Fi Phi Fo Fun
Since publishing some concepts about grid ratios and the golden mean, I have received various emails and questions from interested people who would like a deeper understanding of its relationship in 2D forms, and how it can be used to create efficient patterns.
First off all, imagine a square piece of paper…
One large square becomes four smaller squares
Tear that piece in half – now you have two equal sections.
Tear those in half again, and now you have four equal squares.
Those squares can be arranged to look like the original square.
Cut those pieces in half again…
Keep repeating.
The smallest piece you can cut can still be reassembled with all of the original pieces to form the original square. It is this property that very few shapes posses. Most of those shapes are symmetrical and regular (such as the super sounding isosceles triangle) and when split up, form smaller variations of themselves which tessellate back together in a fixed order.
The “ratio” in the case of the square is the size of larger piece divided by the smaller piece of paper – or 0.5 – meaning that it halves exponentially.
In this exact same fashion the Golden Ratio can be written as 1.6180339887…ad infinitum. This is a funky number, obscure and irrational, a bit like a baby PI… In fact many people even refer to it PHI.
Each square consists of 2 congruent golden rectangles and 2 squares
The Golden Section follows this exact same theory but rather than chopping the square into four smaller equal squares, you chop it into two different sized squares, and two identical rectangles. Each of these pieces can be chopped again into smaller similar shapes, and ultimately reassembled back into the original.
You can explore this beautiful principle mathematically, simply select any positive real number, as big or as obscure as you like. Now add one and take the square root of the result. Add one and again take the square root. Repeating this a number of times will always tend the answer towards Phi, which if repeated infinitely would equal the irrational golden ratio number.
It is this unique property of self division that makes it a useful tool for creating grids and layouts. For example, A1 paper is two A2 pieces, A3 is two A4 pieces and so on… this is possible thanks to the original A1 piece being designed around Phi. In this way, all paper sizes can be cut from one original piece, with no waste or off-cuts. If you rearrange all of those different “A” sizes of paper into a grid, you can form an always divisible, strangely elegant layout that not only looks good, but is also highly versatile and practical for scalability. Now if you apply the same principle with the actual Golden Rectangle (the two small oblongs in the figure above) it becomes even easier and fun due to the fact that every division creates another golden rectangle and a perfect square.
Famously, the golden ratio is ‘used’ by nature and can be seen in many everyday structures, from broccolli and fern to snail shells and sunflowers.
coutesy of James Michael Hill
The reality is that these are some of the most efficient structures in terms of the organisation of space and growth It is this self-organisation and scalability that leads to these elaborate growing patterns such as the spiral in the snail shell and the interspersed double helixes of the sunflower seeds. This beauty in growth is how a structure adds elements to itself and how those elements find their place. It is these fractal arrangements that often visually approximate to the golden ratio, so there is a general relationship, especially in our 2D and 3D representations where our mediums themselves (ink, paper, eyes) interpolate the small variations so that they look smooth and sharp, much more so than they actually are. For example, the Parthenon has columns that approximate to an aspect ratio of 1 : 1.6 which appears very close, but not identical to the Golden Rectangle, and although they look similar, they are in fact two different shapes, yet for years it seems to get cited as an early example of the use of PHI in Architecture, which really isn’t that true, but it is good enough to fool us – and that is all that matters.
Photo courtesy of TunaBot
Another more fascinating example is how lightning finds it’s way to the earth, which as it has no sense of direction, travels in stages down to the earth, where it favours the fastest, shortest and most direct way as possible towards its unknown target. We see intricate patterns and forks which branch out at irregular intervals, many of which fail to reach earth. At first appearance it seems simple, elegant, almost clever and optimised, like a Blind Bloodhound sniffing out a small sausage, taking a few sniffs left and right then honing in on the kill! If we zoom in on a lightning fork however, we can see many smaller forks and no defined edges to any of the shapes, nor a simple path that the lightning has travelled. These chaotic and wild permeations, flaring out wildly can be considered less successful iterations of the branching which do not quite fulfil their destinies and so fail early. From a distance however, the lightning seems very strong and confident in its path through the sky, rather than the delicate bumbling about in the dark that it actually is.
In the same way, using our eyes to interpret an image, our brains try to understand and comprehend what that image may mean, often by filling in areas and making baseless assumptions. It seems evolution favours certain patterns and numbers, shapes and forms and we favour them as they are things we can innately relate to.
There are many reasons that humans are fascinated by the golden section but it is mainly due to our use of the decimal system and base 10 counting ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ) which although makes perfect sense for our daily lives, PHI is more like PI, an irrational, never-ending number that we only ever can approximate towards that looks awkward when written as a number. Generally, for approximation using traditional maths : Phi can be found using 1 + ( -1/2+sqr(5)/2 ) .
It is this lack of similarity towards our other numbers that makes Phi an oddity worth researching, especially when finding it seems easy… It is everywhere like Helvetica!
So, if it helps you understand it, quantum physics is searching for the smallest possible building block particles that make up the universe… It is a safe bet to assume that the tiniest particle, that all of the other particles are made of, scales according to the ratio of the Golden section, and that is why, scaled up to the macro sized universe that is our world, it is visible everywhere is nature. Mathematicians have been wrestling with determining the relationships between quantum matter for years resulting in the formation of a branch of mathematics called E8 which examines the symmetry of geometry in 248 dimensions. There you can discover all manner of incredible patterns and decorative shapes!
If it helps you to feel more in tune with the universe, feel free to shift from using Base 10 decimal to counting in Base Phi for everyday arithmetic – just like Mother Nature!
Since originally writing this article, it has come to light that scientists have found the golden ratio at a quantum level in the frequency of a resonating quantum string. Find out more information over at Eurekalert.