What similarity asks for that congruence does not

0:00Congruence asks two things of a pair of figures, and it asks them at the same time. They have to agree in shape. And they have to agree in size. The test for it is physical. Pick one figure up and lay it on the other. If it can be made to coincide - every corner on a corner, every side along a side - then the two are congruent.

0:24Turning it round is allowed. Sliding it across is allowed. Flipping it over is allowed. What is not allowed is making it bigger. That last restriction is doing a great deal of work, and this whole video is about what happens when you take it away. So take it away. Lay one figure on the other, and this time allow yourself to scale it as well. Everything else stays. The turning, the sliding, the flipping over. Only the size is now free.

0:58A pair that passes this weaker test is called similar. And notice what similar is not. It is not a new relation standing beside congruence. It is the same test with one demand withdrawn. Which makes congruence a special case of it - the case where the scale you needed turned out to be one. And notice what is still not being asked. Nothing here mentions which way up a figure is, or where on the page it sits, or whether it has been turned through some awkward angle.

1:33Those were never part of congruence either. Position and orientation have never mattered, and they still do not. So every congruent pair is a similar pair, automatically, with nothing left to prove. Start with the easiest family there is. Circles. A circle is fixed by one number. Its radius. Nothing else about it is free. So given any two circles, scale the first by the second radius over the first, and it lands exactly on the second.

2:09Always. There is no pair of circles anywhere that fails. Over twenty-five pairs of circles, every single pair is similar. And exactly five of those pairs are congruent as well - the five where the two radii were the same number to begin with, so the scale came out as one. Now the same story, twice more. A square is fixed by its side. An equilateral triangle is fixed by its side.

2:40One number each. So any two squares are similar, and any two equilateral triangles are similar. Twenty-five pairs of squares. Similar, all twenty-five. Twenty-five pairs of equilateral triangles, carried in exact arithmetic so that no corner is ever rounded off. Similar, all twenty-five. And congruent in exactly five of them, which are the five where the sides agree. A family is safe when one number fixes it. But that is a statement about the family, not about the name.

3:19Rectangles are not fixed by one number. A rectangle needs two. So take five rectangles and put every ordered pair of them through the same test. Out of twenty-five pairs, only seven come back similar. Five of those seven are a rectangle against itself. The seventh is one genuine pair: the two-by-one against the four-by-two. A two-by-one and a three-by-one are both rectangles, and they are not similar. Triangles behave the same way. Five isosceles triangles give seven out of twenty-five. Five scalene ones give seven out of twenty-five.

4:02Being in the same family is not the point. Being fixed by one number is. Now the direction of the thing. Ninety-six figures. Every ordered pair. Every pair laid on the other in every way the corners could be matched, turned over as well as not. Nine thousand two hundred and sixteen questions. The test does not just say yes or no. It hands back the scale it needed. Lay a square on a square twice its size and the answer comes back as two.

4:38Four hundred and sixteen pairs coincide at scale one. Those are the congruent ones. One thousand and twenty-four coincide only at some other scale. Those are similar and not congruent. And the cell that would hold a pair that is congruent but not similar is empty. It stays empty, over all nine thousand two hundred and sixteen, because it cannot be filled. Congruent forces similar. Similar does not force congruent. It is a one-way street.

5:13Some pairs you really can settle by looking, and it is worth being clear about which. A triangle against a square. No amount of scaling adds a corner. Scaling stretches; it does not bend a side into two. In that same pool, four thousand three hundred and twenty pairs had different numbers of corners, and not one of them was similar. Not one. A circle against a square is the same argument taken to its extreme. A circle has no corners at all, and a square has four.

5:49Scaling cannot remove the four, and it cannot smooth them away either. So no circle is ever similar to any square, and no looking hard at them will change that. So when two figures differ in something scaling cannot touch, looking is enough. And now the case that actually matters. Two four-sided figures, side by side, with no length written on either one and no angle marked on either one.

6:20They look alike. Are they similar? You cannot say. And it is worth seeing exactly how badly you cannot say. Take one of them, and build every convex four-sided figure whose corners sit at whole-number places in a window around it. Seventy-five thousand, six hundred and sixty-four of them. Now keep only the ones carrying the cues the bare drawing actually gives you: four corners, bulging outwards everywhere, and the sides in the same order from shortest to longest.

6:56Three thousand, eight hundred and eighty-nine survive that. And of those three thousand eight hundred and eighty-nine, exactly two are similar to the one we started with. Itself, and its double. Every other one is an impostor that looks the part. Here is what similarity looks like when it is real. One photograph of a building, reproduced at three sizes. Same shape, three sizes. Every pair of those three is similar, and no pair of them is congruent.

7:32And here is the trap sitting next to it. Two photographs of the same person, made at exactly the same size. One taken at ten years old, one at forty. Same size. Different shape. That pair fails. Seven figures put through both questions at once: eighteen of the pairs cover exactly the same area and are still not similar. Size agreeing is not the half of the test that similarity keeps. It is the half it threw away.

8:06So what does an enlargement actually do to a picture? A negative thirty-five millimetres across, enlarged to forty-five. Every length in it is multiplied by forty-five over thirty-five, which is nine sevenths. A segment twenty-one millimetres long comes out at twenty-seven. Enlarge the same negative to fifty-five instead and the multiplier is eleven sevenths. The same twenty-one comes out at thirty-three. Run either one backwards and the multipliers are seven ninths and seven elevenths.

8:45And every angle in the picture is left exactly as it was. Not nearly. Exactly. Lengths scale. Angles do not. That is the whole of what an enlargement is, and it is the reason a definition of similarity is possible at all. One warning before that definition. Similar does not mean roughly alike. There is no slack in it anywhere. Take a figure. Scale it properly. Then move one single corner by a tenth of a unit.

9:20Every corner test fails. Every side test fails. Laying one on the other fails. Twenty-four true scalings pass all three of those, and twenty-four near misses fail all three, with nothing in between them. A figure that is nearly the right shape is not nearly similar. It is simply not similar. And the mirror image of that warning is worth saying too. Being bigger is never a reason for a pair to fail.

9:51Size is precisely the thing similarity has stopped looking at. So if looking cannot settle it, what would? Two things, and it has to be both of them. The sides have to be in proportion. All of them, taken in order. And the corners have to agree. All of them. Take away either half and the test breaks - and it breaks in a different place each time. Sides alone. A square and a rhombus have four equal sides each, so their sides are in perfect proportion. They are not similar.

10:28Two hundred and eighty-eight pairs slip through that way. Corners alone. A square and a rectangle have four right angles each. They are not similar either. Eight hundred and sixty-four pairs slip through that way. And no pair at all slips through both. So neither half is a weaker version of the other. Each one is blind exactly where the other sees. So. Similarity is congruence with the size demand withdrawn, which is why every congruent pair is similar and the reverse fails.

11:05A family is safe when one number fixes it. Circles, squares, equilateral triangles. Rectangles and ordinary triangles are not. Looking settles a pair only when the two differ in something scaling cannot touch, like a corner count. Otherwise looking settles nothing, and what has to be measured is the sides in proportion and the corners agreeing, together. And here is why any of it is worth the trouble. A mountain you cannot climb, and a moon you cannot reach, both have a size that no tape measure will ever get to.

11:45You cannot lay a tape along a sight line to a mountain top. But you can build a small triangle down here that has the same shape as the enormous one up there. Same shape means the sides are in proportion, and proportion is a multiplication, and a multiplication is something you can do on paper. Shape without size is the tool that gets to them, and the next thing to build is the definition that makes it usable.