Ultimately, my goal with all this mathematics was to figure out how to design an accurate 3-arm proportional divider like the one posted by David. Please visit his blog post for more information and links.
The first step was to understand the Golden Ratio (post 1), the second step was to figure out the geometry of a simple 4-arm divider (post 2 & 3), and the final step is to apply this theory to the 3-arm divider I want to make:
I just rearranged the two isosceles triangles from the 4-arm divider to produce a 3-arm divider. Now “Z” is not directly measurable, but it can be calculated. Once you select the X of your choice, and then calculate Z, you can use the second equation to calculate Y.
The cross piece (dashed line) is the same length as the line segment W, so you’ll need to calculate that length as well (W = X – Y). Because A=B=C=W, once you know W, you know how far down on the legs to place the cross piece.
Remember that all these measurements are from fastener to tip (or from fastener to fastener for the cross piece). You’ll need to add a little bit of length to accommodate the fasteners…
Now I have to do something with all this. I’ll be putting together models for both the 4-arm and 3-arm dividers in Sketchup, and then I’ll try to actually construct them! I’ll post as I can…
-- -- "there are many good reasons to use old hand tools, but moral superiority is NOT one of them..."