Symmetry.

Step 1 of 8

Counting symmetries

A symmetry of a shape is a map that keeps distances and leaves the shape exactly as it was. On the plate one motif, a curled frond, a bead and a small leaf, is copied by every turn and every mirror. A turn by $\frac{360^\circ}{n}$⁠ leaves the pattern as it is, and so does every turn by a multiple of that angle, including the turn by zero, which does nothing. With Mirrors on, so does a reflection in each of the $n$⁠ mirrors.

The collection of a shape’s symmetries is called its group: two symmetries one after the other make a symmetry again. Without mirrors it is $C_n$⁠, and with mirrors $D_n$⁠. Here the pattern is a snowflake, $D_6$⁠. Count its symmetries.

Step 2 of 8

Same size, different shapes

The size of a group is its number of symmetries. $C_n$⁠ has $n$⁠, and $D_n$⁠ has $2n$⁠. Change Rotation order and Mirrors to find a pattern with exactly $8$⁠ symmetries. There is more than one way.

Step 3 of 8

The angle between mirrors

A starfish has $5$⁠ arms, and a mirror line runs along each arm. The smallest turn that brings it back is $\frac{360^\circ}{5} = 72^\circ$$\frac{360^\circ}{5} =$$72^\circ$⁠. Find the angle between two neighbouring mirrors.

Step 4 of 8

Two reflections

Each reflection turns the pattern into its mirror image; in the language of this Unit its determinant is $-1$⁠. What happens when you reflect twice, in different mirrors? Think about what happens to a point lying on the first mirror.

Step 5 of 8

Tilings

Step through Find the symmetry: the same rules turn up in a snowflake, a crystal and a tiled floor. A floor repeats itself by sliding, and only certain turns fit a lattice of slides. If a tile’s turn does not fit, the tiles around it leave gaps or overlap, and the plate shows where. Predict which orders a floor can repeat.

Step 6 of 8

Eight times round

How far must you turn it for the pattern to come back to itself?

Step 7 of 8

A pinwheel

Some patterns can be turned but not reflected: they spin one way.

Step 8 of 8

The largest that tiles

You have seen which orders tile. It remains to check whether any order above $6$⁠ does.