geometry. In art and architecture, the concepts derived from studying such tilings inspire designs that incorporate complex geometric patterns, especially in contexts that value non-repetitive, intricate designs.

See Also

• Hyperbolic geometry
• Tiling
• Uniform tiling
• Polygon