Regular tessellation

Definition

A regular tessellation is a way of covering a surface using one type of regular polygon in a repeating pattern without any gaps or overlaps. In the context of geometry, regular tessellations occur in both Euclidean and non-Euclidean spaces, including hyperbolic geometry, where the rules for tiling can differ significantly from traditional flat surfaces.

5 Must Know Facts For Your Next Test

1. In Euclidean space, the only regular polygons that can tessellate are equilateral triangles, squares, and regular hexagons.
2. Regular tessellations can create interesting visual patterns and are often used in art and architecture to create aesthetically pleasing designs.
3. In hyperbolic geometry, it is possible to have regular tessellations using different types of polygons due to the unique properties of hyperbolic space.
4. The angles around each vertex in a regular tessellation must add up to exactly 360 degrees for a proper fit; in hyperbolic space, this sum can be less than 360 degrees.
5. Regular tessellations are fundamental in studying symmetry and group theory as they reveal underlying geometric properties of shapes and spaces.

Review Questions

• How do regular polygons contribute to the concept of regular tessellations, especially in Euclidean spaces?
  • Regular polygons are crucial for understanding regular tessellations because only certain types can fit together without gaps or overlaps. In Euclidean spaces, only equilateral triangles, squares, and regular hexagons can achieve this perfect tiling. Each polygon must have internal angles that allow them to come together neatly at vertices while maintaining the overall structure of the tessellation.
• Discuss the differences between regular tessellations in Euclidean versus hyperbolic geometry.
  • Regular tessellations in Euclidean geometry are limited to specific regular polygons like triangles, squares, and hexagons due to their angle sum properties. In contrast, hyperbolic geometry allows for more complex arrangements and a variety of polygons because the rules governing angles and distances differ significantly. This leads to unique and intricate patterns not achievable in flat geometries.
• Evaluate the implications of regular tessellations on the study of symmetry and geometric transformations across different geometries.
  • Regular tessellations serve as a foundational concept for studying symmetry and geometric transformations. They highlight how different shapes can maintain symmetry through various transformations while occupying space. In both Euclidean and hyperbolic geometries, these patterns reveal how angles and distances interact differently, leading to a richer understanding of geometric principles that apply across multiple mathematical frameworks.

Related terms

Regular polygon: A regular polygon is a polygon with all sides and angles equal, such as equilateral triangles, squares, and regular hexagons.

Tessellation: Tessellation refers to the covering of a plane with one or more geometric shapes, called tiles, arranged in a repeating pattern without gaps or overlaps.

Hyperbolic geometry: Hyperbolic geometry is a non-Euclidean geometry characterized by a space where the parallel postulate does not hold, allowing for unique properties regarding angles and distances.