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 the original pieces to form the original square.

The ratio in this case is the size of larger piece divided by the smaller piece of paper – or 0.5 – meaning it halves exponentially.

Each square consists of 2 congruent golden rectangles and 2 squares

The Golden Section is this exact same theory but instead of chopping the square into four equal square pieces, 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 orginal.

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 will 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… that could only be possible if the original A1 piece was based around Phi. This way all paper sizes can be cut from one original piece, with no waste or offcuts. 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 the most efficient structures in terms of space and it is this self-organisation that often leads to elaborate patterns. The 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 approximate to  the golden mean so there is some 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.

Photo courtesy of TunaBot

An example is lightning, which as it travels in the fastest, shortest and most direct way as possible, it is not surprising that we see intricate patterns and forks which suggest the use of Phi for pathfinding and arrangement. It is simple, elegant and optimised for finding similar ratios between forms. If we zoom in on a lightning fork however we can see many smaller forks and no defined edges to any of the shapes, these chaotic and wild permeations can be considered less successful iterations. It seems evolution favours certain numbers.

The reason humans are fascinated by the golden section hinges mainly on 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, like PI, we are stuck with an irrational, never-ending number that we only ever can approximate towards. We cannot create a simple formula for using these numbers in our equations without approximation, for example,using traditional maths : Phi = 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 is as easy as plotting y=(x^2)-1 and noticing that the axis are both  intersected at Phi and -Phi… 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, is shaped 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.

And 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!

More like this golden section|howto|layout|phi.