Shapes come in sequences too, with rules of their own

0:00Triangle. Square. Pentagon. Hexagon. What comes next? You know the answer, and notice how you got it. You didn't add anything. You didn't multiply anything. You worked out how the shapes were being made, and then you made the next one. That's a sequence. It's just not made of numbers. Your book calls these shape sequences, and it gives you five of them in one table. And they're governed by rules, exactly the way the number sequences were. What changes is the kind of rule.

0:35A number sequence's rule is a step you do to a number. A shape sequence's rule is a construction. Something you build. Before we start, your book names the branch of mathematics these belong to, and it does it in a single line. Geometry. The study of patterns in shapes. So the last few videos were number theory — patterns in whole numbers. This one is geometry. Same subject, same habit of mind. Find the pattern, then find the reason. Different raw material.

1:14Your book also slips in something quite large here, and it's easy to miss. It points out that shapes live in different numbers of directions. A line segment needs one direction. That's one dimension. A square needs two. Length and width. Two dimensions. A cube needs three. Length, width and height. And then — and this is the bit — the book says shapes can live in more than three dimensions too.

1:44Not as a joke. Just quietly, as a fact. If you watched the video on drawing the powers of two, you've already met one. That fifth figure, two cubes with matching corners joined, with sixteen corners. It didn't fit in ordinary space, and now you know why. It wasn't trying to. Right. Table three, page ten. Five rows. Let's take them one at a time. Row one. Regular polygons. A polygon is a closed shape made of straight sides. Regular means all the sides are the same length, and all the corners are alike.

2:26Three sides, a triangle. Four sides, and because it's regular, that's a square. Five, a pentagon. Six, a hexagon. Seven, a heptagon. Eight, an octagon. Nine, a nonagon. Ten, a decagon. The rule is about as simple as a rule gets. Add one more side, and keep it regular. One thing to be careful with. That four-sided one is a square only because we asked for a regular one. The word quadrilateral by itself means any closed shape with four straight sides. A squashed one still counts. Your book says regular on purpose.

3:10Row two, and this one has an odd-looking name. Complete graphs. Forget the name for a second and watch the rule, because the rule is lovely. Put down some dots. Then draw a line between every pair of them. Every single pair. That's it. Two dots. One line between them. The book calls this one K two. Three dots. Now each pair gets a line — and you get a triangle. K three.

3:40Four dots. Join every pair, and you get a square with both its diagonals drawn in. K four. Five dots. K five. It's a pentagon with every diagonal drawn. Six dots. K six, and it's starting to look like a spider's web. Now, you might be itching to count those lines. Hold on to that. It's the whole of the next video. For now the rule is all we want. Add a dot, and join it to everything already there.

4:13Row three. Stacked squares. One little square. Then a block two along each side. Then three. Then four. Then five. And you've seen this shape before, in a different costume. Two videos ago these were dots in a square array. Here they're little squares tiling a big one. Same construction, different material, and the same rule. Grow the side by one, and fill the block in. Drawing the next one is easy. Six along each side.

4:47Row four. Stacked triangles. Same idea, with triangles instead of squares. One little triangle. Then two rows. Then three. Then four. Then five. And here's a trap, and it's the reason I'm slowing down. Look carefully at the two-row triangle. How many little triangles are in it? Most people say three. Three triangles pointing up. But look at the gap between them. There's a fourth little triangle sitting there, pointing downwards.

5:20It's easy to miss because it's upside down, but it's the same size, and it's really there. So a two-row triangle has four little triangles, not three. Three up, one down. Three rows? Six pointing up, three pointing down. Nine. This matters more than it looks. If you only count the upward ones you get one, three, six, ten — the triangular numbers. If you count them all, you get one, four, nine, sixteen. The squares.

5:52Two completely different answers, from the same row of the table. And the next video depends on getting this right. Row five. And this one is different from everything else in the table. It's called the Koch snowflake, and it's built by replacing rather than adding. Here's the rule. Take a straight piece. Cut it into three equal parts. Now push the middle part out into a bump — a little triangle sticking up where the middle third used to be.

6:26Your book has a nice name for that shape. It calls it a speed bump. So one straight piece becomes four shorter pieces, with a point in the middle. Now do that to every side of a triangle, all at once. Three sides, three bumps. And you get a six-pointed star. Now do it again. Every straight piece of the star — and there are twelve of them — gets its own bump.

6:52Twelve becomes forty-eight. And again: forty-eight becomes a hundred and ninety-two. Each round the pieces get shorter, and there are four times as many. The outline gets crinklier and crinklier. One round of doing that is called an iteration, and it's a word worth knowing. Now, your book asks two questions about this table, and the second one is the one that matters. The first is: can you see the pattern in each row? Which, honestly, you can.

7:27But the second question asks for four things. Redraw the row. Draw the next shape. Say why or why not. And describe the rule in your own words. And it carries the Math Talk badge, which means say it out loud, to somebody. That phrase — why or why not — is doing something. It's warning you. It's telling you that for at least one of these rows, the answer is going to be not straightforward.

7:55So let's actually try it. Five rows, one request: draw me the next shape. Regular polygons. Next shape? Eleven sides. Easy. Complete graphs. Next? Seven dots, every pair joined. K seven. Fiddly, but easy. Stacked squares. Six by six. Easy. Stacked triangles. Six rows. Easy. Four out of five, no trouble at all. And then the snowflake. Can you draw the next one? Well — the rule tells you exactly what to do. Take all seven hundred and sixty-eight straight pieces, and give each one a bump.

8:38Seven hundred and sixty-eight bumps, each one a third the size of the last lot. And at some point your pencil is thicker than the bump you're trying to draw. The ink just merges. So here's the question that actually matters. Has the sequence stopped? No. Absolutely not. The rule works perfectly. There is a next shape, and the one after that, for ever. What stopped is your pencil. Those are completely different things, and mixing them up gets the lesson backwards.

9:12A rule can keep going long past the point where you can draw the result. That's the whole point of that row being in the table. So let's do the last part of the book's request. Say the rules out loud. Regular polygons: add one more side, keeping every side the same length. Complete graphs: add one more dot, and join it to every dot already there. Stacked squares: make the side one longer, and fill it in with little squares.

9:46Stacked triangles: add one more row along the bottom. Koch snowflake: replace every straight piece with a bump. Read those back. Every single one is an instruction you could hand to somebody with a pencil. That's the test. If your rule is vague, the person can't follow it. If it's a real rule, they can produce your next shape without ever having seen it. And notice something. Not one of those rules mentions a number.

10:18Which brings us to what happens next, and it's the nicest turn in the chapter. So far these are shapes. No numbers anywhere. But every one of them has parts you could count. Sides. Corners. Lines. Little triangles. And the moment you choose something to count, a number sequence falls out of the shape sequence. Count the sides of the regular polygons, and you get three, four, five, six. Count the little squares in the stacked squares, and you get one, four, nine, sixteen. The squares, again.

10:55Count the little triangles — all of them, upward and downward — and you get one, four, nine, sixteen as well. Which is strange, and worth explaining. So here's your homework, and it's a real one. How many lines are there in K six? Count them off the picture. Then work out what K seven would have, without drawing it. Put both in the comments. Next time, we count the parts of every one of these, and see which number sequences come out.