Regular tessellation

Definition

A regular tessellation is a pattern formed by repeating a single type of regular polygon without any gaps or overlaps. These shapes fit together perfectly in a plane, creating a visually appealing design that exhibits both symmetry and regularity. Regular tessellations can be categorized into three types based on the polygons used: triangles, squares, and hexagons, which are the only regular polygons capable of tessellating the plane on their own.

5 Must Know Facts For Your Next Test

1. Regular tessellations can only be created using three types of regular polygons: equilateral triangles, squares, and regular hexagons.
2. The internal angles of the polygons used in regular tessellations must add up to exactly 360 degrees at each vertex for the tessellation to work.
3. Regular tessellations can be seen in nature and architecture, often found in honeycombs or tile designs.
4. When visualizing a regular tessellation, it's important to note that all tiles are congruent and are arranged in a way that they completely cover a surface.
5. Unlike irregular tessellations, which can use multiple polygon types, regular tessellations maintain uniformity by using only one polygon type throughout.

Review Questions

• How do the properties of regular polygons determine the ability to create a regular tessellation?
  • The ability to create a regular tessellation depends on the internal angles of the regular polygons being used. For a tessellation to form without gaps or overlaps, the angles at each vertex where the polygons meet must sum to 360 degrees. This is why only equilateral triangles (60 degrees), squares (90 degrees), and regular hexagons (120 degrees) can create regular tessellations; their angle measures allow them to fit together perfectly at each vertex.
• Discuss how symmetry plays a role in the aesthetic appeal of regular tessellations.
  • Symmetry is central to the aesthetic appeal of regular tessellations as it creates a sense of balance and harmony within the design. Each shape used in a regular tessellation maintains uniformity and repetition, contributing to an overall symmetrical appearance. This symmetry not only enhances visual interest but also allows for predictable patterns that are pleasing to the eye, making regular tessellations popular in art and architecture.
• Evaluate the significance of regular tessellations in both mathematical theory and practical applications.
  • Regular tessellations hold significant importance in mathematical theory as they illustrate concepts such as geometry, spatial reasoning, and symmetry. They serve as fundamental examples in understanding tiling patterns and their properties. Practically, regular tessellations find applications in various fields including art, design, architecture, and even nature. For instance, they can be observed in tiling floors or walls and are crucial in optimizing space in architectural designs while also providing visual interest.

Related terms

Tessellation: The covering of a surface with a pattern of one or more geometric shapes, called tiles, without overlaps or gaps.

Polygon: A closed geometric figure with at least three straight sides and angles, commonly classified as regular or irregular based on the equality of its sides and angles.

Symmetry: A property where a shape or pattern remains unchanged under certain transformations, such as rotation, reflection, or translation.