Why some resized images look right and others look stretched

0:00Here is one photograph of a tiger, and here it is five times over, at five different sizes. Look at them and you can tell, without measuring anything, that three of the five are the same picture at different sizes. The other two are wrong. One of them looks squashed and the other looks stretched. Your eye is certain about this and your eye cannot tell you why. So the question is not which ones look right. It is what the right ones have in common that the wrong ones do not.

0:34And it will turn out to be a single number. One number, carried by each picture, that decides whether you changed its size or changed its shape. Here is the first explanation anybody offers. One of the bad ones came out square, and the tiger is not square. That is why it looks wrong. It is a good explanation. It fits. And it is not the answer. Because the other bad picture is not square either. It is a rectangle, wider than it is tall, exactly like the good ones.

1:11So squareness explains one of the two failures and says nothing at all about the other. An explanation that survives one example and dies on the second is not an explanation. It is a coincidence you have not caught out yet. So stop looking, and measure. Picture A is sixty millimetres wide and forty tall. B is forty and twenty. C is thirty and twenty. D is ninety and sixty. And E is sixty and sixty.

1:45Now, ten numbers is not an answer. Ten numbers is just the measuring done. The answer is what you do with each pair. Divide the width by the height, and every picture hands you one number instead of two. A is sixty over forty, which is three halves. C is thirty over twenty. Three halves. D is ninety over sixty. Three halves again. B is forty over twenty, which is two. And E is sixty over sixty, which is one.

2:19There it is, and it took no judgement at all. A, C and D carry three halves. B carries two, E carries one. The three your eye picked out are exactly the three that share a number, and the two it rejected are exactly the two that do not. Your eye was measuring this all along. It just could not tell you what it was measuring. Now look at how you get from A to the ones that worked.

2:48A is sixty by forty. C is thirty by twenty. Halve the width, halve the height. One factor, used twice. A to D: sixty becomes ninety and forty becomes sixty. That is three halves on the width and three halves on the height. One factor again, and it does not have to be a whole number. Now A to B. Sixty to forty is two thirds. But forty to twenty is one half.

3:17Two different factors. There is no single number you can multiply A by to get B, and that is the whole of B's problem. And you can see why one factor has to work. The number a picture carries is a division. Width divided by height. Multiply the top and the bottom of a division by the same thing and the division does not notice. Ninety over sixty is three halves because the extra three halves is sitting in the top and the bottom at once, and it cancels.

3:53That is the entire mechanism. A common factor is precisely the thing a quotient cannot feel. So proportional does not mean the picture got bigger. It means both measurements were multiplied by one and the same number. Different factors, different shape. Every time, with no exceptions to remember. Now here is the mistake that feels fair and is not. Take twenty millimetres off the width of A and twenty off the height.

4:27Sixty becomes forty, forty becomes twenty. That is B. That is exactly the picture your eye rejected. And the reason is arithmetic, not aesthetics. Twenty is half of the height. Twenty is a third of the width. The same subtraction is a much bigger event for the shorter side than for the longer one. So the shorter side loses proportionally more, the number goes from three halves up to two, and the picture comes out stretched.

5:01Predict this before you look at the next one: does equal subtraction make a picture squarer or longer? Almost everybody says squarer. It is always longer. Not usually longer. Always, and here is the proof in one line. Compare width over height with width minus k over height minus k. Cross-multiply and the difference between them comes out as k times width minus height. If the width beats the height and you took something positive off, that is positive. So the second number is always the bigger one.

5:41It never repairs a mismatch and it never leaves one alone. It exaggerates, always. Which is why a wide picture shrunk this way comes out looking stretched, and never comes out looking square. Now the part that is easy to skip, and should not be. That proof needs the amount you take off to be smaller than the shorter side. Not because it is tidy. Because it is false otherwise.

6:10Swept over every width and height and every amount below the shorter side, the rule holds one thousand one hundred and forty times out of one thousand one hundred and forty. Take off exactly the shorter side and the second comparison has nothing left to divide by. It does not say anything. And take off more than the shorter side, and the conclusion flips. Swept over another one thousand one hundred and forty cases, it exaggerates in none of them.

6:43The same algebra, and the opposite answer, on the other side of one number. A theorem that does not carry its condition is not a theorem. It is a rule that happens to work where you tried it. There is a second way to wreck a picture, and it is even easier. Hold one side still and move the other. Keep A's width at sixty and push its height from forty up to sixty, and you have E.

7:14The width factor is one. The height factor is three halves. Two different numbers again, so the shape had to move. And notice the direction. This time the number went from three halves down to one, and the tiger came out squashed rather than stretched. Both failures are the same failure. Two factors instead of one. Squareness was never the problem. It was only ever a symptom, and it was not even a reliable one.

7:46One warning, and it matters the moment you leave photographs. Here are five rectangles drawn at different angles. Which of them are the same shape? Measure the first: six millimetres by eighteen. Measure the fourth: thirty-nine by thirteen. Width over height gives you a third for one and three for the other, so you would say they are different. But turn one of them a quarter turn and they sit on top of each other. Longer over shorter is three for both.

8:19Width means the horizontal one, and horizontal is not a property of a shape. It is a property of how the shape happens to be lying. Pair your two numbers by naming what each one measures, not by which way it is pointing. Two more things worth knowing before you use any of this. The first: shapes read off a drawing carry measurement error, so agreement is a claim about how closely.

8:48Two of those five rectangles agree to better than one part in a hundred. Within a fiftieth they group cleanly into a three-to-one pair, a five-to-two pair, and one that matches nothing. Loosen the tolerance far enough and everything matches everything, which tells you nothing. The second: a tolerance does not chain. Three shapes can each sit within a fiftieth of the next while the two ends sit further apart than that.

9:19So when you draw a rectangle to a shape rather than to a size, name the factor you used. Everyone's drawing will be a different size and every one of them can be right. That is what proportional buys you: a shape you can hand to somebody else without handing them a ruler.