Symmetry

Three quick questions from earlier lessons.

A square is turned about its center. Which is the smallest turn that makes it look exactly the same?

Symmetries are rigid transformations that map a figure onto itself.

A figure has reflection symmetry across a line if reflecting maps it onto itself, and rotational symmetry if a rotation of less than

360^\circ

does. A regular polygon with

n

sides has

n

lines of symmetry and rotational symmetry by multiples of

\tfrac{360^\circ}{n}

A regular hexagon.

6 lines of symmetry

One through each vertex (pairs with the opposite one) and each edge midpoint.

Smallest rotation:

60^\circ

360° ÷ 6.

Your turn: the smallest rotation for a regular pentagon.

72^\circ

360 ÷ 5.

What is the smallest positive angle of rotation that maps a regular polygon with 8 sides onto itself? (degrees)Not to scale
What is the smallest positive angle of rotation that maps a regular polygon with 3 sides onto itself? (degrees)Not to scale

How many lines of symmetry does a regular polygon with 4 sides have?Not to scale

A rectangle 4 cm by 6 cm (not a square). Which transformations map it onto itself?
1. Reflections across its two midlines, and a 180° rotation about its center
2. Reflections across its diagonals
3. A 90° rotation

Three questions to check you have got it.

What is the smallest positive angle of rotation that maps a regular polygon with 8 sides onto itself? (degrees)Not to scale
What is the smallest positive angle of rotation that maps a regular polygon with 3 sides onto itself? (degrees)Not to scale
What is the smallest positive angle of rotation that maps a regular polygon with 11 sides onto itself? (degrees)Not to scale

Symmetry

Regular $n$ -gon: $n$ lines; rotations by $360^\circ/n$ .

Watch out: diagonals of a rectangle are not lines of symmetry.

End of the lesson