I'm a Golden Meanie

As in I am a huge fan of the Golden Mean. Or the Golden Ratio. Two quantities are in the Golden Ration if their ratio is the same as the ratio of their sum to the larger of the two quantities. Wikipedia has a good illustration of this. Look at the diagram at the top of their page, on the right hand side. If you look just below that diagram you'll see a Golden Rectangle. I love those. The Golden Ratio or Mean is commonly expressed with the Greek letter φ (phi). φ is an irrational number, like pi and has a value of 1.61803398874989..., usually written as 1.618.

Actually, I love geometry. I sucked at maths at school, partly because my brain is not wired that way, partly because I couldn't see any relevance for it. I still glaze over when I hear sine, cos and tan (and don't try and explain them, please. Lalalalalalalala). I can do long multiplication and long division, I can keep accounts and manage money. I don't need any more. And I love geometry.

Not trigonometry, geometry. The way the Greeks did it. With a straightedge and pair of compasses. You can draw any geometric shape or relationship with a straightedge and compass. No measurement marks, no protactor or set square. Fabulous. And I love sacred geometry. It is kind of nuts, but it is fun.

When I was in University way back when, I came across a series of articles by Robert Stevick on the use of geometry in Insular manuscripts. Dr Stevick is a mathematician, so that was his focus. I already knew a little of sacred geometry (it is used in quite a slab of Renaissance paintings) and I recognised the forms he was writing about. I looked at the pages he detailed and realised that the geometry was telling a story. It was more than just about beautiful structure (although that was in there too), it was about narrative. Suddenly something mathematical was relevant and my mind lapped it up.

I got really excited, I mean really excited, and ended up researching and writing a paper on the subject in my third year. If you have access to Project MUSE you can actually read it here (it is weird seeing your own name on a paper on the internet). I have been hooked ever since.

Recently I had an idea for a set of drawings based on a Nautilus shell that is in the Macleay Museum at the University of Sydney. I decided to do a little research on Nautilus shells and found a lot of people saying it forms a Golden Spiral based on the Fibonacci Sequence. In other words, it is a 90degree rotation Golden Spiral. Which it isn't. You can see that by looking at it. Its curve is much more gradual and fluid, amongst other things.

Nautilus xray by Bert Meyers, found on Pintrest

There are lots of videos on YouTube and articles on the net about how to draw a Fibonacci Spiral and about the Golden Mean and Golden Rectangles, some better than others. Most show Nautilus shells.

The Fibonacci sequence is 1,1,2,3,5,8,13,21,34,... Basically you add the first two numbers to get the third, then the second and third to get the fourth and so on. To make a Fibonacci spiral you construct squares with sides equal to each number in the sequence, rotating though 90 degrees as you go. Draw an arc through each and you make the spiral. What a lot of sites don't tell you is that the spiral is also made up of nested Golden Rectangles, although it is not a true Golden Spiral as it starts with two identical squares (a true one uses only Golden Rectangles). Close, but no banana.  It's not the Golden Spiral I am looking for.

So I tried to work out what was going on in a Nautilus shell and found this article. A 180degree rotation Golden Spiral. It all made sense. Except that when I looked at the diagram for the spiral I couldn't work out some of the numbers.

Eventually, after taxing my spouse's maths wiz brain, and then playing with compasses and straightedge, I worked out the obvious. All the rectangles are Golden Rectangles. All the ratios are Golden. It all fell into place. And I am suddenly drawing Nautilus shells like there is no tomorrow. They are no longer the Macleay shell (no real shell is a perfect Golden Spiral - nature likes order but it doesn't believe in straightjackets), but they will serve my purpose well.

The original diagram by Gary Meisner

Spouse initially gave me complicated decimal numbers to divide or multiply each line by to give me the lengths of other lines. But I got to thinking about that. There had to be an easier way. After all, Pythagoras knew nothing of Silicon Heaven. Hence the playing with compasses and the dawning realisation of the Golden Rectangles.

Here's an amazing thing about phi. Phi is 1.618, as I said at the beginning. If you add one to 1.618 you get 2.618 (big deal), which is also phi squared. Phi squared is the same as phi+1. So that large rectangle for the first four rotations of the Nautilus spiral is 2phi:2phi+1 or 2phi:phi+phi squared. I'm going with the first version because I can see the Golden Rectangle expressed in it. Gives me goosebumps just looking at it. I know, I know, I am weird.

Here's my reworking of Meisner's diagram.

On the diagram I have marked more or less the order in which points were made. I've deliberately left in some of the construction lines. The length of lines is marked in green. Look at the number of times certain measurements repeat themselves and the way everything expands but continues in proportion. The curves are labelled as rotations. They aren't arcs, they are not that regular. I have not marked "rotation 1" because it isn't. There are clearly others before it, constructed in smaller and smaller Golden Rectangles (don't hassle me, I'm not putting them in. I can draw them well enough without). The original square, from which all this was drawn, with just a straight edge and compasses, is marked in red. Its length is designated 1 and represents one unit rather than a measurement of something particular. This could go on forever, spiralling in and out, but that's the size I want.

There's so much you can do with geometric construction. Draw squares from circles, make perpendicular lines, draw Golden Rectangles (you sick of those yet? I never am), make hexagons, triangles, pentagons, octagons, fractals, carpet pages. You name it. Anyone interested?