 Pythagorean mysticism was based around the idea of number, which they defined number is a collection of units. It's convenient to represent these units as dots. You can think about them as pebbles or something like that that you can manipulate. While number is a collection of units, we don't have to leave them in a disorderly pile. We can arrange a collection in a regular pattern. They can form figures. A number that can be formed into a figure is called a figure-at-number. So what type of figure do we want to form? So a very simple thing to do is we might try to put our number into a rectangular array. Let's make the number larger to make this more interesting. And if we experiment a little, we find out very quickly that some numbers can be formed into rectangular arrays, and sometimes several rectangular arrays are possible. But some numbers can't be put into a rectangular array. And you know where this is leading to, but not for a while. We'll get there when we get there. Somewhat simpler, we can also form numbers into triangular arrays. So this is a triangle. So the whole triangle is a number. It's a collection of units. If we take a look at the sides, we also see that the sides are also numbers. They're collections of units. This side is a collection of three units. We can say this side is three. Likewise, this side is a collection of four units. This side is four. This side is a collection of five units. This side is five. And so this is a three, four, five triangle. What's important to realize here is that geometric configuration is subject to some variation. So while this is a three, four, five triangle, so is this. Now if you think about that, almost any number can be put into some triangular array. So let's try and be a little bit more specific. We'll define a triangular number as one that can be arranged into a triangle where all three sides are equal. And we might do this as an equilateral triangle, but remember the geometric appearance of the array is irrelevant. So what are the triangular numbers? Since all three sides of a triangular number have to be equal, we might try to organize them by the length of the sides. So what would a triangle look like if each side had length of one? So let's put down a side. And the problem is if we add any more units to this, we get a side that is not equal to one. So this triangle with the side length of one is a little peculiar, so let's ignore that for a moment. Let's see, if I have two, it's not clear how I could make this into a triangle with equal sides, but if I go up to three, I can put these into a triangular array where each of my three sides has length two. And so we might say the first real triangular number is three, and this produces a triangle with sides two. Well the next triangular number would have sides of length three. So let's see if we can put that together. So we'll start with the side of length three. Let's attach another side of length three. And when we do that, notice that we formed that third side of length three. And so our next triangular number is six. Now the next triangular number has sides four. And so it is, well, actually there's a problem. We have two possibilities for what that triangular number will look like. Maybe it's an open frame like this, or maybe they're arranged like a set of bowling pins. To decide, we have to introduce a new idea that of a nomen. Given a figure at number, the next larger figure at number can be described by adding a nomen. The nomens are going to have different shapes and sizes depending on the figure. So for the triangular numbers, if I want to add something to a triangular number to make it into another triangular number, the nomen is going to look like a line. And so if I start with my triangular number three, I can add a nomen, a line, to get my next triangular number six. And I can make the next larger triangular number by adding a nomen, a line. And that gets me the next triangular number ten. And I can take my triangular number ten and add a nomen, a line, to get the next larger triangular number 15. And we can continue this as far as we want. How about squares? So let's start off with a square with a side length of two. So here's one side, second side, third side, and our fourth side. And this is a collection of four things, so four is our first square number. Now if I want to get a square with a side of three, we want to add a nomen. And in this particular case, we should add an L-shaped nomen. And in fact, a real nomen is part of a sundial and it is L-shaped. So this is probably the origin of the term nomen. Now if you look at what we need to add, this nomen corresponds to the number five and so the next square number will be nine. The next nomen will add an L-shaped piece to get us to the next larger square. And this nomen will have seven. And so the next square number is going to be 16. Notice that our square numbers are not produced by multiplying a number by itself, they're produced by adding a nomen to an existing square number. And this is an important observation. Incidentally, we get number multiplied by itself, but their creation is based upon addition and not multiplication. This is important for understanding what comes next. For it leads us to the Pythagorean theorem, but not the one about triangles. Instead, it's this one. Start with any odd number a, square and subtract one, divide by two to get another number b, then add one to get c. And if you do this, these numbers a, b and c satisfy a remarkable relationship, a squared plus b squared equals c squared. The numbers a, b and c are what are called a Pythagorean triplet. For example, we might find a Pythagorean triplet starting with seven. So we've started with an odd number, check. We square and subtract one, that's seven squared minus one gives us 48. We take half of the number, that's 24, and that's b. And then we take one more, that gives us 25. And seven squared plus 24 squared is in fact 25 squared. And seven, 24, 25, form a Pythagorean triplet. So how did the Pythagoreans find this? We don't know, but one way is by considering the moments. Since we add a nomen to a square to produce the next larger square, it's enough to make the nomen itself a square number. And the thing to notice here is the nomens are always odd numbers. So if our nomen is an odd square, one less, then half will give us the side of the square to nomen two. And because we're adding the nomen to produce a larger square, then one more will give us the side of the larger square.