The two conditions polygons must both meet

0:00Two figures, side by side, and the question is whether they are the same shape. Looking will not settle it. That is not a criticism of anybody's eyes - it is a fact about what a drawing carries. So we need a test. Something you can carry out on paper, with numbers, that returns yes or no. And a test has to come from somewhere. It cannot just be announced.

0:27The place it comes from is the enlargement, so start there. Take a small photograph and print it larger. Two things happen, and only two. Every length in the picture is multiplied by one number. The same number, everywhere. And every angle in the picture is left exactly as it was. Not nearly. Exactly. That is the whole of what an enlargement does, and those two facts are going to become the two halves of a test.

1:00Lengths: all scaled by one number. Angles: untouched. So if two figures really are the same shape, they must agree on both counts. And a definition is just that sentence, said carefully. Here it is. Two figures with the same number of sides are similar when two things hold at once. First: their matching corners are equal. Second: their matching sides are in the same ratio. Both. Not one. Not one and then the other implied.

1:35The rest of this video is about that word - both - because it is the only part of the definition anybody ever argues with. And notice the second clause said the same ratio, not similar ratios, and not roughly the same ratio. One number, shared by every pair of matching sides. That number has a name. It is the scale factor. If a figure's sides are 3, 4 and 6, and the other's are 6, 8 and 12, the ratios are 2, 2 and 2, and the scale factor is 2.

2:13The other way round it is a half. Every scale factor has a partner, and the two multiply to one. You have used it without the name. A map that says one centimetre to five kilometres is telling you a scale factor. A floor plan, a model aeroplane, a photographic print - each is a figure with a scale factor attached. The important word is one. Not one number per side. One number for the whole figure.

2:45A test that let the number drift a little from side to side would be a different test, and a much weaker one. Where does a figure with a scale factor actually come from? Here is a way to make one. Put a lamp at a point above a table. Hold a flat card between them - a quadrilateral, cut out of cardboard. The card throws a shadow on the table. Trace it.

3:12Now look at how the shadow was built. Every corner of the shadow sits on the straight line from the lamp through the matching corner of the card. One point, four rays, four corners placed. That is the whole construction. Measure the shadow against the card and you find it: every corner the same, every side multiplied by one number. Which is exactly the definition, arriving without anybody imposing it.

3:44Why does that work? Because the rays all come from one point. Pushing every corner out along its own ray from a single point, by the same factor, is one operation, and it is the only thing happening. Four different lamps and four different factors were tried. Every time: no corner moved at all, every squared side multiplied by the same number, and the shadow could be laid exactly on the card at that scale.

4:15Compare that with something that is not a construction from one point. Slide every point of the card sideways by an amount proportional to its height. A push, rather than a stretch. Three different amounts of push were tried. Every one moved the corners, gave no single scale factor at all, and produced a figure that cannot be laid on the card at any scale. So the one point is doing the work. Rays from a single point scale without distorting; almost nothing else does.

4:51One thing has to be said before the definition can be used, and it is the thing most often skipped. The definition says matching corners and matching sides. That word carries a whole clause of its own. A four-cornered figure can be laid against another in eight ways. Four rotations, and the same four with the figure turned over. Take a quadrilateral and its exact double. Under the pairing the figures are written with, both clauses hold.

5:25Under the pairing that starts one corner along, both clauses fail. Not one of them - both. And that is not an edge case. For that pair, exactly one of the eight pairings works. For a square against its double, four of the eight work, because a square can be turned onto itself. So the number of pairings that work is a measure of how symmetric the figure is - and the claim of similarity always includes which pairing you meant.

5:57Now a pair that passes, worked all the way through. One quadrilateral has sides of 1.5, 2.5, 2.4 and 2.1 centimetres. The other has 3.0, 5.0, 4.8 and 4.2. Four ratios. Three over one and a half is 2. Five over two and a half is 2. Four point eight over two point four is 2. Four point two over two point one is 2. One number, four times. So there is a scale factor, and it is 2.

6:36Read the other way, every one of the four is a half. The corners are marked 105, 100, 70 and 85 degrees on the first figure, and 105, 100, 70 and 85 on the second. Equal, corner by corner. And they total 360 degrees, which any four-cornered figure's must. Both clauses hold, under one pairing. The pair is similar, and the whole of the evidence is those eight numbers. Now the two arguments that make the word both necessary.

7:15First: could equal corners be enough on their own? A square with every side 3. Beside it a rectangle, 3.5 by 3. Every corner of the square is a right angle. Every corner of the rectangle is a right angle. So the corner clause holds perfectly. Four right angles against four right angles, nothing to argue about. Now the sides. Three point five over three is seven sixths. Three over three is one.

7:48Two different numbers. So there is no scale factor at all - not a bad one, none. And the pair is not similar. You can see it: the rectangle is a stretched square, and stretching in one direction only is exactly what similarity forbids. Corners alone is not enough. Second: could the sides be enough on their own? A square with every side 2.1. Beside it a rhombus with every side 4.2.

8:20Four ratios, and every one of them is exactly 2. A perfect scale factor. The side clause holds as cleanly as it possibly could. And the pair is not similar. The square's corners are right angles; the rhombus is leaning over, and no pairing of its corners can be made to agree with the square's. A rhombus is a square that has been pushed sideways. Every side keeps its length, and the shape is gone.

8:51Sides alone is not enough. And the same counter-example turns up again at a different size - a rhombus of side 1.5 against a square of side 3 - which is worth noticing, because it means the objection is about the shapes and not about the numbers. Two counter-examples is two counter-examples. So here is the same question asked exhaustively. A hundred and eight figures, built by putting nine shapes under a lamp at various scales, then turning them, then reflecting some of them.

9:30Every ordered pair of those: eleven thousand, six hundred and sixty-four questions, each asked four ways. Laid one on the other, allowing a scaling. Both clauses. The corner clause alone. The side clause alone. Twelve hundred and ninety-six pairs are genuinely similar, and on every single one of the eleven thousand six hundred and sixty-four, the two clauses together agree with laying one figure on the other. Not almost always. There is no pair anywhere in the census where one of them says yes and the other says no.

10:11That is what makes it a definition rather than a guess. And the two halves, run on their own? The corner clause alone lets 288 pairs through that are not similar. Every one of them is a square against a rectangle, in one order or the other. The side clause alone lets 864 pairs through that are not similar. Every one of them involves a rhombus. Different numbers, and more importantly different pairs. Neither half is a weaker version of the other.

10:49Each one is blind exactly where the other sees. And no pair anywhere passes both halves and is still not similar. Which is the same sentence as before, read from the other end: both halves together is the test. So the definition is not being cautious. It is the minimum that survives its own counter-examples. Two consequences, quickly, because they fall out of what we already have. First: similarity chains. If a first figure is similar to a second, and that second to a third, then the first is similar to the third.

11:30And the scale factors simply multiply. A factor of 2 followed by a factor of 3 is a factor of 6. Four such chains were run, each step thrown from a different lamp, and the first figure reached the third at exactly the product every time. Put a push in the middle instead of a stretch and the chain breaks, exactly where you would expect it to. Second: some families never need the test at all.

12:00Any two circles are similar. Twenty-five pairs of circles, all twenty-five similar, and five of them congruent as well. Any two equilateral triangles are similar, carried in exact arithmetic so that no corner is ever rounded off. Twenty-five pairs, all twenty-five. But not any two isosceles triangles. Twenty-five pairs of those, and only seven come back similar. One number fixes a circle, and one number fixes an equilateral triangle. Two numbers are needed for an isosceles one, and that is the whole difference.

12:45So, the test. Same number of sides. Matching corners equal. Matching sides in one ratio. All three, and the pairing is part of the claim. The corner clause alone passes a square against a rectangle. The side clause alone passes a square against a rhombus. Neither can be spared. And now the question that the next stretch of this subject is really about. That is the definition for polygons in general - four sides, five sides, any number.

13:20What happens if the figures have only three? A triangle is the smallest polygon there is, and it turns out to be a special case in a way nothing above it is. For triangles, checking one clause turns out to give you the other for free. Either one. Which is a strange thing for a definition to do, and it is worth finding out why.