More Squares and Circles

I recommend to first read Squares and Circles, before reading this Thought. As I have already mentioned in previous Thoughts, squaring the circle has a number of peculiar features. A few of those features really stand out. First the impossibility to construct squaring the circle with just using compasses and a straight edge as tools. Second the absence of it in nature. I recommend to first read Squares and Circles, before reading this Thought. As I have already mentioned in previous Thoughts, squaring the circle has a number of peculiar features. A few of those features really stand out. First the impossibility to construct squaring the circle with just using compasses and a straight edge as tools. Second the absence of it in nature.

While other known geometries like the Phi ratio and the Fibonacci sequence can be found everywhere around us, squaring the circle is totally absence. It is perhaps for this reason that many people find squaring the circle awkward to look at while these same people find the Golden Mean and the Phi ratio beautiful and often even peace giving. Soothing for the Soul.

What is it with squaring the circle? Is it the core essence of our being? Is it the glue that binds matter and Spirit together? Is it the ultimate key to the Theory of Everything? Since squaring the circle has captivated my mind, I have tried to find it in all my surroundings, but have not been successful so far. Can it be hidden and I just don’t recognise it? I have tried to come up with different ways of ‘encoding’ squaring the circle without squaring the circle being recognisable in the end pattern, in the hope I will recognise it when I come across it somewhere out there. Just the other day I came up with the following relatively simple encoding.

I first added a circle with its centre on the perimeter of the ‘big’ Circle and its own perimeter going through the intersection of the Square and the Circle. I did this at both sides. This sounds complicated but the diagram above on the left shows how simple it all is. I added two further circles now going through the other intersection of the Square and the Circle. A final circle that exactly fits in the Square concludes the ‘encoding’. We now have three rings that fully lock the Square and the Circle in place. See diagram below on the left. The Square and Circle cannot escape and they only fit together in one way, but … would you have recognised ‘squaring the circle’ in the diagram below on the right? For myself, likely not!

After doing all the drawings I sat back and stared at the paper in front of me for a while. The last diagram, the black rings, looked familiar to me, but I could not exactly pinpoint where I had seen it before. Later that day – what a 'coincidence' – I browsed through the beautiful book of Dirk Möller, Peter Möller and Eckhard Weber, ‘Der Kornkreis-Code’. Such a treasure of a book, but unfortunately only available in German. I was looking for a persons name and started in the back of the book. I was in for a shock. On page 392 I saw a diagram (number 313) that looked very similar to the diagram I had constructed earlier that day. The diagram in the book represented the crop circle that had come down near South Harting on June 23, 2001. I immediately tried to trace down photos of this formation, but I could find only a very few.

On the left you see the original photo of Lucy Pringle, on the right the turned, stretched and skewed version. Although my diagram and the crop circle differ in size, in ‘wideness’, they have their general idea in common. It was only after I studied the crop circle thoroughly, that I recognised its beauty and intriguing geometry. Let me share my findings with you. My first step was realising that, although my diagram was beautiful and powerful, it contained more information than necessary to 'encode' squaring the circle. The diagram below on the left shows how little is needed for the encoding. Only three circles. The middle circle defines the Circle and the side circles define the size of the Square. If we now superimpose these three circles on the crop circle we see a 100% perfect match. Amazing! Squaring the circle in its most simple way encoded in the crop circle and neither I, nor anybody else ever noticed it.

I was still recovering from my surprise and amazement, when I realised that the crop circle was hiding another extraordinary feature. The diagram below on the left shows how a new square fits in the 'big' Circle. Two new circles can be added that intersect the 'big' Circle there were the new square touches this Circle. And now follows the most amazing feature. The newly added square can in its turn be used for a new 'Squaring the Circle'. The diagram below on the right shows this. The 'rings' we have constructed in the previous steps are of the identical size as the rings in the crop circle. Truly amazing.

The outer perimeters of the rings in the crop circle are defined by 'Squaring the Circle' and the inner perimeters are also defined by Squaring the Circle, whereby the second Squaring the Circle exactly fits in the first Squaring the Circle. It is a fractal. A Squaring the Circle Fractal.

The method I used to encode Squaring the Circle is perhaps beautiful, but the method used by the 2001 South Harting crop circle is brilliant. Squaring the Circle to define the outer perimeters and a fractalised Squaring the Circle to define the inner perimeters. Again we are presented with the ultimate merging of Matter and Spirit in a very elegant way. The ultimate Oneness! Only a phenomenon as the crop circles are, can come up with 'solutions' like this. And be honest, did you ever suspect so much intricate geometry in such a 'simple' crop circle? After this discovery, I have to ask once again, ‘What on Earth is Going On?’ © Bert Janssen, 2008.

Join me on my and lets find out together what is going on.