Ok, while I’m messing with the Golden Ratio and mathematics, I thought I’d also try to figure out how to make a simple cross beam proportional divider. It’s been a while since High School Geometry, so I had to go back and reprove the relationships between proportional isosceles triangles to myself. I also couldn’t find this specific formula for making a divider anywhere on the net in a reasonable amount of time, so I just re-derived it.
(I believe this is all correct, but don’t use this for any mission-critical or potentially injurious tasks until you’ve tried it out! grin)
What we’re talking about is essentially two arms of the same length, joined somewhere between their ends so that they can pivot around that point, and the distance between the arms at one end is some proportion of the distance between the arms at the other end. E.g. given a ratio (proportion) of 1:2, if I make the arms 2 inches apart at one end, the arms are 1 inch apart at the other end. The result is a divider having 4 arms in the shape of an “X”
(For those who are interested, this divider creates two similar isosceles triangles, and since the legs of the triangles are fixed, the law of cosines says that the bases of the triangles must be proportional however far apart they are set.)
This formula is generalized so that you can make a divider of any length and any proportion, including a cross-beam Golden Ratio divider. Just decide how long you want the overall divider to be, decide what proportion you want, and plug those numbers into the formula to determine how far from one end you need to put the pivot on the arms of the divider.
BTW, if you use ratios, greater than 1, you are actually calculating the distance Y, not X…
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