A tessellation is a tiling pattern in which two-dimensional shapes fit together with no overlaps or gaps. A regular tessellation uses congruent shapes.
Below are three examples of tessellations, using right triangles, parallelograms, and two slightly different birds, respectively. The first two are regular tessellations, the third is not.
A circle is an example of a shape that does not tessellate, since circles cannot be tiled without creating overlap or gaps:
For each shape below, name the shape and then decide whether or not it can be tessellated. Note that shapes may be rotated, reflected, and/or translated in order to make them tessellate.
Hint: For no gaps, what must the total angle be where vertices meet?
On a piece of cardboard or sturdy construction paper, draw an irregular quadrilateral (that is, a quadrilateral with all four sides of different lengths) and cut it out. Trace it on a piece of paper, then continue to move and retrace it to see whether it will tessellate or not.
Suggestion: If desired, use a digital tool instead of tracing by hand.
A key requirement for a tessellation is that the total angle where vertices meet must be \(360\degree\). Otherwise there will either be a gap (if it is less) or an overlap (if it is more).
The shape is an equilateral triangle. An equilateral triangle tessellates. A tessellation of six equilateral triangles is shown.
The total angle where the six vertices meet is \(6\times 60\degree=360\degree\). The tessellation can be extended by adding copies of the tessellation shown.
The shape is an isosceles triangle. An isosceles triangle tessellates. By rotating three copies of the triangle \(180\degree\), these three rotated triangles along with three other copies of the triangle can be arranged so that they meet at a common vertex, as shown.
Since the three angles in an isosceles triangle sum to \(180\degree\), the total angle at this vertex is \(2\times 180\degree=360\degree\). The tessellation can be extended by adding copies of the tessellation shown.
The shape is a square. A square tessellates. A tessellation of four squares is shown.
The total angle where the four vertices meet is \(4\times 90\degree=360\degree\). The tessellation can be extended by adding copies of the tessellation shown.
The shape is a regular pentagon. A regular pentagon does not tessellate. This is due to the fact that each angle in a regular pentagon is \(108\degree\), and \(3\times 108\degree=324\degree\), which is less than \(360\degree\), and \(4\times 108\degree=432\degree\), which is greater than \(360\degree\).
The shape is a regular hexagon. A regular hexagon tessellates. A tessellation of a regular hexagon is shown.
The total angle where the three vertices meet is \(3\times 120\degree=360\degree\). The tessellation can be extended by adding copies of the tessellation shown.
The shape is a regular octagon. A regular octagon does not tessellate. This is due to the fact that each angle in the octagon is \(135\degree\), and \(2\times 135\degree=270\degree\), which is less than \(360\degree\), and \(3\times 135\degree=405\degree\), which is greater than \(360\degree\).
Any quadrilateral will tessellate. To understand why, consider a quadrilateral with sides of length \(a\), \(b\), \(c\), and \(d\). By rotating two copies of the quadrilateral \(180\degree\), these rotated quadrilaterals along with other copies of the quadrilateral can be arranged to meet at a common vertex, as shown.
Since the interior angles in a quadrilateral sum to \(360\degree\), the total angle where the four vertices meet in the tiling shown is \(360\degree\). The tessellation can be extended by adding copies of the tessellation shown.
Extension: Can you use your solution to part (b) to show that any scalene triangle will tessellate?