Line symmetry and rotational symmetry are independent of each other

0:00There are two completely different questions you can ask about a shape, and people use the same word for both of them. First question: is there a line you can fold along, so that the two halves land exactly on each other? Second question: is there a turn, short of a whole turn, that leaves the shape looking untouched? Both of those get called symmetry. And here is what matters: they are separate questions.

0:29A shape can pass one and fail the other, and it can happen either way round. So saying a shape is symmetric tells you nothing at all until you say which test you ran. Watch all four answers happen. Start with a paper windmill. Fold it upright: no. Flat: no. Corner to corner, either way: no. In fact no line anywhere on this page works, and that has been checked properly rather than guessed at.

1:00Zero fold lines. By the folding test, this shape is not symmetric at all. Now hold the centre and turn it a quarter of the way round. It lands exactly on itself. Four angles of symmetry: ninety, a hundred and eighty, two hundred and seventy, three hundred and sixty. Four angles, no lines. Turns without a mirror. That is the first of the four cases. Now the other way round.

1:31This strip has two parallel edges, the bottom one longer, and two ends leaning inwards. Fold it straight down the middle and the halves land on each other perfectly. One fold line. Now mark the centre and turn it a half turn. It comes back upside down, wide edge on top, sitting off the original. Turn it a half turn again and it is restored — but that is a whole turn in total, and a whole turn works on everything.

2:03So no turn short of a whole one works. One line, no angles. A triangle with two equal sides behaves the same way: one fold line down the middle, and not a single turn. Third case, and this one is the familiar one. A square. Fold it upright: yes. Fold it flat: yes. Along one diagonal, and the other: yes and yes. Four fold lines. Now turn it a quarter: yes. A half: yes. Three quarters: yes. Four angles.

2:38Four and four. It passes both tests, and it is because shapes like this one are the ones everybody meets first that people expect the two tests to travel together. They usually do. They do not have to. Hold on to that difference. Fourth case. A triangle with three different sides. There is no fold line: fold it whichever way you like and the halves never match up. And there is no turn either, because for the shape to land on itself a corner would have to arrive where another corner already is, and no two of these corners are alike.

3:19Zero and zero. It fails both tests. Which is the ordinary situation, by the way. A cloud, a coastline, a scribble on a page — almost anything drawn without thinking about it lands in this box. Symmetry is the special case, not the usual one. Now put all four of those into a square of four boxes. Across the top: does it fold? Down the side: does it turn? Yes to both is the square. Turns only is the windmill. Folds only is the strip. Neither is the three-sided shape with nothing equal about it.

4:01And every single box has something in it. That is the whole point of this video, and it is the one thing to carry away. The two properties are independent. Neither one forces the other, in either direction. Which is exactly why the word symmetric, on its own, is not a useful answer to anything. You have to say which test. Now a shape that breaks every pattern in this topic. A circle.

4:32Turn it about its centre by a quarter turn: it lands on itself. By a fifth of a turn: yes. By one degree: yes. By seventeen degrees — and seventeen was ruled out for every other shape there is. By any angle you can name at all, including angles that are not whole numbers of degrees. Here is why. Every point of the rim is exactly the same distance from the centre, so turning slides each rim point along to another rim point.

5:06Nothing can move off the rim, and nothing else was ever drawn. So the picture cannot change, whatever angle you use. Now try folding it. A line straight down the middle works, and the two halves land on each other. So far, so ordinary. But now pick any point at all on the rim. Join it to the centre, and carry the line on to the far side. That line is a diameter, and it is a fold line too, for the very same reason as before.

5:41Folding across it keeps every point at the same distance from the centre, so every rim point lands on the rim again. And you can start from any point on the rim you like. So there are endlessly many diameters, and every single one of them is a fold line. Which makes the circle an exception to something we relied on earlier. For every other shape, the angles of symmetry are the multiples of one smallest angle, and that smallest angle goes into a whole turn exactly.

6:18The circle has no smallest angle at all. Try to name one. Say a tenth of a degree. But then a twentieth of a degree also works, and it is smaller. Halve it again. And again. And again. There is never a least one to find, because halving never runs out. So a circle has angles of symmetry, endlessly many, and no smallest angle of symmetry. It is the one shape the rule cannot be applied to.

6:50Now count both numbers for the same shape, and something starts to happen. Here is the outline of a building seen from above: a triangle with equal sides, with its three corners cut off. Three fold lines, and three angles. Three and three. Be careful with that one, though. Cut the corners at exactly the thirds and all six sides come out equal, and the answer jumps to six and six. Where you cut decides it.

7:21A regular five-sided shape: five fold lines, five angles. Six-sided: six and six. Seven, eight, nine, ten — the same number twice, every single time. A wheel with twenty-four spokes: twenty-four fold lines and twenty-four angles, the smallest of them a whole turn shared twenty-four ways, which is fifteen degrees. Every one of those came out the same number twice. That is not a coincidence, and the reason is something you can follow.

7:59Take a shape that has at least one fold line, and fix on one of them. Call it your fold. Now do your fold, and then do any turn that the shape has, one after the other. Both motions land the shape on itself, so the two of them together do as well. And that combination turns out to be a fold — across some different line. Different turns give you different folds. So every turn hands you a fold, and no two turns hand you the same one.

8:31That pairs them off, one for one, which is why the two counts are equal. But look at what the argument needed: at least one fold line to start from. The windmill has none, nothing pairs off, and its counts are four and zero. One last shape, to test everything at once. Start with a triangle with equal sides. Three folds, three turns. Now push a bump out from the middle of every edge. What you get is a six-pointed star. Count again: six folds, six turns.

9:09Do the same thing to every one of its twelve edges, and then again to all of those. The shape gets more and more jagged, and the counts stop moving. Six and six, from the second shape onwards, for ever. That is the Koch snowflake, and it behaves because every edge is treated exactly alike. So: two tests, four possible answers, one word doing the work of both, and one shape with no smallest angle at all.

9:41Ask which test. It genuinely matters.