LEGO Techniques - On Angles

If you don't follow Caperberry's New Elementary, you should.  He posts about new elements that are being released.  Of special interest are the new molds or designs.  He recently highlighted the new A-frame plate that will be in the upcoming Ninjago sets.  This piece will allow for a rigid 45° angle without the hinge pivot that would otherwise get in the way.  We've got a smattering of rigid 30° connections, but very little that is adequate at 45°.  This will be a fun new piece.

The 1x2-1x2 plate and brick hinges have a distinct advantage though.  They can be made to swivel in any angle from 0 to 180°.  But for their multi-lingual advantage, they suffer one problem.  The pivot can often get in the way structurally or aesthetically.  And not just on one side but on both sides of the hinge.  On small builds, there's not much to do about it.

But there is a way to lock plates together and achieve angles.  A while ago I posted some brick built solutions for an equilateral triangle.  I'm going to open up that can of worms (Pandora's box?) and look at some of the other simple ways to create brick built angles.

The premise is this; touch two 1x8 plates at their corners.  Then take a 1x3 plate and attach at each set of points going out.  By the time you get to stud 7 and 8, there's not much difference in the created angle.  The result is the same as using a plate hinge, just without the pivot.  The problem with this construct is that the two studs closest to the origin are still a touch more than 2 studs wide.  So it cannot effectively be connected.  The better solution would be to use 1x8 technic plates.  These should be a staple of any MOCers collection anyway.

Since a system and technic 1x8 would be so close in angle, I chose not to feature both.  You can see though how to achieve some different angles with this technique.  But notice how quickly the angles drop at first before tapering to a slow crawl.  This is a good visual example of a hyperbola.  If you were setting out to create a circle, the only useful one would be the last one.  The rest of them don't evenly divide into 360°.  If you've seen my slope and wedge charts, you can see how these might be useful in conjunction with those pieces.

Of course there are many more uses to these than making large circles.  I imagine wings or gangways with jogs in them.  There are a plethora of other options as well.  You could run this exercise with a 1x4 plate instead of a 1x3 as the spacer.  Or you could lock the 1x3 plate on the number 1 studs and use a 1x4 for the angler.  Or a 1x6.  Or do all that further out on a 1x12.  I suppose someone could create a whole matrix of possibilities, or possibly a little GUI script that will calculate it all out for you.

Let me know when you're done with that, Ace.  I'm headed to bed.

LEGO Tips, Tricks, and Techniques - Equilateral Triangles

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?