When I was writing the Seurat post I mentioned that, in that incidence, the difference between one number, 1.618 (the golden section) and another, 1.414 (DIN paper system), didn’t make much difference. In other examples, though, it will, and below shows how.
I’ve used the Fibonacci sequence for a long while as a guide to my type sizes because, when I was just starting out in using type in my designs at uni, I read it would give me the most pleasing proportions. It has stuck with me since but I now know much more about it. The sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 etc, is made by adding the previous number onto the current number (starting with 1 and adding 0 to get a second 1). If you take a number in the sequence and divide it by the preceding one you’ll get something close to 1.618 (golden section). Notably, 8 and 13 are in the sequence (13:8=golden section ratio). The further you go in the sequence, the closer you get to 1.618 in dividing each number by the one before. In other words, the Fibonacci sequence is pretty much the golden section (it’s a rounded up/down version).
Not only that, you can start with any number, like 10, and add, say, 5 and repeat the process, (so 5, 10, 15, 25, 40, 65, 105, 170, 275, 445, 720, 1165) and, if we divide 1165 by 720 we get 1.61805555…. Golden section. Spooky! What’s strange is that you’re not using a fixed number to enlarge the previous – 10 is 2x5 but 445 is 1.618181818x275.
The two sets of lines below do use a fixed number to increase the type sizes. One is the golden section (right side) and the other uses 1.414 (the ratio of DIN paper measurements). Although the golden section sequence looks better hands down I wonder if it would still be most suitable to use the 1.414 sequence if you were working on A4/A3 (DIN) paper. Even lettering would be a consideration – lots of lettering is based on the square, but would it not be more logical to design type whose character shapes were similar to the shape of A4 paper?