Does anyone - consciously or unconsciously - make use of Pythagorean quadruples in his/her Technic builds? Or does anybody know examples that make use of these kind of quadruples?

 

As we all know triangular connections help making stiff constructions. And as Technic is particularly well suited for building right angle connections on a one stud grid, it is most convenient to find right triangles that only have whole-integer side lengths. Such triangles in fact make Pythagorean triples. Triples that we often use: (3, 4, 5) and (5, 12, 13). You could say that with Pythagorean triples we can make connections that span two dimensions: The (3, 4, 5)-triangle spans 3 studs in one dimension, 4 studs in the other dimension and together that makes a 5 stud span. The (3, 4, 5)-triple is even supported by Technic's angled liftarms.

 

However, with LEGO we build in 3 dimensions, right? So we may also want to make triangular connections that span all three dimensions. And if we want to do so on a one stud grid we can make use of Pythagorean quadruples. For instance the well-known triples (3, 4, 5) and (5, 12, 13) can be combined in a quadruple (3, 4, 12, 13) where the 5 in the second triple is obtained using the first triple. This would look like this:

Other useful quadruples are: (1, 2, 2, 3), (2, 3, 6, 7), (1, 4, 8, 9), (4, 4, 7, 9), (2, 6, 9, 11), (6, 6, 7, 11) and (3, 4, 12, 13). There are more, but they exceed the max liftarm length. Now what I would like to know is whether you know about LEGO Technic examples that make use of these kind of quadruples. Do you?

Edited December 8, 20168 yr by Didumos69

WOW! Very useful! Thank you for posting!

 I got to know about  Pythagorean triples this year, didn't even know about quadruples... 

Very useful, many thanks for posting it!!

I no longer need my maths teacher,with you here

Edited December 8, 20168 yr by LXF

I guess the reason why it's not used commonly is that you need more complex hinges (two pins instead of one) which are more flexible, so you loose with the joint what you gain with the beam. Or dunno. The pinhole-pin part may overcome this issue better than older alternatives. Have you tested the chassis stiffness with and without the beam?

It's a cool theory, but I doubt whether it's of much use.

 

I mean, you can easily span a diagonal of a simple triangle with liftarms where the holes are in line with the rest of the structure. A 3-4-5 triangle is easily secured. But the beam in the picture has to be connected with two hinges on each side (the black pin, and the black pin-with-hole), which decrease the stiffness and increase the amount of play in the structure. Also, the holes of the red beam can't be connected to anything. Thirdly, as you can see, the red beam is offset by 1 stud due to the black pins-with-holes. So the connection isn't actually in-line with its endpoints. So when compressed, the beam simply bends. So in this particular case, you might be better off just building with a 5x9 double-bent liftarm or a 5x7 box liftarm instead.

So I very much doubt it brings us much structural strength. Maybe over very large distances it would be useful (for people wanting to build large towers). For most normal-sized builds, I think a robust structure with axis-aligned parts is, I expect, stronger. It's certainly cool though, for angled structures on modern vehicle designs or whatnot, but in such cases, the approach I usually use is: whatever fits.

The only use-case I can imagine where this would be useful, is when connecting two universal joints with a diagonal axle. With the 5.5 axles, many such number quadruples can be divided by 2.

2 minutes ago, Lipko said:

I guess the reason why it's not used commonly is that you need more complex hinges (two pins instead of one) which are more flexible, so you loose with the joint what you gain with the beam. Or dunno. The pinhole-pin part may overcome this issue better than older alternatives. Have you tested the chassis stiffness with and without the beam?

You indeed need more complex hinges, and indeed the pinhole-pin part sits so tight that it partly overcomes that drawback. I tested the chassis stiffness before and after and it made quite a difference, it especially reduced twist. The chassis was already quite stiff when it comes to normal bending, but the narrow console between the seats did allow for some twist. It feels much more rigid now.

32 minutes ago, Erik Leppen said:

It's a cool theory, but I doubt whether it's of much use.

 

I mean, you can easily span a diagonal of a simple triangle with liftarms where the holes are in line with the rest of the structure. A 3-4-5 triangle is easily secured. But the beam in the picture has to be connected with two hinges on each side (the black pin, and the black pin-with-hole), which decrease the stiffness and increase the amount of play in the structure. Also, the holes of the red beam can't be connected to anything. Thirdly, as you can see, the red beam is offset by 1 stud due to the black pins-with-holes. So the connection isn't actually in-line with its endpoints. So when compressed, the beam simply bends. So in this particular case, you might be better off just building with a 5x9 double-bent liftarm or a 5x7 box liftarm instead.

So I very much doubt it brings us much structural strength. Maybe over very large distances it would be useful (for people wanting to build large towers). For most normal-sized builds, I think a robust structure with axis-aligned parts is, I expect, stronger. It's certainly cool though, for angled structures on modern vehicle designs or whatnot, but in such cases, the approach I usually use is: whatever fits.

The only use-case I can imagine where this would be useful, is when connecting two universal joints with a diagonal axle. With the 5.5 axles, many such number quadruples can be divided by 2.

That makes sense. When you use the tightly sitting pinhole-pin part as in my example you can partly overcome the connection not being in-line with its endpoints. With the drawbacks you describe, like not being able to attach anything to it, I agree that this is not something that one would use as part of the core structure of something. I guess it's more of a means to make auxiliary/secondary connections, just like in my example. The fact that it turns out to reduce twist in my chassis - even with the double hinges at the endpoints - is probably because it is used quite far away from the twist axis and works as a kind of stabilizing lever. I have to admit that I was also excited about how it looked .

The discussion about the practical value of this is very useful, but I also remain to be interested in whether anyone knows a practical example .

Edited December 8, 20168 yr by Didumos69

Not knowingly. My usual technique is "if it fits without breaking a piece, go with it" . I know they exist and have probably used them at some point, but I quite often form imperfect triples and quadruples.

I have used the triples from time to time, but not the quadruples. Might be good for boom structure/reinforcement on cranes... Thanks for posting these!

I have used (and seen) them a couple of times, but not only for chassis stiffness/rigidity but also because I needed the complex angle for attaching e.g. parts of the bodywork or headlights. Or like tubular chassis for a monster truck.
Where I think probably everyone has used them is in suspension links (theoretically then, not for stiffness).

Anyway, it does indeed come in handy for what you're using it for; rigidity in chassis.

Tibivi

Very useful! Thank you for posting it. I hadn't even thought of using these.

Might be obvious to a great deal of you, but you can also use mutliples of these quadruples or triples. For example: (1, 2, 2, 3)*2 = (2,4,4,6) which is still a valid solution.

This is because you can multiply all the terms in the equation ( a^2+b^2+c^2=d^2 ) with a constant. 

 

I made a few of these (consciously, before you ask...).  Here are two versions of 1-2-2-3 (which looks as if it might be particularly useful, although I don't know for what).

Here are 2-3-6-7, 1-4-8-9, and 4-4-7-9:

Finally, just as in the two-dimensional case (see topic "a perfect fit"), even when the diagonal is not an integer length, we can still get some interesting angles by making the same diagonal in two different ways.   Here it is the diagonal of a 2-2-4 box:

 

Edited February 1, 20178 yr by aeh5040

While this is for sure a very interesting finding and discussion, I am not sure of applications. Let's face it, most of the designs/machines we use out there (in real life) are not truly 3D, but more like multiple 2D layers stacked on top of one another. If you think of an automobile engine, it consists of several crank mechanisms (2D) stacked on top of one another. The only two real world applications I can think of in real life include trusses (as in large cranes, static) and multi-link automobile suspensions (mechanisms) where the various links actually articulate around one another. Yes, again, very nice finding/discussion.

I made use of them in my LMP1 prototype. It was used to create a  strong and reliable frame. I had  connected  the suspension system on it