Counting the parts of a shape sequence produces a number sequence

0:00Here are the five shape sequences from last time. Polygons. Complete graphs. Stacked squares. Stacked triangles. And the snowflake. Not a single number anywhere. Just shapes, and the rules that build them. So where do the numbers come from? They come from you. From a decision you make. You pick something on the shape that you can count. And the moment you do, a number sequence appears. Which means a shape sequence doesn't have one number sequence. It has as many as there are things worth counting.

0:38And by the end of this video, two of these are going to land on exactly the same numbers, for reasons that have nothing to do with each other. That's the thing I really want you to see. So let's start counting. Your book's own example is the polygons, and it picks the obvious thing. Count the sides. The triangle. One, two, three. Three sides. The square. Four. The pentagon. Five. The hexagon, six. The heptagon, seven. The octagon, eight.

1:13The nonagon, nine. The decagon, ten. Three, four, five, six, seven, eight, nine, ten. That's a row of table one. It's the counting numbers — just started at three instead of one. Which makes sense. You can't close a shape with two straight sides. Three is where polygons start. Now something you might have half-noticed already. The names aren't decoration. Each one is telling you the count. Pentagon. Penta means five. Hexagon, six. Heptagon, seven. Octagon, eight.

1:51You already know that last one from somewhere else. An octopus has eight arms. Same word. Nonagon is nine, decagon is ten. And a decade is ten years. So the names were never arbitrary labels you had to memorise. They were the counts all along. Which is a small thing, but it's the sort of small thing that makes a subject stop feeling like a list of words. One quick note about that word regular, because your book is careful about it and I'd like to be too.

2:27Regular means two things. All the sides are the same length. And all the corners are alike. The first half you can check with a ruler. The second half — all corners alike — needs a way of measuring corners, and we haven't got one yet. Your book says so, quite openly. It says angles are taken further in the chapter that follows this one. So think of it as a promise. Half the definition is cashed in now. The other half arrives in chapter two.

3:01I'd rather tell you that than pretend the word is fully defined when it isn't. Back to counting. Now let's count something else on exactly the same shapes. The corners. Triangle. One, two, three corners. Square, four corners. Pentagon, five. Hexagon, six. And on it goes. Seven, eight, nine, ten. Three, four, five, six, seven, eight, nine, ten. The same numbers. We changed what we were counting, and nothing moved.

3:37Now, is that a coincidence? It really isn't, and the reason is worth a minute. Take any closed shape made of straight sides. Any at all. It doesn't have to be regular. It doesn't have to be in our table. Now walk round the edge of it. Start at a corner, and go. You walk along a side, and you arrive at a corner. Along the next side, and you arrive at the next corner.

4:04Every side you walk ends at exactly one corner. And every corner you reach was the end of exactly one side. So the sides and the corners pair off. One each. Nothing left over on either team. That's why the counts are equal, and notice what the argument did. It never mentioned three, or four, or ten. It works for any closed figure at all, including squashed and lopsided ones we've never drawn.

4:36That's a reason, not an observation. And we got it by walking round a shape. Right. Different row. The complete graphs, and this time we count the lines. K two. Two dots, one line. That's one. K three. Three dots, and every pair joined. Count them. Three. K four. Four dots. Now we've got the four sides of the square, plus both diagonals. Six. K five. Take your time with this one. It's ten.

5:09K six. Fifteen. One, three, six, ten, fifteen. And those are old friends. The triangular numbers. Now why would joining dots up produce the triangular numbers? Let's watch it happen. Here's K five, with its ten lines. I'm going to add a sixth dot. The new dot has to join to every dot already there. There are five of them. So that's five new lines. Ten plus five is fifteen. And that's K six.

5:42Now run the same argument backwards. Going from K four to K five added four new lines. K three to K four added three. K two to K three added two. So the jumps are two, then three, then four, then five. Each jump is one bigger than the last, and that is exactly how the triangular numbers grow. So it isn't a coincidence that these are triangular numbers. It's the same reason as the triangles.

6:15Adding a dot means adding a longer row of lines each time — just like adding a longer row of dots to a triangle. Next. The stacked squares, and we count the little squares inside each one. One. Then four. Then nine. Then sixteen. Then twenty-five. The square numbers, and the reason is right there in the picture. The fourth block is four rows, with four little squares in each row.

6:46Four rows of four is sixteen. That's not a discovery, that's just what a rectangle is. So: stacked squares give the squares, because the block is n rows of n. Easy, and completely solid. Hold on to that reason, because we're about to get the same numbers a completely different way. The stacked triangles. And now we count the little triangles. Carefully — including the ones pointing downwards, which is the trap we found last time.

7:20One. Four. Nine. Sixteen. Twenty-five. Which are the square numbers again. From a row of triangles. And you might think, fine, same reason as before. It isn't. Look at what the rows actually hold. Take the four-row stack. Top row: one little triangle. Second row: three. Two pointing up and one pointing down. Third row: five. Three up and two down. Fourth row: seven. Four up and three down. One, three, five, seven. Those are the odd numbers.

7:57And add them up: one plus three is four, plus five is nine, plus seven is sixteen. Which is a fact we proved two videos ago, with a completely different picture. Running totals of the odd numbers are the squares. So the answer is sixteen — but it got here by a totally different road. Put those side by side, because this is the most important minute in the video.

8:26Stacked squares: one, four, nine, sixteen, twenty-five. Stacked triangles: one, four, nine, sixteen, twenty-five. Identical. Every single value. But the reasons have nothing to do with each other. One of them is: a block is n rows of n. That's a rectangle argument. The other is: the rows hold one, three, five, seven, and odd numbers add to squares. That's a completely different argument, with a completely different picture. Same landing place. Different roads. And here's the point.

9:05If somebody hands you the numbers one, four, nine, sixteen, and asks you why — you cannot answer. The numbers don't know why. The numbers came out of a shape, and the reason stayed behind in the shape. That's the lesson of this whole chapter, sitting in one example. The count is never the explanation. One row left. The snowflake, and we count the straight pieces. The first shape is a triangle. Three straight pieces.

9:37Then every piece gets replaced by four. So three becomes twelve. Twelve becomes forty-eight. Forty-eight becomes a hundred and ninety-two. And that becomes seven hundred and sixty-eight. Three, twelve, forty-eight. Now be careful — it's tempting to say this doubles. It doesn't. It's multiplied by four every time, because one piece becomes four. Your book describes it as three, multiplied by four over and over. And then it says something I want you to notice. This row is not in table one.

10:16Table one had powers of two and powers of three. It never had this. Which tells you what table one always was. A sample. Ten sequences the book picked to get you started. Never a complete catalogue. You can build a shape sequence tomorrow that lands on numbers nobody has tabulated, and you'd still be doing mathematics. So here's your question. Take the stacked triangles, and count only the ones pointing downwards. Nought, one, three, six.

10:50What sequence is that, and can you say why? Put it in the comments. That's the end of chapter one. Patterns, reasons, sequences and shapes. Next time we start chapter two, and we finally get to measure a corner.