Symmetry and tessellationIB MYP Maths Standard: Revision notes

A shape has line symmetry if a mirror line splits it into two halves that are reflections of each other. The mirror line is a line of symmetry. Fold the shape along the line and the two halves match exactly. A rectangle has $2$ lines of symmetry (through the midpoints of opposite sides) but its diagonals are not lines of symmetry. A regular polygon with $n$ sides has $n$ lines of symmetry.

A shape has rotational symmetry if it fits exactly onto its own outline after turning less than a full turn about its centre. The order of rotational symmetry is the number of times it fits in one full turn. A shape that only fits after $360^\circ$ has order $1$, which means no rotational symmetry. The angle of the smallest turn is $\frac{360^\circ}{\text{order}}$. A square has order $4$ (turns of $90^\circ$). A parallelogram has order $2$.

Learn the common cases (lines of symmetry, order):

• Equilateral triangle: $3$, order $3$.
• Isosceles triangle: $1$, order $1$.
• Square: $4$, order $4$.
• Rectangle (not a square): $2$, order $2$.
• Rhombus (not a square): $2$, order $2$.
• Parallelogram (not a rhombus or rectangle): $0$, order $2$.
• Kite: $1$, order $1$.
• Isosceles trapezium: $1$, order $1$.
• Regular hexagon: $6$, order $6$.

A tessellation is a pattern of identical shapes that fit together with no gaps and no overlaps, covering a surface. The shapes can be turned or flipped. Regular shapes that tessellate on their own are the equilateral triangle, the square and the regular hexagon. Irregular shapes can tessellate too: every triangle and every quadrilateral tessellates, even one with no equal sides.

Where shapes meet at a point (a vertex) the angles must total $360^\circ$ exactly. The interior angle of a regular polygon is $180-\frac{360}{n}$ (or $\frac{(n-2)\times180}{n}$). Triangle $60^\circ$, square $90^\circ$, pentagon $108^\circ$, hexagon $120^\circ$, octagon $135^\circ$. Check that a whole number of equal angles fits into $360^\circ$: $360\div60=6$ triangles, $360\div90=4$ squares and $360\div120=3$ hexagons all work.

A regular pentagon has an interior angle of $108^\circ$ and $360\div108=3.33\ldots$, so three leave a gap and four overlap: it does not tessellate. For a regular polygon with more than $6$ sides, each angle is more than $120^\circ$, so three overlap and two leave a gap. Different regular polygons can tessellate together if the angles at the vertex total $360^\circ$: one square and two octagons give $90+135+135=360$. An irregular quadrilateral tessellates by putting one of each of its four angles at a vertex, because its angles total $360^\circ$.