LEGO is great for building square shapes. There are a plentiful amount of slopes and angled wedge plates to get out of the grid. But everything still works off of a square grid. Therefore it is sometimes difficult to get away from that and make any angle you want. After my less than satisfactory review of the TIE fighter I felt it was appropriate to share some tips on how to make equilateral triangles. For the non-nerds, an equilateral triangle is perfect in that all sides are the same length and all the angles are the same pitch. Think of a hazard symbol:

LEGO asplode! Part number 892, one of the few equilateral triangle LEGO pieces |

Working with triangles in general means using the Pythagorean Theorem. This states that:

a² + b² = c²

This means that if line a and b are drawn in an 'L' shape, the angled line or, hypotenuse, must be length c. The most common set of lengths is 3, 4, and 5. Plugging them into the formula results in:

3² + 4² = 5²

9 + 16 = 25

Hey, it works!

This formula can be used for all sorts of numbers, not just round numbers. You could do something completely nerdy and useless such as:

π² + φ² = 3.533785... blah blah blah ²

Back to the application with LEGO bricks. Let's say you wanted to build a TIE fighter with true hexagonal wings. A hexagon is nothing more than 6 equilateral triangles nestled together. And an equilateral triangle is two mirror image right triangles with their backs to each other. We're going to tackle this from the perspective of studs out.

So your modest TIE fighter will have wings that are 10 studs across each side. Your triangles will therefore have sides of ten studs each. We need to use the formula in reverse because in this case we know the distance of the angled side but not the vertical side. So:

5² + x² = 10² or:

x² = 10² - 5²

x² = 100 - 25

x² = 75

x = 8.66

Your triangle needs to be 8 and 2/3 studs high. Good luck champ. First start by creating your triangle:

Actually, this is pretty strong on it's own or would be with another layer of plates. However you'll want to probably fill in the middle. That 8.66 studs distance between the peak and the middle of the bottom translates into 8.66 x 8mm = 69.3mm. Can we fill this with plates?

Nope. As projected it's off by 2/3 of a stud, about 5.25mm. Maybe we could come up with a SNOT solution.

This appears to be a lot closer. The distance between the SNOT studs on this column works out to 68.8mm. We need 69.3mm. Turns out it's pretty close. This stack of bricks might have some tiny gaps in it (like 0.5mm) but that can be acceptable when shape is the priority.

If we consider that 1.6mm is the lowest common denominator we can easily create, then how many units is 69.3mm? 69.3mm / 1.6mm = 43.3 units of 1.6mm each. 44 units would be 44 x 1.6mm = 70.4mm which would be about 1.1mm of difference. Personally I would settle for this solution shown above. Your job is to figure out how to connect 6 of these together. I only promised the triangles. Maybe something like this for your hub? Yes, they come in black.

What other solutions have you used for a perfect equilateral triangle?

There are other triples that work out to even studs and 60 degree angles. In particular 3-7-8 and 5-7-8 work. Using that you can construct inner braces though you don't get a vertical element out of it.

ReplyDeleteHere are two pictures demonstrating this: http://neilwebber.com/notes/?attachment_id=1122 and bottom view

The bottom view shows it better including the "pylons" I used to make sure the studs don't interfere -- these constructions are easier to make out of technic but can be done with system bricks too if you work at it.

Shameless self-promotion: I did a write-up on triangles and angles which people might (I hope) find useful: http://neilwebber.com/notes/2015/07/25/lego-technic-triangle-geometry/