One of the places I sometimes have trouble is remembering all the useful angles or construction techniques that are available. So I made a template of a few of the “good” ways to join parts together. This includes things like the old “beam-plate-plate-beam” spacings that allow you to cross-brace a conventional studded structure, but here I’ve included a bunch of studless options as well. The fact that the long double-bent liftarm allows right angles is obvious, but now I can see that I can pin it easily into a “beam-plate-plate-beam” structure as well, using for instance the 1st holes to either side of the 45° section. And the single-bent liftarm will also fit handily – in fact, if you look at the spacing you’ll see it allows a 3-4-5 right triangle construction. Also look at the little odd bit in red and yellow, up under the curve of the double-bent liftarm: the distance between the top-left and bottom-right holes is 1.5 units, very handy for certain gear combinations. Almost hidden in the back are two black beams: these sketch out the geometry for more 3-4-5 right triangles (in this case, double-sized).

These are certainly not all the interesting geometries that are possible, but a simple hybrid structure that shows some of the common geometries. How many more interesting ones can you find?

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Brian Davis

## 6 comments:

If you like pythagoras triangles, there are indeed a lot of other possibilities; and you don't always need the long side to be an integer.

For example, you could hook 2 3x5 liftarms by the ends and have yet another angle. The diagonal isn't an integer (2²+4²=20), but it doesn't matter. Think of gluing two square triangles one to another by their long side.

Another example: you can attach a 7-1 triangle to a 5-5 one (1²+7²=5²+5²) (I learned that one when checking if all studs from 4x4 round corners were indeed at exact same distance from the circle centre)

All these should be easy to do with studless beams. I just hope I'm clear enough; sorry I don't have any pictures

I tihnk I understand, although I hadn't thought of it before. Thank you! I wonder how many "coupled pairs" there are like your "7-1 5-5" pair there are to discover. The other thing is I'm not sure how to prevent these from becoming mechanisms: as an example, if I take beams of length 5, 5, 7, & 1 and connect them together, It get a four-sided figure, which can actually flex into many different shapes... unless I can pin one diagonal firmly to turn the mechanism into a structure.

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Brian Davis

Well to prevent the movement, you have to force two right angles: between the two 5's and between the 7 and 1. (Well, if they were beams, they'd need to be 6,8 and 2, actually). Think of it as of two L-shaped beams, one which would be 6x6 and the other 8-2; their ends would match.

By the way, the set 8453 (4WD) uses a lot of these 3x4x5 triangles; and in the alternate model they use the two-triangles-attached-by-long-side technique to have something (the steering wheel if I remember correctly) at a non-right angle. I suppose other studless sets do that also, but I don't have many of them yet.

I just found a nice illustration here of the use of similar right triangles to create angles.

Here the triangles have as sides: 1 plate high, 3 bricks width, so a 2-15 ratio. the diagonal isn't integer (2²+15²=229 which isn't a square) but that doesn't matter since the same triangle is used on the green and red parts.

Not studless though, so I'll stop wandering off that topic.

Something to keep in mind when talking about triangles is that the key to stability isn't necessarily a triangle - it's three rigid sides connected to each other.

So... pieces that don't have much flex but aren't necessarily straight can be used as one of the 'sides'. Likewise, two pieces connected via an axel joiner are typically 'rigid enough' to allow the two newly joined pieces to act as a single side.

This opens up a great deal of possibilities.

Another thing to realize is that there is often a little play in some pieces -- often enough to let you fudge things.

I suspect that with a little finagling you could convince your lego pieces to make a 5x5x7 triangle that was 'almost' a right angle.

All of this is static structure though -- hopefully Brian intends to head into the magic realm of linking static structure with axels/gears & pulleys to make things move. With only 3 motors and no way to split the electrical signal (you used to be able to power two motors off of a single output port with the RCX), mechanical solutions are going to be more prevalent.

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