Searching For Water

One abiding pre-occupation of mine has been to capture the surface of water using pattern. The movement I observe every time I gaze into a stream or at the sea suggests an organised formation of fantastic complexity, but it exists so fleetingly I am unable to capture it.
I began this quest with the idea of a perfectly flat millpond and dropped a stone into it. I observed that the ripples move outward in regular concentric circles. Then, dropping two stones in simultaneously I see two sets of concentric ripples and an interference pattern where the ripples meet. It seems but a few steps further to suggest that however complicated the forces on the ripples they will all have been generated mathematically at some time past.
In my search for the 'water pattern' I chose as my model a scene common to nightime revellers in Portsmouth. Close to the mouth of the harbour there is a pub called the Still and West (it's very name seemed appropriate) and standing outside on the sea wall we can look across the harbour at the lights on the promenade at Gosport. Because they are only a few hundred metres away, each light casts a broken line of light almost to our very feet. At night there seem to be only two colours on the main body of water which suggests a duo-tone pattern with the lights from the prom being picked out in fragments.
But where to find the pattern that would do the scene justice. The answer seemed to lie in a quite extraordinary direction.
For some months previously I had been working on Magic Squares where what appears at first glance to be a randomised system gives rise to a mathematical beauty. In particular I was searching for a geometrical pattern that could define the layout of all magic squares.In this I have failed but along the way have been exposed to some intriguing theories.
To start with I had to chose a single Magic Square on which to work. I chose the 5x5 square partly because there are hundreds of variations, but also because this is of a size to generate enough interest without becoming overly complicated. I tried many of the variations and eventually settled on the one at Figure 1. At this point I have to assume that the reader is aware of the properties of a Magic Square.
To create a pattern I constructed another grid of the same size and numbered the cells in a logical sequence - Figure 2. These two will be used to determine the formation of the pattern which is drawn onto a third grid of similar size - Figure 3. I will illustrate two methods of doing this.
In the first, the centres of cells are linked by straight lines such that the connected cells contain the same numeral in either Figure 1 or Figure 2. Where a number occupies the same cell in both Figure 1 and Figure 2 no line is drawn. The resulting pattern is shown at Figure 3.
The second method looks at the set of five numerals in the first horizontal line of Figure 1, then, noting which cells contain these numbers in Figure 2 connect those cells in a continuous loop as in Figure 4. I repeated this for the other rows. (You will note that four of these lines are shown broken. This is because they, rather annoyingly, terminate at points which are surrounded by three areas, which would necessitate the use of three colours which I didn't want to do. So I applied a little artistic licence and omitted them). This is the design that is carried forward.
The resulting pattern is then placed in a larger grid with alternate drawings reversed. Colour is then applied to create Figure 5. I felt I was getting close, but the image lacked the necessary perspective. So it was back to the drawing board for Figure 6. And that's about as far as I've got so far.


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