Tessellations

Definition

Tessellations are patterns formed by fitting together shapes without any gaps or overlaps, covering a surface completely. They can be created using various geometric shapes, including regular polygons, and have significant implications in different areas of mathematics, particularly in hyperbolic geometry, where the rules for creating tessellations differ from those in Euclidean space.

5 Must Know Facts For Your Next Test

1. In hyperbolic geometry, tessellations can use an infinite number of shapes to fill the space without gaps, unlike in Euclidean geometry, where only specific shapes work.
2. The angles in hyperbolic tessellations must sum to less than 360 degrees at each vertex, which allows for more complex arrangements compared to flat surfaces.
3. Tessellations can be created using combinations of regular and irregular shapes, and this versatility is crucial in hyperbolic settings.
4. Art inspired by hyperbolic tessellations often showcases intricate patterns and designs that visually represent the unique properties of hyperbolic space.
5. Mathematically, tessellations in hyperbolic geometry can be represented using models like the Poincaré disk or the hyperboloid model, providing tools to visualize these complex patterns.

Review Questions

• How do the properties of tessellations differ between hyperbolic geometry and Euclidean geometry?
  • Tessellations in hyperbolic geometry have distinct properties compared to those in Euclidean geometry due to the negative curvature of hyperbolic space. In hyperbolic tessellations, shapes can fit together in ways that allow for infinite combinations without gaps or overlaps. Additionally, the angles around each vertex in hyperbolic tessellations sum to less than 360 degrees, enabling more complex patterns that cannot occur on flat surfaces. This fundamental difference leads to unique artistic and mathematical applications.
• Discuss the significance of angles in hyperbolic tessellations and how they impact the overall structure of the patterns.
  • In hyperbolic tessellations, the sum of angles around a vertex is always less than 360 degrees, which allows for a greater variety of configurations than found in Euclidean spaces. This means that multiple shapes can meet at a point without causing overlaps or leaving gaps. The lower angle sum leads to intricate structures where numerous identical shapes can connect at vertices, creating complex designs that highlight the uniqueness of hyperbolic geometry. As a result, this property has important implications for both theoretical mathematics and practical applications such as art and architecture.
• Evaluate the role of models like the Poincaré disk in understanding hyperbolic tessellations and their properties.
  • Models like the Poincaré disk are crucial for visualizing and comprehending hyperbolic tessellations because they provide a way to represent the infinite nature of hyperbolic space within a bounded area. By mapping points on the hyperbolic plane to points within a circular disk, it becomes easier to study how shapes can tessellate this space without gaps or overlaps. These models illustrate how traditional Euclidean concepts need to be adapted in hyperbolic contexts, enabling mathematicians and artists alike to explore complex patterns and relationships inherent in hyperbolic geometry.

Related terms

Polygon:

A closed figure made up of a finite number of straight line segments connected end to end.

Symmetry:

A property where a shape or object is invariant under certain transformations, such as reflection or rotation.

Hyperbolic Plane: A surface that exhibits a constant negative curvature, allowing for unique geometric properties and tessellation possibilities.