A line of symmetry is a fold that makes the halves coincide

0:00Look at these four. A flower. A butterfly. A pattern drawn on a floor. A pinwheel. Something about all four is satisfying, and most people reach for the same word. Symmetry. Now look at this. A photograph of clouds. Nothing about it is satisfying in that way. Nothing in it comes back anywhere. So there is clearly something the first four have and this one does not. The trouble is that so far, the only evidence is that they look nice.

0:34And looking nice is not something anybody can argue with. By the end of this we will have replaced it with a test — one that can answer no. Start with the best description you can manage without measuring anything. In each of the four, parts of the picture come back. The flower's petal turns up again, and again, all the way round. The butterfly's wing turns up twice, once on each side.

1:04The pinwheel's blade turns up six times. And not scattered anywhere — in a settled pattern, at regular places. That is a real observation, and it is the right thing to start from. It is also still only a description of what you noticed. Which is fine, right up until somebody disagrees with you. Here is why that matters. Suppose I show you a shape and claim it is symmetric, and you say it is not.

1:36What happens next? With nothing but an impression to go on, nothing happens next. You look at it again. I look at it again. Neither of us can show the other anything. A description tells you what you already noticed. A test tells you something you did not know. And the real difference is that a test can come back and say no. So we need a procedure. Something you could carry out with a sheet of paper and no opinions at all.

2:08Here is one. A triangle, with a coloured disc at each of its three corners. Two yellow ones at the bottom, one magenta at the top. And a line dotted down the middle of it. Now fold along that line. The left half swings over onto the right half. And it fits. The two sloping sides land on each other. The bottom left corner lands on the bottom right corner, yellow onto yellow.

2:38The magenta disc sits on the crease itself and does not move at all. Nothing sticks out. Nothing is left over. That is the test, and you have just watched it pass. The fold has cut the triangle into two pieces, and those two pieces have a name. Mirror halves. They are called that because each one is what you would see if you stood a mirror upright along the fold.

3:06Look at one half together with the fold line, and the other half is not extra information. It is already settled. There is nothing left to be told. Which is worth pausing on, because it is what symmetry actually buys you. A line of symmetry means half the figure is enough. Give me one side and the crease, and I can build the rest without asking you a single question.

3:35Now the case that matters more. Here is a square made of four interlocking pieces, the way a jigsaw is. Each piece has a tab sticking out of one edge and a matching hollow cut into the next. And here is a line dotted straight down the middle of it. It really is down the middle. It cuts the square into two halves of exactly the same area. Nobody could accuse it of being off-centre.

4:04Now fold. The tabs do not land on hollows. They land on other tabs. Two pieces of the picture stick out past the edge, and two gaps are left uncovered. The fold fails. And the line looked perfect. So here is the definition, and it is worth saying slowly. A line of symmetry is a line you can fold the figure along, so that the two parts land on each other exactly.

4:34Exactly is doing all the work in that sentence. Not roughly. Not almost. Not close enough that nobody would notice from across the room. The jigsaw's line runs down the middle, it splits the area in two, and it looks entirely reasonable. It is still not a line of symmetry, because a tab the width of your fingernail is left hanging over the edge. A fold that leaves a sliver over has failed. There is no partial credit here.

5:08One habit gets in the way at this point, and it is worth breaking early. People look for folds that stand upright. Take a square, sitting flat. It has four fold lines: one upright, one across, and the two that run corner to corner. Now stand the same square on one of its corners. It is the same square. It still has four. But they have all turned with it. The corner to corner lines are now the upright one and the flat one, and the other two run slanting across the page.

5:43Nothing about the square changed. Only the way it was put down. So look for slanted folds as well, or you will miss them. Now take the test back to the four pictures we started with. The flower, with six petals round its middle. Fold it down through a petal, and it fits. And there are six different lines that work. Six. The butterfly. One fold, down between the wings, and it fits. Exactly one.

6:14The floor pattern, with four alike shapes set round the centre. Four folds. And the clouds. Try every line you like. There is not one. Zero. Which is a real answer, and the picture is not defective for giving it. But the pinwheel is where this gets interesting. Six blades, evenly spaced, every one identical. It is about as orderly as a picture can get. Try to fold it. Down the middle, no. Through a blade, no. Between two blades, no.

6:50Slanted, at any angle you like, no. The pinwheel has no line of symmetry at all. Not one. And the reason is that every blade leans the same way round. A mirror would have to make them lean the other way, and then they would not match. What the pinwheel does have is turning. Give it a sixth of a turn and it comes back looking untouched. That is real, and it is a different property. And the jigsaw has it too — turn that a quarter of the way round and it also comes back.

7:25Both of them turn beautifully, and neither one folds. Which is the whole point of owning a test rather than an impression. Here are five plain outlines. No colours, nothing to like or dislike. A dart, with a notch cut into its base. One fold, straight down through the point. A kite. One fold, along its long way — and not along the other way. That one fails. A four-sided figure with an upright side and a slanted top. None at all.

8:00An L with two arms the same length. One fold, and it runs corner to corner, on the slant. Look only for upright lines and you would have said none. A long thin triangle. None. Five figures, and no opinion was required for any of them. One last thing, and it is only about words. The line you fold along has two names in common use, and they mean precisely the same thing.

8:31Line of symmetry. And axis of symmetry. You will meet both. Neither one is more correct than the other. And the thing they name does not depend on anybody having drawn it. The dotted lines in all of this were an aid to you, not part of the figures. A figure has whatever fold lines it has, drawn or not, noticed or not. The test finds them. It does not create them.