Fractals – a glimpse into divine relationships
‘God is love’ (1 John 4:16) in which case 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 – have a go - build your own fractal …. it’s 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 though 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.
A4 sheet of paper
T
Random positioned
first dot in top right
corner of paper
T
So next dot placed in
top right corner of
chosen rectangle
T
T
Rectangle
chosen
randomly
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 corner of the whole sheet, your next dot is near the top-right corner of the chosen
rectangle.
Do that last paragraph 100+ 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 that
will slowly form a fractal dot-picture – often something beautiful, entirely determined by the
relationships between the rectangles and the sheet of paper.
T
2nd random
rectangle
choice
Dots eventually
cluster to make
the fractal
picture
Next dot
T
Beauty comes more spectacularly when a computer is used to render them 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 computer game writers 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 a feather:
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 forever, 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 to
both the resolution of a computer screen and to what we can understand – it’s like seeing the
infinite through a glass, darkly (1 Cor 13:12). Yet, what we see, we recognise as having some
meaning, some pattern. The glorious picture is a human-comprehensible incarnation of the vast and
inconceivable. Literally 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 showing where to put the next dot, where the next crumb of activity adds 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 (and web site) illustrate this feature – 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 in 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 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 Cor 12:27) and God in Christ (Rom 6:11) all resonate the self-similarity of the
persons of God and of us in Christ. And Jesus’ words I am the vine and you are the branches (John
15:5), speak of us becoming similar to him, contained in him, structurally part of him, being him in
the world, just as a 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.
Freewill is also important. The chaos game above, needs you, each time you place a dot to make a
random choice about which of the two rectangles you are going to use. Wherever you start from
(your first dot), eventually your dots will become part of the shape – the 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 whole thing does not work if you keep choosing the same rectangle. Then there is no beauty
and the picture fails. It is the chaotic activity we have in love with relations, with neighbours, with
ourselves and with God – the whole community around us - that creates the beauty. Life needs
some chaos, we need to keep all the balls in the air!
Experiencing the infinite, overwhelms 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: 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 my ecclesiastical (!) fractal
website: splat.co.nr/chaos
Adrian Low
Revd. Adrian Low is emeritus professor of computer education at Staffordshire University and
Assistant Curate of Abbots Bromley, Blithfield, Colton, Colwich and Great Haywood.
Currently about 1000 words.