The Idea

Researching recursion brought up a lot of references from a range of sources. I worked my way through them. Intuitively there seemed to be connections centring on the idea of self-similarity, a function that expressed itself by referring to itself, like a thing that made itself, like way experiential learning and art make themselves. There was a lot of maths, code, logic, and some geometry. I wanted to start to explore recursion by making something, so geometry seemed a good place to start. I decided to start by making a thing called the Golden Ratio as a thing that makes itself. As the quote below from Wiki indicates, the study of the Golden Ratio has spanned millenia.

‘Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. … Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.’

Mario Livio — The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number.

I cannot do this material service so the Wiki entry is here

But first a bit of maths. A ratio is a thing that exists only as one value as expressed in relation to another value. It is one thing divided by another. 1 divided by 2 is a ratio. It is 1/2, a half. The golden ratio is a mathematical and geometric expression of a particular ratio. As a ratio, it is described as (a+b)/a=a/b where a and b are numbers.

As a geometric form it is expressed thus…

As a decimal number is it 1.618033988749…. so it is an irrational number, it goes on to infinity. The ratio is the same for 1 and 0.618… as it is for 1.618… and 1.

But as many sources show, the golden ratio is also a thing that exists in nature.

It is also a thing expressed as a mathematical symbol, Phi, a letter from the Greek alphabet.

It is also the basis for a mathematical phenomenon called the Golden Spiral which expresses a thing called the Fibonacci Sequence found in art and nature and mathematics. It can be expressed in many ways.

The Golden ratio is recursive and as such seems to reiterate the feeling of researching recursion. A feeling of finding some deeper connection between things, between experiential learning, art-making, crow intelligence, coding, philosophy, on and on. It feels like all the different references may emerge from a single or underpinning referent. This interests me. But this was all a bit abstract. It also evoked a feeling of caution, like I was maybe seeing things that were not there, things not connected. I wanted a more concrete experience. Something embodied not just imagined.

The Intention and the Action – Making the Golden Ratio

You can make a Golden Ratio with nothing more than a pencil, compass and ruler. I decided to make a golden ratio for myself. My intention was to have a concrete experience of making it rather than just having the abstract experience of thinking about it. I was already exploring the recursive behaviour of crows through poetry, and this changed and deepened my connection to them. So I set out to draw the Golden Ratio with a pencil a ruler and a compass. As shown above, there are a great number of discussions of, investigations of, theories of, and expressions of, the golden ratio by very learned people spanning millennia. This does not really interest me. Google Golden Ratio and you will get many varied references thus…

My idea was that the golden ratio has something to do with recursion, which has something to do with making art and experiential learning and this interests me. All the serious ideas generated by other people can help me pursue this interest in ways my brain could never imagine, like Fine Art made by proper artists can help my art-making. But most helpful is the art I make. My intention was to simply make it and pay attention to what I make and what happens when I make it. As always. Don’t Google it do it yourself, the punk ethic. My preparation was to get hold of a big compass and a joiners pencil. I use wallpaper lining paper. It is a heavy gauge 300gm/metre weight paper like fine art paper, but a nice off-white and is as cheap as chips. It remains curved off the roll so needs to be fixed to a board with masking tape. This also fits with a punk ethic. So I made it myself. I went into action.

What Happened?

I found some instructions in a book I got from a charity shop about making the golden ratio¹. I bought the big plastic compass from Amazon here. But this proved a bit bendy. My arcs drawn were a bit wobbly and uneven. So I did some practice on another bit of paper and found a way to make a nice steady arc. I got my ruler and joiners pencil. I photographed my making of it…

Or click here

It proved bizarrely nerve wracking. Hippasus of Metapontum was reputedly drowned at sea by the Pythagorians for discovering irrational numbers here, and according to wiki ‘Irrationality, by infinite reciprocal subtraction, can be easily seen in the golden ratio…’ I hoped Pythagoras and the gods were not watching.

I measured the two sections of the golden ratio I made and did the maths to see how accurate I was. I was not drowned at sea. Nothing happened. I drew a line with two segments of differing lengths. It seemed mundane. What was I expecting? I let it go and let it incubate, or percolate, whichever you prefer. I went to work. I made other art. The board with the line on sat on my desk for ages. I lost my measurements done at the time of making. So I returned to the drawing and the following measurements were taken.

On measuring this, the two segments were 22.3cm and 13.7cm giving 1.62773723. The longer segment divided by the shorter segment should be 1.61803398. So this can be applied to the sides of the triangle I drew, so (22.3 x 13.7) / 22.3 = 1.61434977, this applies to the bottom and side segments of the triangle. This gives an error of about 0.009, mostly due to my hamfistedness and a wobbly compass. I was pleased with this outcome. It is interesting that the Fibonacci sequence moves towards 1.618… with each iteration of the recursive sequence. My second iteration was closer. Maybe it was just noise error. It seemed mundane and magic all at once, which is what magic and reality really are. Perpetually dichotomous. Certainty is certain in its uncertainty.

I did lots more reading about recursion. Eventually, I reflected on the experience for this page. This is my reflection.

Reflection

My immediate reflection after standing away from the image for a while, was that the further you stand from the mirror, the smaller becomes your image, but the setting gets bigger. Whilst there are no unequivocal written accounts, it is suggested that Plato knew about a mathematical phenomenon that went on to be named ‘The Golden Section’ by Martin Ohm in 1826. But the (a+b)/a=a/b formula was written in Euclid’s The Elements 2300 years ago. Maths has moved on a long way since then. My understanding is that maths was then, more of a geometric thing. It was understood as objects, albeit in many senses imaginary objects, it was also in a real sense about things you can make and draw.

Making this object put me in some ways back in that time. The line drawn and segmented could be any length, and the segments, with simple tools, could make an object that made itself. All the actions root back to and emerge out of that line. Then that line produces a form which relates only to itself. I could see how Plato could be Platonic, how he could contemplate pure form which could transcend or sub-ordinate matter. To make the golden ratio, on reflection, was at once transcendent and mundane. The act was mundane, but the emergence of a thing that made itself, utterly without any other referent in this world now seems quite remarkable. It reinforces my belief in nominalism. That all that is written about the golden section are just signs pointing to some thing that exists outside of our experience, and this thing is mundane and transcendent. I can see why Pythagoras and the Pythagoreans felt like they were in touch with some deeper reality. And all it is is a pencil line on a piece of paper.

This reflection also puts me back again in a place I have been before. I have an image of a path, in partial darkness, by some garages near my home. I was doing a series of algorithm walks, llr, rlr, rrrl, etc to make my daily wellness walk more interesting. At some point, I realised that each walk was an act of creation, unique and unlike any other walk I had done. Because the path was varied, but because the time varied, I did just this walk, just today, it could never have been done tomorrow or yesterday. I was strongly struck by the realisation that creativity is mundane. We make a sandwich, we drive to work, we make stuff happen. But through connecting to the innate creativity in the seemingly mundane act of making a sandwich, watching endless TV cookery programmes by celebrity chefs showing me how to make a sandwich become mundane.

The act of making the mundane into art, finding some deeper structure in drawing a line and forming a ratio, has risks. If it can render everyday life mundane, it risks becoming escapism, the evasion of everyday life. For some reason, Irving Welsh sprang to mind writing about the risk and trap of escape. The film version of Welshe’s Trainspotting starts with a great, but disturbing diatribe on the seemingly simple escape from the complexity of life…

‘Make a choice…” I think. I did not, like Renton and Iggy, choose some external internal stimulant, although in the past it was a choice I made, it made me unwell, so I made art, eventually after periods of unwellness that showed TV shows and stimulants to be singularly unstimulating.

So like the Glorious IG, I moved on and chose not to “Choose Life; Choose a job; Choose a career; Choose a family…” or at least I chose to be more choosy over what I chose. The lines at the start of the film are highly ambiguous. Iggy Pop sang about a lust for life. Maybe that was what drove Renton to become a junkie as a way out of the mundanity of his diatribe. Renton chose junk, but it kind of becomes a cypher for an escape from choosing ‘…a f***king big television,’ and ’stuffing f***king junk food into your mouth’ as acts of self-destruction which can be ameliorated by recognising the mundanity of self-creation, making art with a pencil or a walk on a morning past a neighbours garage as opposed to shooting up. The line as a walk or as a pencil mark is transformational. Making them is the key. It is not the same as reading about somebody else’s walk or seeing somebody else’s line. But the purveyors of f***king big televisions and f***ing junk food don’t want us to know that. They want us to buy, to buy into a drug of choice, TV, food, holidays, hair products, an online persona, booze, weed, speed, junk, to take Soma, the drug of choice in Huxley’s Brave New World. In an article from Medium by Yash Deshmukh the author explores Soma as a metaphor for current experience, not a future society. The content of Renton’s rant is the drug. Consumerism as Soma. Work to earn money, to buy things to help you escape the mundanity of working to buy things to help you escape… The endless loop. Recursion as a closed loop is a risk. But the recursion of making the golden mean feels different.

I am still working on how recursion as creativity, exposing deeper forms, and recursion as an endless infinite loop reinforcing the same shallow forms of mundane life, differ. I am still working on what it is about making things as art that makes the content of Renton’s rant mundane, but the experience of drawing a pencil line on paper becomes rich and revelatory. Am I mad, or is the world mad? I will make more stuff as art as research. The journey goes on. The path is revealed by the walking of it.

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1. The Secret Code – The mysterious formula that rules art, nature, and science. By Priya Hemenway. 2008  ↩