Rotational Symmetry

Worked example

1. 1

   $\text{turn the tracing }90^{\circ }$ — Test the smaller turns first. After 90° the long sloping sides point the wrong way, so it does not fit.

2. 2

   $\text{turn }180^{\circ }:\text{it fits}$ — Half a turn swaps the two pairs of sides, so the tracing lands exactly on the outline. That is fit 1.

3. 3

   $\text{turn }180^{\circ }\text{ more: }360^{\circ }$ — A full turn brings it back to the start, and fitting there counts as the second fit.

4. 4

   $\text{order}=2$ — There are 2 fits in one full turn, so the order is 2.

Worked example

1. 1

   $\text{regular pentagon: }5\text{ sides}$ — Regular means all sides and angles are equal, so it looks the same from every corner.

2. 2

   $360^{\circ }÷5=72^{\circ }$ — One corner moves onto the next corner every 72°. This is the smallest turn that fits: a full turn divided by the number of corners.

3. 3

   $72,144,216,288,360$ — Keep adding 72° to list every turn that fits, up to the full turn.

4. 4

   $\text{count the turns: }5$ — There are 5 fits in one full turn, counting the one at 360°.

5. 5

   $\text{order}=5$ — The order is that count, so a regular polygon has order equal to its number of sides.

Worked example

1. 1

   $\text{quarter turn clockwise}$ — The top row moves to the right-hand column, the right-hand column to the bottom row, the bottom row to the left-hand column, and the left-hand column to the top row.

2. 2

   $(1,2)\to (2,4)$ — The top row square moves to the right-hand column. (2, 4) is shaded.

3. 3

   $(2,4)\to (4,3)$ — The right-hand column square moves to the bottom row. (4, 3) is shaded.

4. 4

   $(4,3)\to (3,1)$ — The bottom row square moves to the left-hand column. (3, 1) is not shaded.

5. 5

   $(3,1)\to (1,2)$ — The next turn brings it back to the start, which closes the ring of four squares.

6. 6

   $\text{shade }(3,1)$ — It is the only square in the ring still missing. Now every turn of 90° fits.