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Tessellations and Tile Patterns
Regular and Semi-Regular Tessellations
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A tessellation is a pattern of repeated shapes that covers a flat surface with no gaps and no overlaps. Tessellations appear in floor tiles, brick walls, quilts, and many works of art. They matter in geometry because they connect angle measures, symmetry, and transformations in a visual way. Studying them helps students see how simple rules can create complex patterns.
To make a tessellation, a shape must fit together with copies of itself around each point. Regular tessellations use one kind of regular polygon, while semi regular tessellations combine more than one type in a repeating arrangement. The key geometric idea is that angles meeting at a point must add to 360 degrees. Tessellations also show translations, rotations, reflections, and glide reflections acting on shapes across the plane.
Key Facts
- A tessellation covers a plane with no gaps and no overlaps.
- Interior angle of a regular n-gon: ((n - 2) x 180 degrees)/n
- For shapes meeting at a point in a tessellation, angle sum = 360 degrees.
- A regular tessellation uses only one type of regular polygon repeated everywhere.
- Only 3 regular polygons tessellate by themselves: equilateral triangles, squares, and regular hexagons.
- A translation slides a tile, a rotation turns it, and a reflection flips it to continue a pattern.
Vocabulary
- Tessellation
- A repeating arrangement of shapes that covers a flat surface completely without gaps or overlaps.
- Regular polygon
- A polygon with all sides equal and all interior angles equal.
- Interior angle
- The angle formed inside a polygon by two adjacent sides.
- Symmetry
- A property of a figure that stays unchanged after a transformation such as a reflection or rotation.
- Transformation
- A movement of a figure, such as a translation, rotation, or reflection, that changes its position or orientation.
Common Mistakes to Avoid
- Assuming any regular polygon can tessellate, which is wrong because the interior angles must fit exactly around a point to total 360 degrees.
- Adding side lengths instead of angles at a vertex, which is wrong because tessellation around a point depends on angle measure, not perimeter.
- Leaving tiny gaps or overlaps in a drawing, which is wrong because a true tessellation must cover the plane exactly with repeated tiles.
- Thinking a pattern is a tessellation just because it repeats, which is wrong because some repeating patterns still leave empty spaces or require distorted shapes.
Practice Questions
- 1 A regular hexagon has interior angle 120 degrees. How many regular hexagons can meet at one point in a tessellation?
- 2 A regular octagon has interior angle 135 degrees. Can regular octagons tessellate the plane by themselves? Show your angle reasoning.
- 3 Explain why equilateral triangles tessellate the plane but regular pentagons do not, using the angle sum around a point.