Building Angles with Pythagorean Triple

For the next few posts, I’d like to talk about something a bit math-heavy. But all you really need is some simple Trigonometry. Based on Pythagorean theorem we know that a2 + b2 = c2, and the most well-know Pythagorean triple is 3-4-5. This triple gives you two non-90º angles, arctan(4/3) and arctan(3/4), or roughly 36.87º and 53.13º. This triple is in some sense the most useful one, because it’s the smallest. If we are to build with this triple (65, 72, 97), we’ll need a surface that’s 65 studs x 72 studs. For a surface this big, two-studs-connection is too little “clutch power”. Therefore large triples are useless for our purpose.

A way to reduce the size of the build is to scale down the triple by 2, and use 1.5-2-2.5. We can easily achieve this with jumper and open stud with hole in the middle. One thing to keep in mind is that brick length doesn’t equal to studs. A 5 stud-long side doesn’t mean a 5 stud-long plate. You need to measure from stud center to center, since that’s where the triangle is formed. So 3-4-5 triple is formed with 4-5-6 long plates.

Here’s an example of 3-4-5 triple in action. Half triple is on the left.

 

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You can also utilize this fact in SNOT.

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To achieve even smaller size, we’ll have to refer back to the post on brick dimensions. The structure below is a 1.2-1.6-2.0-stud triangle.

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Another triple that’s also useful is 5-12-13. If we use the half version 2.5-6, 6.5, its size can be fitted in a lot of mocs. It gives a smaller angle 22.62º, that might be more useful, since smaller angles are usually harder to create with bricks.

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