An angle of symmetry: turning a figure onto itself

0:00Here is a paper windmill. Four blades, all the same, arranged round a centre. Try to fold it in half so the two parts land on each other. Upright: no. Flat: no. Corner to corner, either way: no. In fact there is no line anywhere on this page that works. Not one. By everything we have said so far, this shape is not symmetric at all. And yet it obviously is. Look at it.

0:30So stop folding it, and turn it instead. Hold the centre still, and rotate the whole windmill by a quarter of a full turn. Watch what happens. It lands exactly on where it started. Every blade sits where the blade before it was, and the picture is untouched. That is not the fold test. Nothing was flipped over. The paper never left the table. It is a second, completely different way for a figure to come back to itself, and this shape has it while having no fold lines whatsoever.

1:09Turning needs one thing that folding does not: a point to turn about. For the windmill it is the spot where the four blades meet, and it is the only point that works. Turn about any other point and the whole shape swings away and lands somewhere else entirely. That fixed point has a name. It is the centre of rotation. It does not have to be marked on the figure. It has to exist.

1:37For a square, where would you put it? At the point where the two diagonals cross, which is the middle of the square, and nowhere else. Now, a quarter turn worked. What about something smaller? Try a small turn. The blades move off, and nothing lines up. Try a bit more. Still off. Keep going and nothing works at all until you reach exactly a quarter, and then everything snaps back into place at once.

2:08The four blades are what forbid anything smaller. To land on itself, a blade has to arrive where another blade already is, and the nearest one is a quarter of a turn away. So the angle is not approximately a quarter. It is exactly a quarter, and nearly right is wrong. This is worth being fussy about, because the eye is easily satisfied here. A shape turned by a nearly-right angle looks nearly right, and nearly right is not a symmetry at all.

2:44An angle of turn that leaves a figure looking untouched is called an angle of rotational symmetry, or just an angle of symmetry for short. And that is the shift worth noticing. With folding, the question was: is there a line? Yes or no. With turning, the question is: through what angle? A number. Because turning is measured, not just spotted. And once you are counting angles rather than answering yes or no, a figure can have several of them, and how many it has becomes a fact about the figure worth knowing.

3:23Before listing any angles, there is one that comes free. Turn any figure through a full turn, all the way round, and every single point of it goes back exactly where it started. That is true of the windmill. It is true of a square. It is true of a triangle with three different sides, and of any scribble you like. So a full turn is an angle of symmetry of absolutely every figure there is.

3:52Which means it never distinguishes anything. It is on everybody's list, so it carries no information at all. Hold on to that. It comes back at the end and it decides where a definition has to be drawn. Now list the windmill's angles properly. A quarter turn, which is ninety degrees, works. So does a half turn, one hundred and eighty degrees, because that is just a quarter turn done twice.

4:23So does three quarters, two hundred and seventy. And the full turn, three hundred and sixty, which we already knew. Four angles. Ninety, one hundred and eighty, two hundred and seventy, three hundred and sixty. And between them, nothing. Do the same for a square, and this time put letters on the corners so you can see what moves where. A at the top left, B top right, C bottom right, D bottom left. Reading round, that is clockwise.

4:56Now turn the square a quarter turn anticlockwise. A has gone down to the bottom left. B has come round to the top left. C is at the top right and D at the bottom right. Turn again: C is now top left. Again: D is top left. And once more, a full turn in all, and every letter is back where it started. Four turns, four positions, and the shape looked identical every single time.

5:26So the square has the same four angles as the windmill. But the square has four fold lines and the windmill has none, which tells you these are two separate measurements of the same shape, not one measurement said twice. There is a practical problem hiding in that last scene. Take the letters away and turn the square a quarter turn. Now tell me whether anything happened. You cannot. That is the whole point of a symmetry: it leaves no trace.

6:00The before and the after are the same picture, and no amount of staring will separate them. So draw a line sticking out from the middle of one side. Nothing mathematical, just a marker. Now turn it. The marker was pointing right, and now it is pointing up. The square looks untouched, and the marker says exactly how far it went. That is how you watch a turn that by definition cannot be seen.

6:31Here is a half turn doing something other than being measured. A game on a six by six grid. Two players take turns. A move is drawing one line across a neighbouring pair of squares, upright or flat. No two lines may share a square. If you cannot move, you lose. The second player has a strategy that always wins, and it is a half turn. Whatever the first player draws, the second player answers with the same line turned half a turn about the middle of the grid.

7:06Because six is even, that middle is a corner shared by four squares, and no line can be its own half-turn image. So the answer never clashes with the move it is answering. After every reply the whole position is symmetric again, so if the first player had somewhere to go, the second player does too. The first player runs out first, every time. And on a five by five grid the same idea fails, because there the middle is a square and a line can sit right on it.

7:43Now a shape where the counting comes out differently. This strip has two parallel edges, the bottom one longer than the top, and two ends leaning inwards. Mark its centre and turn it a half turn. It comes back as the same strip upside down, with the wide edge on top. It does not lie on the original. Turn it another half turn and it is restored. But that is a full turn in total.

8:12So the only angle of symmetry this strip has is three hundred and sixty degrees. Nothing else works. And notice: this shape does have a fold line, straight down the middle. It flips beautifully. It just will not turn. Which brings back the thing to hold on to. The full turn is on every figure's list, so a shape whose only angle is the full turn has nothing that anybody else does not have.

8:43So the definition is written to exclude it. A figure has rotational symmetry when it has an angle of symmetry strictly between zero and a full turn. Strictly. Not zero, which is doing nothing. Not a full turn, which everything has. By that line, the windmill has rotational symmetry and so does the square. The strip does not. And the windmill is the one to remember, because it has no fold line anywhere and it still comes back. Two questions, two answers, and neither one is the other.