Fold, blot, cut, punch: making a figure that is symmetric by construction

0:00So far, symmetry has been something to look for. Here is a figure — does it have a line of symmetry, and where? Now turn that round. Instead of hunting for symmetry in a shape somebody handed you, make a shape that is guaranteed to have it. The trick is one word. Fold the paper first. Everything you then do to the folded sheet, you do to every layer at once. And that is what settles it.

0:30Start with the easiest version, which needs no scissors at all. Take a sheet, fold it in half, and open it flat again. There is a crease down the middle now, and nothing else. Drop ink on one side of the crease. Anywhere. Several drops, in no pattern whatsoever. Now fold the sheet back over along that crease, and press. Open it. The blot you get is symmetric, and its line of symmetry is the crease.

1:00Not roughly. Exactly. It is worth being clear about why, because it is not that ink spreads evenly. It does not. When you press the two halves together, every single drop touches the point directly facing it across the crease. So each drop makes a second drop, in exactly the place a reflection would put it. Every point of the blot has a partner. That is the definition, satisfied by construction, before you have looked at anything.

1:35And notice what happens without the press. The same drops, left alone, are symmetric about nothing at all. It is the fold that does the work. One more thing. The blot has that one line of symmetry and no others, because the drops themselves were irregular. Folding gives you the line you folded along. It does not give you any extra. Now bring in the scissors, and the same idea gets sharper.

2:06Fold a sheet in half. The folded sheet has two upright edges: the crease on one side, and the open edges on the other. Cut a bite out of the open edge, and stop before you reach the crease. Open it. Two bites, one on each outer edge of the sheet, matching each other exactly. Now do it again, but cut the bite out of the crease edge instead. Open it. One hole this time, sitting in the middle of the sheet, straddling the crease, and touching no edge at all.

2:40Same scissors, same size of bite. Which edge you cut decides whether you get two bites or one hole. And in both cases the opened sheet is symmetric about the crease, because whatever the scissors removed, they removed from both layers. That phrase — both layers — is the one to hold on to, so let us count them. An unfolded sheet is one thickness. Fold it once, and every part of it is two thicknesses.

3:11Fold that in half again, and every part is four. Fold once more: eight. The number doubles at every fold. It does not go up by one. Two, four, eight, sixteen. Which means that one snip, or one punch, does not make one hole. It makes as many holes as there are thicknesses under it. So here is a game you can play with a sheet of paper and a hole punch.

3:40Fold the square sheet once, down the middle. Punch a single hole, somewhere off to one side. Open the sheet. Two holes, in mirror positions about the crease. You knew that before you opened it, and you knew where they would be. One punch, two thicknesses, two holes. Now the interesting direction. Somebody hands you an opened sheet with holes in it, and does not tell you how it was folded.

4:10You can read it off. The fold has to be the line that carries one hole exactly onto the other. And it has to be a line of symmetry of the sheet itself, because that is the only way the paper could have folded flat. Two holes side by side, level with each other: the line between them runs up and down, so it was folded upright. Two holes one above the other: the line runs across, so it was folded flat.

4:41Two holes close together and set slantwise, up near a corner: the line running corner to corner carries one onto the other. It was folded along the diagonal. And if the line between the two holes is not one of the sheet's own lines of symmetry, then no single fold could have made them. Somebody punched twice. Here is one that catches people. Four holes, one near each corner, and you are told it was a single punch.

5:13One punch through one fold gives two holes. There is no way round that — two thicknesses, two holes. So four holes means four thicknesses, and four thicknesses means two folds. Fold upright, then fold flat. Punch once, near the corner of the little folded square. Open it out: four holes, one near each corner of the sheet. The other order gives exactly the same four. And the four are symmetric about the upright line and about the flat line, which is precisely the pair of folds that made them.

5:52Once you can read the fold, you can finish the job somebody else started. A figure, its line of symmetry drawn in, and one hole marked. Where does the other hole go? Straight across the line, the same distance out. There is only one place it can be. A square folded on its diagonal: the hole above the diagonal has its partner below, and you get there by counting squares across.

6:21A rectangle folded flat: straight down. And a circle, folded along a slanted diameter. This one is worth a moment, because the partner is not at any obvious spot on a grid — but it is still exactly determined. Same distance from the centre, straight across the diameter, and the diameter cuts the journey in half. Now the payoff. Predict the figure before you open it. A sheet, taller than it is wide. Fold it in half down the middle, so you have a tall narrow strip.

6:59Cut a rectangle out of the strip's open edge, halfway up. Then cut a rectangle out of the strip's crease edge, at the same height. The folded strip now has a waist, like a capital letter I. Stop. Say what the opened sheet looks like, without opening it. The cut at the open edge becomes two bites, one on each outer edge of the sheet. The cut at the crease becomes a single rectangular hole in the middle, touching nothing.

7:31So: a tall rectangle, a bite out of each side, and a window in the centre. Now open it, and check. Here is a harder one to aim for. Put a square hole in the middle of a square sheet, using a few folds and one straight cut. Fold upright, then flat. The middle of the sheet is now the corner of the little folded square. One straight cut across that corner, at half a right angle, takes off a triangle.

8:03Open it, and there is a square hole in the middle — standing on one of its corners. To get a hole that sits square to the sheet instead, fold along the two diagonals rather than the two mid-lines. Now the middle of the sheet is the sharp point of a wedge, and one straight cut across that point gives a square hole sitting square. Same cut, same number of folds, and the hole turns by half a right angle. And do check that it really is a square. Four equal sides is not enough — a squashed diamond has four equal sides too. You need the corners square as well.

8:45Last idea, and it turns all of this into a way of designing. Take a large grid and fill in one quarter of it with a pattern. Any pattern. This one has no symmetry of its own whatsoever. Now draw the upright mid-line and the flat mid-line, and let those two lines do the work. Reflect the quarter across the upright line. Reflect the whole lot across the flat line.

9:13One quarter has become four copies, and the finished design has exactly two lines of symmetry — the two you chose. Not one extra copy. Three. That is the part people get wrong. And the count is yours to set. Build a design with exactly one line of symmetry, or exactly two, or four. You are not looking for symmetry any more. You are deciding how much of it to put in.