Tessellations Using Rotations

A tessellation is a pattern made of repeated shapes that cover a flat surface completely, with no gaps and no overlaps. There are several ways to build a tessellation, and one of the most common is by rotation: take a single shape, turn it around a fixed center point by some fixed angle, and repeat. Each turned copy locks into place next to the last one, and the pattern grows outward to fill the plane.

This is different from building a tessellation by sliding a shape sideways (a translation) or flipping it (a reflection). If you want a refresher on those methods, see the lesson on tessellation reflection techniques. Rotation tessellations rely on a very specific idea: the angle you rotate by has to fit evenly around a point.

Any full turn around a point is \(360^\circ\). If several copies of a shape are going to meet at a single point without leaving a gap or overlapping, the angles they contribute at that point must add up to exactly \(360^\circ\). That means the rotation angle \(\theta\) you use must satisfy:

\( n \times \theta = 360^\circ \), where \(n\) is a whole number (the number of copies meeting at that point).

For example, rotating by \(120^\circ\) works because \(3 \times 120^\circ = 360^\circ\), so three copies fit perfectly around a point. Rotating by \(90^\circ\) works too, since \(4 \times 90^\circ = 360^\circ\). But rotating by \(100^\circ\) would leave a \(20^\circ\) gap after three turns and a \(60^\circ\) overlap after four, so it will not tessellate cleanly.

This is closely tied to rotational symmetry: a shape's own order of rotational symmetry tells you which angles will map it onto itself, and shapes with high rotational symmetry (equilateral triangles, squares, regular hexagons) are the easiest starting points for rotation tessellations.

Here is the general process for tiling a plane using rotations:

1. Choose a starting shape and a center point, usually a vertex of the shape.
2. Pick a rotation angle \(\theta\) such that \(360^\circ\) divides evenly by \(\theta\).
3. Rotate the shape by \(\theta\) around the center point to create the next copy.
4. Repeat the rotation until the copies have gone all the way around the point (a full \(360^\circ\)).
5. Move to a new center point on the outer edge of the pattern and repeat, letting the tessellation grow outward.

Because every copy is the exact same shape and size, just spun to a new orientation, the side lengths and angles always match up along shared edges.

Suppose you start with a triangle that has one vertex sitting at a center point, and the angle of the triangle at that vertex is \(120^\circ\). Since \(360^\circ \div 120^\circ = 3\), rotating the triangle by \(120^\circ\) twice more (so three copies total) will exactly surround the center point with no gap and no overlap. The diagram below shows the original triangle together with the two rotated copies, forming a pinwheel around the shared center point.

Notice that each triangle is identical in shape and size — only its orientation changes. This is the core idea behind every rotational tessellation: repeat the turn, check the angles sum to \(360^\circ\) at each meeting point, and the pattern tiles perfectly.

Regular polygons whose interior angle divides evenly into \(360^\circ\) are natural candidates:

• Equilateral triangles (interior angle \(60^\circ\), six meet at a point)
• Squares (interior angle \(90^\circ\), four meet at a point)
• Regular hexagons (interior angle \(120^\circ\), three meet at a point)

More artistic and irregular shapes can also tessellate by rotation, as long as the rotated copies still add up to \(360^\circ\) around every point where they meet — this is how many decorative and pinwheel-style tile patterns are constructed. Before working through rotation tessellations, it helps to be comfortable with the basic idea of turning a figure around a point; see Introduction to transformations for that foundation.

• Forgetting to check that the rotation angle actually divides \(360^\circ\) evenly — if it doesn't, gaps or overlaps will appear.
• Rotating around the wrong point; the center of rotation must stay fixed for every copy in that cluster.
• Mixing up rotation with reflection: a rotated copy keeps the same handedness, while a reflected copy is mirrored.
• Assuming any shape with rotational symmetry will automatically tessellate the whole plane — the angle condition must hold at every point in the pattern, not just the first one.

Related lessons

• Geometry Help