Golden ratio circles

I’m not sure that anyone will ever buy this apparently simple design but that’s OK because I love it. I love it because in a few lines which can easily be created using a ruler and compass it reveals many of the secrets of the so-called ‘golden ratio’ and its relationship to the Fibonacci series.

It begins with a triangle whose purpose is to provide a line representing the value. We’ll treat the vertical side as representing a value of 1. The horizontal line is twice the length of the vertical and so represents the value 2. Pythagoras theorem tells us that the length of the hypotenuse of the triangle is therefore or The next step is to extend the hypotenuse by the length representing 1, which is simply accomplished by drawing a circle with the radius of the triangle’s vertical and drawing a straight line out from the hypotenuse to meet the circle. The length of the extended line now represents Finally, we find the mid-point of the extended line, so now we have two segments, each of which represents the valueIf you don’t recognize it, that’s the strange value known as the ‘golden ratio’. Its approximate value is 1.618 and its strangeness and wonder is that its inverse – the number it is multiplied by to get the value 1 – is 0.618. It is the only number that has this strange quality and it is precisely this quality which underlies the Fibonacci sequence – 1,1,2,3,5,8,13… – where each number is formed by adding together the two previous members of the series. How does this relate to the golden ratio? Well 0.618*1.618=1 and, obviously 1.618*1=1.618. So we have a series of values: 0.618, 1, 1.618.

If we were applying the principles of the Fibonacci series the next number would be 2.1618, which happens to be 1.618*1.618. The next number in the series would be 1.618+2.618=4.236, which just happens to also be – you’ve probably guessed – 2.618*1.618. Of course, all these values are truncated and so approximate but the underlying principle is spot on – a series where every succeeding value is composed of the sum of the previous two in the series and the last number in the series multiplied by 1.618.

The relationship to the Fibonacci series is that as the series goes on and the numbers get bigger, the relationship between each number in the series and its predecessor approaches closer and closer to the golden ratio. In fact you can start a series with any two numbers at all and build a series by adding the last two numbers to create he next and after a while the relationship between the last number in the series and its predecessor will approach the golden ratio.

So if we return to the design and draw a circle based on the midpoint of the extended hypotenuse, its radius isor 1.618 times the upright of the triangle, which represents 1 for us. And the smaller circle we already drew has a radius of 1, which 0.618 that of the larger circle. If we add a third circle to fill the space left on the hypotenuse it will be smaller again by a factor of 0.618. And just because the values are based on the golden ratio, they all fit together in a Fibonacci-like way, two smaller units adding up to the next larger one.So with a few lines we’ve demonstrated that scary numbers (like and the golden ratio, neither of which can be expressed exactly in our number system) can be created with simple geometric techniques and the resulting shapes used to provide straightforward examples of complex concepts.

Oh yes, and the design is beautiful.

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Where mathematics becomes an object of desire.