Semi-regular tessellation

Definition

A semi-regular tessellation is a pattern formed by two or more types of regular polygons that fit together without gaps or overlaps, creating a repeating structure in a plane. These tessellations maintain a consistent arrangement of angles and vertices, allowing for aesthetic diversity while preserving the mathematical properties of the shapes involved. The unique feature of semi-regular tessellations lies in their ability to incorporate different polygons, making them distinct from regular tessellations, which use only one type of polygon.

5 Must Know Facts For Your Next Test

1. Semi-regular tessellations can only occur with combinations of regular polygons whose angles fit together perfectly at each vertex, such as triangles, squares, and hexagons.
2. There are eight unique semi-regular tessellations, each characterized by the arrangement and combination of polygons at their vertices.
3. Semi-regular tessellations exhibit symmetry and can create visually appealing patterns that are used in art, architecture, and nature.
4. The discovery and study of semi-regular tessellations helped in understanding the relationships between different geometric shapes and their properties.
5. Mathematicians like Escher have famously used semi-regular tessellations to create intricate artworks that challenge perceptions of space and form.

Review Questions

• What distinguishes semi-regular tessellations from regular tessellations, and how does this impact their geometrical properties?
  • Semi-regular tessellations are formed using two or more types of regular polygons, while regular tessellations consist of only one type. This distinction affects their geometrical properties by allowing for a variety of angles and vertices to come together in a structured way, resulting in diverse patterns that maintain specific symmetry and tiling characteristics. The presence of different polygons enables more complex designs that are visually striking yet mathematically coherent.
• Describe the process by which mathematicians classify and identify the eight unique types of semi-regular tessellations.
  • Mathematicians classify semi-regular tessellations by analyzing the combinations of polygons that can fit together at each vertex without gaps or overlaps. They focus on the internal angles of the polygons involved, ensuring they sum up to 360 degrees at every vertex. By systematically exploring different configurations and rotations of polygons such as triangles, squares, and hexagons, mathematicians have identified eight unique arrangements that fulfill these criteria, each leading to distinctive visual patterns.
• Evaluate the significance of semi-regular tessellations in both mathematical theory and artistic expression, providing examples from both realms.
  • Semi-regular tessellations hold significant importance in mathematical theory as they exemplify the relationships between different geometric shapes and their spatial properties. They serve as a foundation for advanced studies in geometry, topology, and symmetry. In artistic expression, artists like M.C. Escher utilized semi-regular tessellations to create intricate works that play with perception and visual complexity. This blending of mathematics and art showcases how abstract concepts can inspire creativity and how visual patterns can stimulate mathematical inquiry.

Related terms

Regular tessellation: A tiling pattern made up of only one type of regular polygon, where all the angles and sides are congruent.

Tessellation: The covering of a plane with one or more geometric shapes, called tiles, with no overlaps and no gaps.

Voronoi diagram: A partitioning of a plane into regions based on the distance to a specified set of points, creating an arrangement that can be seen as a type of tessellation.