Symmetry and Tessellations (Edexcel GCSE Maths): Revision Notes

Symmetry and tessellations

Understanding symmetry and tessellations is essential for recognising patterns and properties in geometric shapes. There are two main types of symmetry you need to know about, plus the important concept of tessellations.

Line symmetry

Line symmetry occurs when you can draw one or more mirror lines across a shape, and both sides will fold together perfectly. When you fold the shape along these lines, one half will sit exactly on top of the other half.

The number of lines of symmetry varies depending on the shape:

• A regular pentagon has 5 lines of symmetry
• An isosceles triangle has 1 line of symmetry
• A parallelogram has no lines of symmetry
• A rhombus has 2 lines of symmetry
• A kite has 1 line of symmetry

To find lines of symmetry, imagine folding the shape along different lines. If the two halves match up perfectly when folded, then you've found a line of symmetry.

Rotational symmetry

Rotational symmetry describes how a shape can be turned into different positions that look exactly the same as the original. The order of rotational symmetry tells you how many times the shape looks identical during one complete turn.

Here's how different shapes behave under rotation:

• A square has rotational symmetry of order 4 (looks the same 4 times during a full rotation)
• A regular hexagon has rotational symmetry of order 6
• A parallelogram has rotational symmetry of order 2
• A rhombus has rotational symmetry of order 2
• A kite has rotational symmetry of order 1
• A trapezium has rotational symmetry of order 1

When a shape has rotational symmetry of order 1, this means it only looks the same in its original position. You can describe this as either "order 1 symmetry" or "no rotational symmetry."

Tessellations

Tessellations are tiling patterns where shapes fit together with no gaps or overlaps. They create repeating patterns that could theoretically continue forever.

Regular polygons will tessellate if their interior angles divide evenly into $360°$. This is because the angles around each vertex must add up to exactly $360°$ for the shapes to fit together perfectly.

Working out if shapes tessellate

To determine whether a regular polygon will tessellate, you need to:

1. Calculate the interior angle of the polygon
2. Divide $360°$ by this interior angle
3. If the result is a whole number, the polygon will tessellate

Only three regular polygons tessellate by themselves: triangles, squares, and hexagons. Their interior angles ($60°$, $90°$, and $120°$ respectively) all divide evenly into $360°$.