Symmetry as reflection: the fold line acts as a mirror

0:00Fold the paper along the line, and see whether the two parts land on each other. That is the test, and it works. But it needs paper. And most of the time you do not have any. So here is the same idea said a different way. Instead of folding the whole figure at once, take one point of it and send that point across the line, on its own.

0:25Then do the same to the next point. And the next. If every point lands on a point of the figure, the line is a line of symmetry. Nothing has changed except what you are watching. And what you get in return is that each point now has a partner with a name. To see why a name matters, put letters on a square. A at the top left. B at the top right. C at the bottom right. D at the bottom left.

0:56Going round, that is A, B, C, D, clockwise. Now the square has four lines of symmetry, and we know that from the last video. The upright one down the middle, the flat one across, and the two that run corner to corner. Take them one at a time and ask a sharper question than before. Not does it work. Where does each corner go? Start with the upright line, straight down the middle.

1:25Watch A. It is at the top left, and it crosses over to the top right. And what is already at the top right? B. So A takes the place B was in. Now B. It crosses the other way and takes the place A was in. D, at the bottom left, goes to the bottom right, which is where C was. And C comes back the other way to where D was.

1:50Four corners, all four moved, and each one landed exactly where another corner had been. Write down what just happened, because the way it is written is the point. A with B. D with C. Two pairs. Nothing on its own, nothing left over. That is not a coincidence of this square. If a corner had landed somewhere that was not a corner, the line would have failed the test.

2:19And notice something about the letters themselves. Before the fold, reading round the square gave A, B, C, D. After it, reading round the same way gives B, A, D, C. The same four letters, read backwards. That is what crossing a line does, and it is worth holding on to, because turning the square would not do it. Now a question that sounds like a trick and is not.

2:48What happens to a point that is sitting on the line itself? It goes straight across the line. It travels nothing at all. It stays exactly where it is. And that is not the reflection failing. It is the reflection working. The line is made entirely of points that are their own partner. Everything else moves. Take any point that is not on the line, however close, and it lands somewhere new.

3:16One more thing worth knowing. Reflect a point across a line, then reflect it back across the same line, and it returns to precisely where it started. Which is why the two halves are interchangeable. Neither one is the original. Now the diagonal, the one running from A down to C. A is on the line. So A does not move. C is on the line too. C does not move either.

3:47That leaves B and D, and they are the interesting ones. B is at the top right. Send it straight across the diagonal and it arrives at the bottom left, which is D. And D crosses the other way to B. So this fold holds two corners perfectly still and swaps the other two. Which is a different answer from the upright fold, on the same square, and both are correct.

4:16One more, to finish the set. The flat line, straight across the middle. A is at the top left and goes down to the bottom left, where D was. B goes down to C, and the two at the bottom come up to meet them. A with D. B with C. No corner stays put. And there is the fourth line, the other diagonal, which holds B and D still and swaps A with C.

4:42Put all four side by side and a pattern falls out. Every one of them holds two corners still, or holds none. Never one. Never three. There is a name for what a figure has when at least one line does this. Reflection symmetry. A square has it. So does a rectangle, with two lines rather than four. A triangle with three sides all different does not have it at all, and that is a complete answer, not a missing one.

5:16The word reflection is doing real work in that name. It says the two halves are related by crossing the line, not by sliding along it and not by turning round. Turning is a different thing entirely, and it is worth keeping the two apart. A quarter turn of our square sends A to D. The upright fold sends A to B. Both leave the square looking untouched. They are still not the same move.

5:46Now the part that makes all of this useful. Here is a line, and here is a point, and no paper anywhere. Where exactly is the partner? Three things pin it down, and between them they leave only one place it can be. First, it is straight across. Leave the point travelling at right angles to the line, not at any other angle. Second, it is equally far. However far the point sits from the line, the partner sits the same distance on the other side.

6:20Third, and this is the same fact said again, the line cuts the journey exactly in half. Straight across, equally far, half way. That is a construction you can carry out with a ruler, or by counting, and it never needs a fold. Which is exactly what squared paper is for. Here is half a shape drawn on a grid, and a line beside it. The job is to finish the shape so that the line becomes a line of symmetry.

6:53Take one corner of the drawing. Count the squares from it across to the line. Three. Now count three squares out on the other side, staying in the same row. Mark it. That is the partner. The next corner is two squares from the line. So its partner is two squares out on the other side, again in its own row. Do that for every corner and join them up, and the shape is finished.

7:21One warning. The copy is turned over, not slid along. If you slide it, you get a shape with no line of symmetry at all, and it will look almost right. Now the version that catches nearly everybody. Two lines this time, an upright one and a flat one, and a small piece of shape in one of the four regions. Both lines have to end up as lines of symmetry.

7:49So reflect the piece across the upright line. That is one copy. Reflect the original across the flat line. That is two. And you are not finished, because the figure you have now is not symmetric about either line. There is a third copy, diagonally opposite the original, and it is the reflection of a reflection. One piece, three more. Four in all. And with the two lines at half a right angle instead, one piece becomes eight.

8:23Last thing, and it goes back to where we started. Counting lines of symmetry looked like a matter of trying folds until you ran out. The reflection picture says what you were actually counting. A line is a line of symmetry exactly when sending every corner across it lands every corner on a corner. That is why the square gets four and the rectangle gets two. The rectangle's diagonal sends a corner to a place where there is no corner, and you can see where it lands.

8:59And it is why the count never depends on how the figure was drawn or which way up it sits. Fold the paper if you have paper. Send the points across if you do not. It is one fact, and now you have two ways to get at it.