Total rotational symmetry, and why every diameter is an axis of reflection

0:00Watch a wheel go past, and note the point of the tyre that is touching the road. Look again a moment later, and note the point touching the road then. Now try to say which of those two observations came first. You cannot. Nothing about the wheel distinguishes them. That is not a remark about wheels. It is a fact about circles, and it is the strongest tool this whole subject has.

0:28If turning a figure leaves you unable to tell the before from the after, then the turn changed nothing you could measure. So the question worth asking is how many turns do that, and the answer for a circle is unlike the answer for anything else you have met. A square has rotational symmetry. Turn it a quarter of the way round and it sits back down on itself, exactly as it was.

0:57Turn it an eighth of the way round and it does not. So for a square the symmetry is a short list of angles, and between the entries on that list, nothing works at all. For a circle there is no list. Turn it one degree and it lands on itself. Turn it a hundredth of a degree and it lands on itself. Turn it by any amount you can name, and by any amount you cannot.

1:24That is what complete rotational symmetry means. Not many angles. Every angle. The reason is one line long, and it is worth having exactly. A circle is the set of points at one fixed distance from one fixed point. The condition mentions distance from the centre, and it mentions nothing else. It says nothing about which way round the figure happens to be sitting. So take any motion that leaves every distance from the centre exactly as it was.

1:59Every point that passed the test still passes it. Every point that failed still fails. The set comes back identical, and turning about the centre is exactly such a motion. Not because a circle looks round. Because the condition it is built from cannot see a rotation happening. Then the centre had better be the centre. Turn the circle about a point one unit off to the side instead. The image is still a circle of exactly the same size, but it is sitting somewhere else.

2:36Two circles the same size with different middles cross at two points and no more. So a quarter turn about a point one unit out keeps two points of the circle and moves every other one off it. Push the pivot out to seven units and it still just clips, at two points. Push it to eight and the image misses the original completely. Nothing survives. The centre is not a convenience in that definition. It is the only pivot there is.

3:09Reflection next, and this one you can do with your hands. Cut a disc out of paper. Fold it so that the boundary comes down on the boundary. Not roughly. Exactly. Press the crease flat, and open it out again. If you have ever been handed a paper circle and asked to find its middle without measuring anything, that fold is the answer. Now do it again in a different direction. The two creases cross at one point, and that point is the centre.

3:42One more fold, in any direction at all, and the third crease runs through it too. Which is a fine thing to watch, and no reason at all to believe it. So why must the crease pass through the middle? Take one that does not. Reflecting in it slides the whole circle sideways, onto a circle of the same size with a different centre. And those two cross at two points. Two. Not the whole boundary.

4:13So folding along a chord that misses the centre brings exactly two points of the edge back, and leaves all the rest hanging over. You can see the overhang in the paper. Slide the crease closer in and it is still two. A hundredth of a unit off centre is still two. Only when the miss is exactly nothing does the whole boundary come back. So the creases that work are exactly the lines through the centre.

4:44And a line through the centre cuts the circle at two points, with a segment between them. That segment is a diameter. Every diameter is an axis of symmetry, and nothing that is not a diameter is one. Be careful with the wording, though. The crease is a whole line, and it runs off the edge of the paper in both directions. The diameter is the chord. The piece with both of its ends on the circle.

5:15The line is longer than the diameter sitting inside it, and calling them the same thing will cost you later. And since the fold can be started anywhere on the boundary, there are as many axes as there are directions to point in. Put numbers against it. Not counting the turn that does nothing at all, a square comes back to itself after three turns, and it has four lines of symmetry.

5:44A regular five-sided figure: four turns, five lines. A six-sided one: five turns, six lines. The pattern is obvious, and the obviousness is the point. The count is tied to the number of sides, so it is always finite. Now run twenty-seven different exact turns past a circle. All twenty-seven work. Run those same twenty-seven past a square, and three of them work. Same test, same turns. A circle is not a polygon that happens to have a lot of sides.

6:23The symmetry earns its keep immediately. Take a circle of radius five and ask for its longest chord. The answer is ten, and it is a diameter, and you can see why without any algebra at all. Any chord that misses the centre gets cut short at both ends. So push a chord away from the middle and watch it shrink. Three units out from the centre, the chord measures eight.

6:50Four units out, it measures six. Four point nine out, almost touching the edge, and it is under two units long. Every step outwards, shorter. The diameter is the one chord that gives nothing away, because it is the one that does not miss. So is there a shortest chord? The tempting answer is the radius, which is not a chord at all. It has only one end on the circle.

7:20The honest answer is that there is no shortest one. Name any chord you like. Move it half of the remaining distance out towards the edge. That is still a chord, it still has a length, and it is shorter than the one you named. You can do that from anywhere, and you can do it again to whatever you get. The lengths run down towards nothing without ever arriving, so there is no smallest one to find, and looking for it is looking for something that is not there.

7:54One more set built out of a condition, and a warning attached to it. Mark two points, and ask which points stand equally far from both. The middle of the segment joining them qualifies, but it is not the answer. The answer is a whole line: the one cutting that segment in half at right angles. And here is the warning. Showing that every point on that line stands equally far from both is only half of the job.

8:24You have also to show that nothing off the line does. Skip the second half and all you have shown is that the line is inside the answer, not that it is the answer. Those are different claims, and only one of them is what was asked for. Which brings us to what symmetry is for, and what it is not for. A fold is persuasive. It is not a proof. Here is a case in point.

8:52A crease two and a half units off the centre crosses the circle at exactly two points. But neither of those two lands on a whole number. So if you were checking by marking the whole-number points on the boundary, you would find nothing at all coming back, and conclude the crease misses the circle entirely. Move the crease out to three units and the very same check finds both of them.

9:20The counting was right once and wrong once, and it looked equally convincing on both occasions. What settled it each time was the argument, not the count. Symmetry tells you what to expect. It does not tell you that you are right.