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.
Understanding Tessellations and Tile Patterns
The angle rule comes from the way a polygon is built. A regular polygon with a larger number of sides has corners that become wider. For example, an equilateral triangle has an interior angle of 60 degrees, a square has 90 degrees, and a regular hexagon has 120 degrees.
Students can calculate these values by splitting the polygon into equal triangles from its center. Each central angle equals 360 degrees divided by the number of sides. This leads to the interior angle rule, which is 180 degrees minus 360 degrees divided by the number of sides.
This calculation explains why most regular polygons cannot form a repeating vertex on their own. Three triangles make 180 degrees, so six are needed to complete a full turn. Four squares make 360 degrees.
Three hexagons make 360 degrees. A regular pentagon has an interior angle of 108 degrees. Three pentagons make 324 degrees, leaving a gap, while four make 432 degrees, causing an overlap.
Polygons with more sides have even wider angles, so they fail for the same reason. The three successful cases are not arbitrary. Their angle sizes divide evenly into a full turn.
Semi-regular patterns use a fixed order of different regular polygons at every vertex. One example places a triangle, a square, a triangle, and a square around each meeting point. Their angles total 60 plus 90 plus 60 plus 90 degrees, which makes 300 degrees, so that arrangement does not work.
A triangle, triangle, square, triangle, square arrangement totals 360 degrees and can work when the same order repeats everywhere. The order matters as much as the total.
Two arrangements may have the same angle measures but create different patterns if the shapes appear in a different sequence. Geometers describe a vertex by listing the numbers of sides in order, such as three, six, three, six.
Transformations explain how a pattern continues beyond one small section. A translation moves a chosen tile by the same distance in one direction. A rotation can repeat a motif around a central point.
Reflection symmetry creates mirror-image parts, though a pattern may have no reflection symmetry at all. In manufacturing, these ideas help designers make repeating wallpaper, fabric prints, paving blocks, and circuit layouts.
Small errors matter. If a tile angle is slightly wrong, the error can build up across many repeated joins and produce visible gaps.
When drawing or checking a tessellation, focus on one vertex before filling a whole page. List every angle that reaches that point and add them carefully. Then inspect the edge lengths and the order of the shapes.
Matching angles alone is not enough if edges of different lengths are forced together. It helps to mark corresponding edges with colors or short strokes.
Students should separate the local rule at one vertex from the larger symmetry of the full pattern. A design can fit perfectly at each vertex while having a more complicated repeating structure 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.