Tuesday, 6 April 2010

New numbers

In my last post I pointed out how the ratio 1:1.618, manifests itself in segments of a circle – of any angle. Three segments of a circle – of the relative sizes 1:1.618:2.618 – can be placed in such a way that the two smaller segments will sit inside the larger to create a curve with their edges that will touch the straight edge of the largest segment. I thought it would only be 1.618 that could create such an effect but I found that the numbers below do the same thing.

The first number I found was 1.465. The Fibonacci sequence, which creates numbers relative by 1.618, relies on your adding the two previous values to get the next (in the sequence 3, 5, 8, we get 8 by adding 5 and 3). If you keep calculating the Fibonacci sequence into the hundreds of thousands, you‘ll find that the relation between the numbers gets closer and closer to the complete number of Phi – though only the first few decimals really matter. 8/5 is only 1.6, so it’s necessary to keep going through the sequence until it gets a more accurate value for the relation between numbers that are the sum of their two precedents. If 1.618 is gained by adding the first and second values in a sequence, 1.465 is made by adding the first and third. If you started with 1, 2, 3, then, the next value would be 4 (3+1). If you continue the sequence you eventually get the numbers 872, 1278, 1873 and 2745 (2745=1873+872) and if you divide 2745 by 1873 you get 1.4655632…. You could round it to 1.466 or just leave it at 1.465 – you’d have to keep going in the sequence to check if it was going up or going down as it averaged out.

Anyway, the first image shows what 1:1.465 does with circle segments relative by its value – remember the second smaller version is ditched – its the first and third smaller versions of the largest that we use.

Adding the first and fourth previous numbers in a sequence will give you proportions of 1:1.382 and if you use the first and fifth previous numbers, you get 1:1.324. Interestingly, in the 1:1.324 sequence, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 106, you’ll find that, as expected, 34 = 26 (first previous) + 8 (fifth previous) but if you add neighbouring numbers, as you would in the Fibonacci sequence, you get a similar effect to the Golden String – 3+4=7, one over 6, 4+5=9, one over 8, 5+6=11, equal or 0, 6+8=14, one under 15, 8+11=19, one under 20, 11+15=26, equal or 0, 15+20=35, one over 34…and so on. In other words it goes +1, +1, 0, -1 -1, 0, +1, +1, 0, -1, -1 etc.

There must be some significance with these numbers if they act similarly to Phi with regard to circle segments within related segments…but I’ve yet to find it out. I added 1:1.414 (the useful proportions of DIN paper edges) as well, to demonstrate how it conforms with the seemingly special ability of these numbers to make segments of any angle able to relate the way they do in the images here.

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