Parallel illusions: why the eye is not a proof

0:00Count how many times you have been asked to look at a picture and say whether two lines were parallel. Do these appear parallel. Does this pair keep its distance. Have a look and decide. Every one of those was a fair question at the time. But notice what kind of question it was. It was a question about you, not about the lines. And that is a very strange thing to build a subject on.

0:27So here are three pictures that punish it. In every one of them there are two lines that are genuinely parallel. And in every one of them, they do not look it. Before we start, one promise, and I would like you to hold me to it. Nothing in these pictures is bent. No line is curved, no line is tilted, nothing has been distorted. The only thing that changes is what has been drawn around them.

0:58Hold on to that, because you are about to stop believing it. The first picture is a fan of straight lines through a single point. The second is a field of small tiles, laid in rows. The third is a dense burst of rays out of a bright centre. Three pictures, and three different ways of being wrong about the same thing. So. Sixteen straight lines, all passing through one point.

1:27And two long verticals, one on each side of that point, running down through the fan. Look at the verticals. Not at the fan, at the two verticals. They splay. They bulge outwards in the middle and pinch in at the ends. Or that is what I see. What you see may be a little different, and that is already interesting. Now here is how those two verticals were actually put on the screen.

1:54One of them is the other one, moved sideways. That is the entire operation. So every point on one is exactly the same distance from the other. At the top, in the middle, at the bottom. Which is easy for me to say and hard for you to believe, so let us just take the fan away. Watch what happens to the two lines. Straight. Both of them, perfectly straight and perfectly parallel.

2:24Now put the fan back. And they bow again. But nothing about the verticals changed. Not one dot of them moved between those two pictures. Only the background came and went. So the bend was never in the drawing at all. It arrived somewhere between the screen and you. The second picture. Rows of small dark tiles on a light background. Each row is shifted a little way along from the row above it.

2:55And the rows slope. Some tilt down to the right, some tilt up. They fan apart. They do not. Every row is dead level, and every one of them is level in the same way. But this picture is different from the other two, in a way worth being careful about. There is no line in it. Not one. Nothing here has been drawn as a line. What you are reading as a sloping line is the top edges of a row of separate tiles.

3:26Which means there is nothing here to lay a straight edge along, and that will matter shortly. The third picture. A bright centre, with rays bursting out of it in every direction. And two long horizontal lines across it. One above the centre and one below. They bow away from the centre. They arch, like something is pushing them apart. They are straight. They are level. And they are parallel.

3:56Take the burst away, and there they are. Now, this is the first picture again, turned through a right angle. A dense fan of straight lines through one point, and a pair of lines crossing it. That is twice the same arrangement has caught us, which is a clue about the arrangement. Which brings us to the obvious question. Why? What is actually going on? And I am not going to answer it.

4:26Not because it is uninteresting. It is one of the more interesting questions there is. But it is not a question about lines. Nothing about the geometry is unusual in any of those three pictures. The lines are parallel, the rays are straight, the tiles are square, and every distance is exactly what it looks like it should be. So whatever is happening is happening in an eye and a brain, and that is somebody else's subject.

4:58What this subject can tell you is what an illusion is not evidence of. And that turns out to be a great deal. Because look at that first picture again, and count what is in it. Sixteen straight lines through a point that sits between the two verticals. Which means every single one of them crosses both of the verticals. And a line crossing two lines is a transversal. We know exactly what to do with one of those.

5:29Take any ray you like. Mark the angle it makes where it meets the left one, and where it meets the right. Matching positions. Compare them. Equal. Do it with the next ray. Equal. And the next. Equal, all sixteen times. The picture that fooled you was carrying sixteen separate proofs that it was fooling you. One thing to keep well apart from all this, because the two get confused constantly.

6:00Photograph a keyboard, or a road, or a row of pillars. The edges converge. They lean in towards each other and meet at a point. That is not an illusion. That is a truthful record of a three dimensional scene. A camera flattens depth onto a plate, and edges running away from you land on lines that meet. Work it out, and the meeting point depends only on the direction those edges point. Nothing else.

6:30Move the edges sideways and it does not budge. Turn them, and it moves. And edges running across the view, not away from you, do not converge at all. So converging in a photograph is a fact about the photograph. There is one more thing these pictures change, and it is about the obvious way out. You might think the answer is to stop looking and start measuring. It is better. It is not the ground either.

7:01A protractor you can read to half a degree cannot separate a pair a fifth of a degree off parallel. Get a finer instrument and the doubt shrinks. It never quite goes away. But look at how different that is from what we have just seen. Measurement error is small, and it goes either way, and it gets smaller the harder you work. An illusion is large, it goes the same way every single time, and staring harder does nothing to it at all.

7:34So let us gather the near misses, because there have been a few. A pair drawn twelve degrees apart, which a quick look would happily pass as parallel. A pair half a degree apart, which meets about five pages away. A pair a twentieth of a degree apart, which meets forty six pages away. All three look parallel. All three are settled by exactly the same move. Lay a straight edge across both, take the two angles in matching positions, and compare.

8:08And the gap you read off is exactly the angle between the two lines. Every time, wherever you lay it. One procedure, and it does not care in the slightest how convincing the picture is. So what survives all of that? Looking does not, and that is not a small loss. It was what you started with. Measuring survives as a check. It never survives as a proof. What does survive as proof is the one thing that never asked you to trust your eyes.

8:43Two lines, one line laid across them, and two angles in matching positions. Equal, and the two are parallel. Unequal, and they are not. Both directions, and no exceptions. That is why we spent so long building it, and it is why it was worth the time. And two of those three pictures were carrying that proof all along, drawn straight across the illusion.