This is a pretty small project compared to most I publish, but I thought it was interesting enough warrant a quick post! I think many people are familiar, at least in passing, with the Golden Ratio, but for those that are not here is a very quick introduction.
Mathematically the Golden Ratio, is represented by the Greek letter φ (Phi). It is, as you will see, a self-describing ratio between two numbers such that the one number is to the second as that second is to the total of the two. I think this picture really is the easiest way to grasp the concept:
Phi is simply the ratio between A and B (and also A + B to A). Rather than take you through the solving for this, I will simply reveal that the ratio is the following irrational number: 
So what? Well, it has been noted that this ratio, also in antiquity called the Divine Proportion, describes many things we see in nature and is said to strongly play a role in that which we find beautiful. It describes naturally occurring spirals, is embodied in a lot of sacred geometry, is found in the human face & body, and observed in the composition of some of the world’s most adored paintings and architecture.
I use the Golden Ratio in a lot of my own art and it occurred to me that it might be fun to try and make a tool to be able to physically measure it when I’m in the shop. My concept was to have a pair of gears that are locked to the same axle sized to a close ratio of phi. Remember, phi is an irrational number so it by definition cannot be described precisely by the ratio of two integers or therefore two gears. Two arms would be extended in opposite directions driven in proportion to those two gears and together they would reflect the golden ratio at various lengths; here is that design:
An interesting note about selecting the number of teeth on each gear: I had intended to just use excel and some brute force to find the best number of teeth, taking into account the physical size constraints, but I stumbled across a much better way and it has to do with the close relationship between the Golden Ratio and the Fibonacci Sequence. That sequence, probably the most famous of all sequences, is simply where each number is the sum of the prior two: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc… You can begin to intuit the deep connection between it and the Golden Ratio by looking at how they both describe the same Golden Spiral:
Simply put, each number in the Fibonacci sequence relative to the prior number in the sequence approaches Phi as the sequence approaches infinity. So in my case, limited by the size of the gears I wanted to have, I selected the two sequential numbers in the sequence: 34 and 55 teeth for the inner and outer gear. This means my ruler is approximating Phi (1.618033988749…) as 55/34 or 1.617647058823…, which is pretty darn adequate for my purposes! So as the gear turns one full rotation, one arm is extended 55 units in one direction and the other, in the opposite direction, 34 units, so at all times the ratio between the two is maintained as Phi and I can slide the ruler in or out to various lengths as a measuring tool to mark Golden Ratios on whatever I am working on.
Here are some pictures, I noticed that in the one where it is fully extended you can see on the tape measure one side is 13 inches and the other side is 8 inches, sequential numbers in the Fibonacci sequence, so I guess it works!
