Tessellations are fascinating patterns that cover surfaces without gaps or overlaps. They're created using shapes that fit together perfectly, like puzzle pieces. These patterns rely on transformations like sliding, flipping, and rotating to arrange shapes in repeating, predictable ways.
Regular polygons, such as triangles, squares, and hexagons, can create tessellations on their own. Their interior angles must divide evenly into 360° to work. Other shapes can be combined to form more complex patterns, opening up endless creative possibilities.
Properties and Creation of Tessellations
Properties of tessellations
- Cover a plane completely without any gaps or overlaps using one or more shapes that fit together perfectly
- Exhibit repeating patterns where the same shape or combination of shapes is used throughout and arranged in a consistent, predictable manner across the entire plane
- Created using transformations such as translations, rotations, reflections, and glide reflections applied to the shapes to allow them to fit together seamlessly
- Translations slide a shape in a specific direction by a fixed distance in a straight line without changing its orientation or size
- Rotations turn a shape around a fixed point by a specific angle clockwise or counterclockwise while maintaining its size and shape
- Reflections flip a shape across a line, creating a mirrored reverse image while maintaining its size and shape
- Glide reflections combine a translation (sliding) and a reflection across a line while maintaining the size and shape of the original figure
- Complex tessellating patterns can be created by combining these transformations
Tessellations with Regular Polygons
Interior angles for regular tessellations
- Regular polygons have equal side lengths and equal interior angles
- Sum of interior angles of a polygon calculated using the formula: $(n - 2) \times 180°$, where $n$ is the number of sides
- For a regular polygon to tessellate a plane by itself, the interior angle must divide 360° evenly (be a factor of 360°) to avoid gaps or overlaps
- Equilateral triangles have 60° interior angles $(180° \times \frac{1}{3})$ which divides 360° evenly $(360° \div 60° = 6)$, allowing them to tessellate a plane by themselves
- Squares have 90° interior angles $(180° \times \frac{1}{2})$ which divides 360° evenly $(360° \div 90° = 4)$, allowing them to tessellate a plane by themselves
- Regular hexagons have 120° interior angles $(180° \times \frac{2}{3})$ which divides 360° evenly $(360° \div 120° = 3)$, allowing them to tessellate a plane by themselves
- Regular pentagons, heptagons, and other regular polygons with more than six sides cannot tessellate a plane by themselves because their interior angles do not divide 360° evenly, resulting in gaps or overlaps when attempting to tessellate
- Convex polygons, where all interior angles are less than 180°, are commonly used in tessellations due to their ability to fit together without creating inward-facing corners
Types of Tessellations
Periodic and Aperiodic Tilings
- Periodic tilings (tessellations) have a repeating pattern that can be translated in two independent directions to cover the entire plane
- Aperiodic tilings lack a repeating pattern that can be translated in two independent directions, yet still cover the plane without gaps or overlaps
- Monohedral tilings use only one shape (tile) to cover the plane, such as the regular tessellations with equilateral triangles, squares, or regular hexagons
- Dihedral tilings use two different shapes to create the tessellation pattern