Nautilus  Complex structures can often be based on very simple actions. This elegant shape, familiar in the form of a nautilus shell, is a good example. It is a simple application of Pythagoras’ theorem to produce a series of triangles of increasing size.

You’re no doubt familiar with Pythagoras’ theorem, which states that if you take any right-angled triangle and draw three squares, based on the three sides, then the combined area of the two smaller squares is equal to the area of the square on the longest side (‘the square on the hypotenuse’). This design starts with a right-angled triangle whose two shorter sides are both 1 unit in length – it doesn’t matter what that ‘1 unit’ is, it could be an inch, a metre or the average size of a banana. That means the squares on the two shorter sides are 1*1 units and 1*1 units. So you have (1*1) + (1*1) = 2 units. In other words, if we were to draw a square on the long side, it would have an area of 2 square units. And just because it is a square with an area of 2, that means each side of the square is equal to √2 (the square root of 2) since √2 * √2 = 2. So the long side of this first triangle is √2 units long.
But what happens if we then build another triangle using the long side of the first triangle and adding a second side 1 unit long at right angles. Now we have a right-angled triangle with the shorter sides 1 and √2, and Pythagoras tells us that the square on the long side will be equal to (√2 * √2) + (1*1) = 3. So the long side of the second triangle is √3.

To make our nautilus shape, all we do is take each hypotenuse and use it as one of the shorter sides of the next triangle in the sequence, combined with another side of length 1 at right angles.

No matter what size you choose for ‘1 unit’,  the nautilus shape that results will always have 17 triangles before the next triangle begins to overlap with the first.

Apart from the simple beauty of the shape that results, it’s a good illustration of the fact that apparently difficult numbers can often be represented quite easily with a little straightforward geometry. So if you ever need to produce a square of, say, precisely 17 square inches, you can either get out your calculator and then try to draw a line 4.123105626 inches long with a ruler, or you can draw a straight line 4 inches long (√16), draw a one inch line at right angles at one end and then join the two ends up to complete a triangle. The long side will be very close to √17 long, and so a square drawn on it with be 17 square inches.  