Counting the angles of symmetry of polygons and radial-arm figures

0:00Here is a figure with four arms, evenly spaced round a centre, at right angles to each other. Turn it a quarter of a full turn and it lands back on itself, so ninety degrees works. So does a half turn, and three quarters, and the full turn. Four angles of symmetry: ninety, a hundred and eighty, two hundred and seventy, three hundred and sixty. Which raises the question this whole video is about. How many angles does a figure get, and what decides that number?

0:35The tempting answer is sitting right there. Four arms, four angles. It is wrong, and the next two figures are going to show you why. Take that same figure and bend two of its arms. The top arm bends over to the right. The bottom arm bends over to the left. The two side arms stay straight. Nothing has been removed. There are still four arms. But try the quarter turn now. The bent top arm swings round to where a straight arm is, and those two do not match, so ninety fails.

1:13Try a half turn instead. The top arm travels down to the bottom, and there it fits, because a turn carries an arm bodily and the side it bends towards goes with it. So this figure has two angles of symmetry. A hundred and eighty and three hundred and sixty. Four arms again. A different answer. Now three arms, spaced badly on purpose. The gaps between them are a hundred degrees, then a hundred and ten, then a hundred and fifty.

1:48Search every turn there is, all three hundred and sixty of them, and only the full turn brings this figure back. Which means it has no rotational symmetry at all. Not one angle. Three arms bought it nothing. So the count of arms is not what settles this. The spacing is. So space three arms evenly, and ask what evenly is forced to mean. There are three gaps here, going round the centre: arm to arm to arm, and back to where you started.

2:22Between them those gaps make one complete circuit, so they add up to three hundred and sixty degrees. That is not a fact about this particular figure. It is what going round once means. And if all three are equal, then each of them is three hundred and sixty divided by three. A hundred and twenty degrees. There is no choice in it. Three of any other whole number misses the target.

2:51Notice what just happened, because it is the heart of this. Nothing was measured. Arithmetic decided the angle, and the figure had to agree. Turn that evenly spaced figure by a hundred and twenty degrees, and every arm lands exactly where its neighbour was. Do it again and each arm has moved on by two places. Two hundred and forty degrees, and it works. Once more and you are back at the start. Three hundred and sixty.

3:24So the list reads a hundred and twenty, two hundred and forty, three hundred and sixty. Three angles, one for each arm. And look at the shape of that list, because the shape is the thing worth carrying forward. Every angle on it is a whole number of the smallest one. One of them, then two, then three. Do the same division again with five arms. A full turn shared among five gaps gives seventy-two degrees, and the list runs seventy-two, a hundred and forty-four, two hundred and sixteen, two hundred and eighty-eight, three hundred and sixty.

4:06Six arms: three hundred and sixty divided by six is sixty, and there are six angles. Twelve arms: thirty degrees, and twelve angles. In each of those I worked the answer out before looking at the picture, and each time the picture agreed. That is what makes this arithmetic rather than observation. The division goes first, and the drawing confirms it. Now seven arms, evenly spaced, because seven is where the tidiness runs out.

4:41Three hundred and sixty does not divide by seven exactly. Seven goes in fifty-one times and leaves three degrees over. So the smallest angle is fifty-one and three sevenths degrees. Not roughly fifty-one. Exactly fifty-one and three sevenths. Check it backwards: fifty-one sevens are three hundred and fifty-seven, and the three left over brings you to three hundred and sixty. Seven of those angles make one complete turn, so this figure has seven angles of symmetry, exactly as the pattern says it should.

5:21It is not one bit less symmetric than the others. What is awkward here is the degree, a unit somebody chose — our problem, not the figure's. None of this needs arms. Here is a plus sign with four equal strokes. A quarter turn sends the upright stroke onto the flat one and everything fits, so all four angles are there. Here is a straight line with a small square drawn at each end, both squares on the same side of it.

5:54Give that a half turn. Each square swings round to the other end, and to the other side of the line as well. But both were drawn on the same side, so nothing fits, and only the full turn works. And here is a capital letter H. A half turn works on it. A quarter turn does not, because the H is taller than it is wide, so the uprights land nowhere. Two angles, and a short list is still a list.

6:26Put the lists side by side now. For three arms: a hundred and twenty, two hundred and forty, three hundred and sixty. For four: ninety, a hundred and eighty, two hundred and seventy, three hundred and sixty. For six: sixty, a hundred and twenty, and so on up to three hundred and sixty. Every one of them has the same shape. Take the smallest angle, and count in it. One, two, three, and the last one lands exactly on a full turn.

7:00That is no coincidence, and the reason takes one sentence. If a turn works, then doing that same turn twice works too, and three times, and so on. So the smallest angle drags every one of its multiples onto the list behind it, which means the smallest angle tells you the entire list on its own. Push that one step further, because it forces something surprising. Start at the smallest angle and keep adding it. Every total you pass through is another angle of symmetry, and the totals climb.

7:39Suppose one of them lands exactly on three hundred and sixty. Then the list closes neatly, and the smallest angle goes into a full turn a whole number of times. But suppose instead that a total overshoots, stepping straight past three hundred and sixty and landing beyond it. Subtract the full turn, which does nothing at all, and what is left is a turn that works and is smaller than the smallest one.

8:09That is impossible. It was the smallest. So nothing can overshoot, and therefore something lands exactly. The smallest angle of any figure has to go into three hundred and sixty a whole number of times. That rule answers questions no amount of drawing could settle. Can a figure have forty-five degrees as its smallest angle? Forty-five goes into three hundred and sixty eight times exactly, so yes, and that figure will have eight angles of symmetry.

8:45Can a figure have seventeen degrees as its smallest angle? Seventeen goes into three hundred and sixty twenty-one times with three left over. So the answer is no. Not no unusual figure, or none anyone has drawn yet. No figure at all, anywhere, ever, and you know it without drawing a single one. In fact only twenty-four whole numbers of degrees can ever be a smallest angle. Everything else is ruled out by a division.

9:19One last thing worth naming properly. The count itself, how many angles of symmetry a figure has, is called its order of rotational symmetry. Be careful with it. A figure of order four has a smallest angle of ninety degrees, not of four. The order counts the angles; it does not measure them. The two are locked together, though. Multiply the order by the smallest angle and you always get three hundred and sixty.

9:51Here is the last picture. A circle cut into twelve equal slices, and you may colour those slices however you please. The orders you can reach are exactly one, two, three, four, six and twelve. The numbers that go into twelve. Five is not available and neither is seven, however you colour it. And you knew that before colouring anything, which is the whole idea. The arithmetic gets there first.