Triangles and squares Mathematicians like numbers with relationships between them. Some-times those relationships are extremely complex but at other times they are obvious aft er a little thought. This design neatly captures two simple relationships between ‘square numbers’, that is numbers where you take an integer (a simple number like 1,2,3 without fractions) and multiply it by itself.

Squares are everywhere in the real world, not least because they’re so predictable. Square packaging is one of the most economical shapes in terms of the amount of material used compared to the space inside. Square components in construction make calculations easy and react predictably to stresses from diff erent directions.

And square numbers keep creeping in whenever things in the natural world are measured. Take the amount of light (or any other radiation) hitting a surface; it will vary according to the square of the distance from the source – double the distance to the source and you end up with a quarter of the light on the same area of surface.
So it’s of en useful to be able to think in squares, and this illustration shows a simple but important shortcut.

First look at the square itself. It’s easy to see that, starting with 1 (which is a square number, 1*1), the way you arrive at the next biggest square is to build out by one row on two adjacent sides. You don’t need to be a maths genius to see that what this means is that if you have, say, a square with a side of length 3, to make the next largest square you add a row of three on two of the sides and then fill in the empty space on the corner, i.e. the next square (2*3) + 1 = 7 bigger, giving 9 + 7 = 16, which is 4 squared.

And since mathematicians love to generalize things, it’s no big leap to a simple piece of algebra. If we say that our starting square has sides of  k units, where k is any integer you like, the next largest square is always going to be 2k+1 bigger.

The triangle above the square illustrates one more nice relationship. Using equal-sided triangles of the same size, the same numbers that increase the size of a square also provide the next line of the triangle as it grows. And since you’re adding the same number of triangles as the number of squares each time you increase the size, the number of smaller triangles in the large triangle will always be the same as the number of small squares in the large square. So if any large triangle is made up of small equal-sided triangles, as in the illustration, the number of small triangles will always be a square number.

Neat.