Vex right triangle diagram

Given a bit of spare time, I thought I would make a diagram of the possible right angles that can be made with the vex structural pieces and their .5" array of holes.

This is similar to a very old thread (Possible lengths for vex right triangles) that the forum suggested when I opened this thread. The flaw in that earlier post is that they overestimate what is “close enough” to fit. #8 screws are just over .01" smaller than the vex holes, so there are many numbers listed in that table that would require some modification of parts to work.

At any rate, I am more visual that mathematical, so approached the problem in diagram form, which results in this:

vex-angles
In reading the diagram, you are counting the spaces between holes (or counting holes, but the first is 0). X and Y are on their axes, H holes follow the arcs. The dots in the .5" array are sized to match the slop between #8 screws and vex holes. If an H arc passes through the center of a dot, you have a perfect fit. If it passes through the dot at all, you have a close fit. If it passes very near the dot, you can force it to fit with some modification.

If you think you might be seeing some patterns, you are right.:

right triangle groups

vex-angles-grouped
I bet there is some math that would describe why these patterns exist. Please enlighten us if you gots the math.

Here are pdf’s if you want to print to scale:
vex-angles.pdf (105.5 KB) vex-angles-grouped.pdf (107.5 KB)

(edit, first post had hole count on graphics wrong.)

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This is a super cool video on Pythagorean triples by 3Blue1Brown. I highly recommend you check out his channel if you haven’t seen it btw.

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Wow this is so cool it will really help we with building.