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Taking a journey into the infinite

02 January 2015

Fractals can express our relationship with the divine, writes Adrian Low

Dot-to-dot: a frac­tal is more than the sum of its (infinite) parts

Dot-to-dot: a frac­tal is more than the sum of its (infinite) parts

IF "GOD is love" (1 John 4.16), then the place to find God is in relationships - within the Trinity, between God and all living things, and within us - or, as David Cunnigham puts it in These Three are One (Wiley-Blackwell, 1998), "God is relation . . . without remainder."

It really is pointless trying to see God as some entity beyond Jesus - a bearded pensioner; a holy icon; a Bruce Almighty. No picture is adequate, no geographic location, no time, no texture or material is God: just relationships. God is relationship.

So are fractals. They are an infinity of multi-dimensional beauty each created by the simplest of mathematical functions, or relationships. But ignore the long words and the maths: fractals are built from simple relationships between shapes - in the example below, between two rectangles and a sheet of A4. And, because of that, they provide a remarkable parallel insight into God's love. In both fractals and love, incredible beauty stems from the activity in simple relationships - "the clouds disperse, the shadows fly, the invisible appears in sight and God is seen by mortal eye" (Charles Wesley).

So you can have a go, and build your own fractal: it is called the Chaos Game.

TAKE a blank sheet of paper and, very lightly, draw two large rectangles on it - any angle, anywhere you like, overlapping or not. Each rectangle should be smaller than the sheet itself, although their corners can disappear off the edge of the sheet. For each rectangle, label one of the sides as the top. Then, with a pen, put a first small dot anywhere on the sheet.

RANDOMLY choose either one of your drawn rectangles. Now the difficult bit: note where that dot is with respect to the whole sheet of paper, and put another dot inside your chosen rectangle, positioning it as if your chosen rectangle were the whole sheet. So, for example, if your dot was near the top right-hand corner of the whole sheet, your next dot is near the top right-hand corner of the chosen rectangle.

Repeat that last paragraph 100plus times, each time randomly choosing one of the two rectangles and using the latest dot. If you persist, your dots will be attracted into some clusters on the sheet which will slowly form a fractal dotpicture - often something beautiful, entirely determined by the relationships between the rectangles and the sheet of paper.

BEAUTY comes more spectacularly when a computer is used to render the images with millions of dots, using some convention to colour each pixel on the screen. Each new dot adds more to the beauty. Each activity, rooted in our loving relationships, builds something more beautiful.

Fractals are only 2D on paper, but their simplicity can provide writers of computer game with a tool to create complex 3D landscapes evolving over time. Depending on the relative position, number, and sizes of the rectangles, the dot-pictures can look like leaves, trees, flowers, corals, ferns, mountains, coastlines, honeycombs, dragons, lichen, flames, clouds, and hundreds more images of nature. The two rectangles (above) generate something like birds flying in formation.

ZOOM in on a fractal, and you discover its complexity is infinite. So, with small enough dots on a large enough piece of paper, you can go on adding dots for ever, colouring and re-colouring. It would take an eternity to explore fully any fractal; we don't have enough light-sensitive cells in our eyes, nor enough neurons in our brains to hold the detail. Instead, we look at the beautiful pictures - limited by both the resolution of a computer screen and to our own understanding, it's like seeing the infinite through a glass, darkly (1 Corinthians 13.12). Yet, what we see, we recognise as having some meaning, some pattern. The glorious picture is a humancomprehensible incarnation of the vast and inconceivable. The logic takes on a shape - becomes flesh - and we behold its glory (John 1.14), all from the relationships between the two rectangles and a sheet of A4. Their position and size show where to put the next dot, where the next crumb of activity can add to the beauty of the picture.

Beauty comes from each activity in loving relationships with God and with others. Do nothing, and you see nothing. Only by engaging with love/doing-the-dots can you really begin to hold the breadth, depth, and height of the infinite.

Also, fractals are beautiful because they are made from ever smaller and similar versions of themselves. The pictures (above) illustrate this feature, which is called self-similarity. Nature often works like that. A fern leaf seems to contain small versions of itself; fragments of coastline look like a whole coastline; a cauliflower is a bunch of cauliflowers. The smaller versions are contained within the larger; yet, because a fractal is infinite, each of the parts is just as wonderful and complex as the whole.

Such self-similarity is also a feature of God and the Kingdom: "I am in the Father, and the Father is in me" (John14.11); "Live in me and I'll live in you" (John 15.4); "This is my body" (Luke 22.19); "You are the body of Christ" (1 Corinthians 12.27); and "God in Christ" (Romans 6.11) all reflect the selfsimilarity of the persons of God and of us in Christ. And Jesus's words, "I am the vine and you are the branches" (John 15.5), speak of our becoming similar to him, contained in him, structurally part of him; of our being him in the world, just as one branch of a vine looks like a whole vine. Self-similarity is key, just as relationships are key. A fractal is self-similar, defined by its relationships.

Free will is also important. The chaos game needs you, each time you place a dot, to make a random choice about which of the two rectangles you are going to use. No matter where you start (your first dot), your dots will eventually become part of the shape - mathematicians use the word "attractor", because the journey of the dots always finds its way there.

Doing love - being active in godly relationships - brings you into the Kingdom. Anyone who loves is a child of God (1 John 4.7).

THE exercise does not work if you keep choosing the same rectangle. Then there is no beauty, and the picture fails. It is the chaotic activity that we have in love, with relations, with neighbours, with ourselves, and with God - the whole community around us - which creates the beauty. Life needs some chaos; we need to keep all the balls in the air.

Experiencing the infinite may overwhelm us, but we definitely need that overwhelming from time to time. Fractals do that. It is as if I am somewhere holy when I zoom into a new fractal: in a place where simple relationships underpin the glorious. Grasp active relationships of love, and suddenly you have in your heart the immensity of God. Try a journey into the infinite for yourself, on the ecclesiastical fractal website: splat.co.nr/chaos

The Revd Adrian Low is Emeritus Professor of Computer Education at Staffordshire University, and Assistant Curate of Abbots Bromley, Blithfield, Colton, Colwich and Great Haywood in the diocese of Lichfield.

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