A tessellation is a pattern that covers a flat plane with shapes so that there are no gaps and no overlaps. Tessellations appear in floor tiles, fabric designs, brick walls, art, and computer graphics. In geometry, they help us understand how angles, symmetry, and repeating patterns work together.
The key idea is that the shapes must fit perfectly around every point where their corners meet.
The most important test for a tessellation is the angle condition at each vertex. The angles around a point must add to exactly 360 degrees, because one full turn around a point is 360 degrees. Regular tessellations use only one type of regular polygon, while semi-regular tessellations use two or more regular polygons in the same repeating vertex pattern.
By calculating interior angles, students can predict whether a set of polygons can tile the plane before drawing the full pattern.
Understanding Geometry: Tessellations in Depth
A useful way to build a pattern is to start with one tile and move it by a transformation. A translation slides the tile without turning it. A rotation turns it about a chosen point.
A reflection flips it across a line, like a mirror image. Many wall patterns are made mainly by translations, while star-like designs often use rotations. These movements preserve lengths and angle sizes.
That is why a tile can be copied many times without slowly changing shape. When drawing a repeating design, students should mark one basic unit first. Repeating that unit carefully is more reliable than placing every shape by eye.
The angle rule is necessary, but it is not the only thing to check. Edges must match in length wherever two tiles share a side. Their directions must line up too.
A set of shapes can appear to fit near one corner yet fail farther away because an unmatched edge creates a narrow gap. Curved boundaries create another challenge. They can tessellate if each outward bump is paired with an inward dent of the same shape.
This idea explains many animal-like tessellations in art. The outline may look complicated, but the repeated tile still comes from a shape that can be copied by sliding, turning, or flipping.
Regular polygons become harder to pack as their number of sides increases. Their interior angles get closer to 180 degrees. Around a point, several large angles quickly exceed a full turn, while too few leave space uncovered.
This gives a fast way to predict which combinations are possible. For example, a triangle with an angle of 60 degrees can place six copies around one point. A square with an angle of 90 degrees can place four copies there.
When using mixed polygons, list the shapes in the order they appear around a vertex. That order must stay the same throughout a uniform semi-regular pattern. Changing the order can alter the design or make it fail.
Tessellations matter beyond classroom drawings. Bricklayers use offset rows because the joints are less likely to form long weak lines. Designers use repeating tile units in wallpaper, textiles, and digital textures.
In computer graphics, a surface is often broken into many triangles because triangles are stable, simple to calculate, and always lie in one flat plane. Maps, engineering models, and video games use this triangular mesh idea.
When studying tessellations, pay close attention to vertices, shared edges, and the smallest repeating region. These details reveal whether a pattern truly continues forever or only looks convincing in a small sketch.
Key Facts
- A tessellation covers a plane with no gaps and no overlaps.
- Angles meeting at each vertex must add to 360 degrees.
- Interior angle of a regular n-gon: A = 180(n - 2) / n degrees.
- Only equilateral triangles, squares, and regular hexagons make regular tessellations by themselves.
- For a regular tessellation, kA = 360, where k is the number of polygons meeting at each vertex.
- Semi-regular tessellations use two or more regular polygons with the same vertex arrangement everywhere.
Vocabulary
- Tessellation
- A tessellation is a repeating arrangement of shapes that covers a flat surface with no gaps and no overlaps.
- Vertex
- A vertex is a corner point where sides of polygons meet.
- Regular polygon
- A regular polygon is a polygon with all sides equal in length and all interior angles equal in measure.
- Interior angle
- An interior angle is an angle inside a polygon formed by two neighboring sides.
- Semi-regular tessellation
- A semi-regular tessellation is a tiling made from two or more regular polygons arranged in the same order at every vertex.
Common Mistakes to Avoid
- Adding the wrong angles at a vertex: Students sometimes add side lengths or exterior angles instead of interior angles, but tessellations depend on the interior angles meeting around a point.
- Forgetting the total must be exactly 360 degrees: A sum less than 360 degrees leaves a gap, while a sum greater than 360 degrees forces an overlap.
- Assuming every regular polygon tessellates alone: Regular pentagons have interior angles of 108 degrees, and 108 does not divide evenly into 360.
- Checking only one part of the pattern: A true tessellation must have no gaps or overlaps across the entire plane, not just in one small cluster.
Practice Questions
- 1 A regular hexagon has an interior angle of 120 degrees. How many regular hexagons must meet at one vertex to make a tessellation?
- 2 Use A = 180(n - 2) / n to find the interior angle of a regular octagon. Can regular octagons tessellate the plane by themselves? Explain using 360 degrees.
- 3 A pattern places one square, one regular hexagon, and one regular dodecagon around the same vertex. Explain whether this combination can form a semi-regular tessellation based on the angle sum.