Subject:
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Re: Pythagorean Triads and Almost-Triads
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Newsgroups:
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lugnet.build
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Date:
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Mon, 24 Jan 2000 03:10:34 GMT
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Viewed:
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1208 times
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John J. Ladasky Jr. wrote in message ...
>
> > I think I don't. You see, my rule of thumb is that the walls that form • the
> > sides of the triangle will not be absorbing ANY deformation stress,
>
> Do we know this for a fact? The LEGO hinge bricks are pretty snug. The • joint
> is a Technic peg, but it is fixed to one brick. And you can't even • disassemble
> the hinge plates. In spite of this, both Ed Boxer and I have built a • 45-degree,
> seven-stud wall, which exceeds your rule-of-thumb error of 0.05 studs. The
> error is 0.071 studs.
No, we don't know this for a fact. When I said "my rule of thumb" I probably
should have said, "the assumption I work under". It is a simplification
designed to make the table more universally applicable... as I have said.
> > because
> > their thickness and construction are unknown (i.e. dependent entirely on • the
> > modeller's application).
>
> Not exactly true. We both provided tables that assume that all the sides • of the
> triangles are measured in studs. That means that we're looking at • triangles
> whose sides are defined by either bricks, or plates. It's not like we know
> absolutely nothing about the thickness and the construction of the walls. • It
> would be interesting to know whether a 2 X N brick is less forgiving to
> stretching/compression than, say, a 1 X N plate...
And what about a 6xN plate? Part of the problem here seems to be a focus on
using this method for outer walls. I intend to use the triangles in
structures where all three sides of the triangles are rigid box structures,
not partially flexible walls. I can't afford to make rash assumptions about
how much stress deformation a 6x18 plate can bear, if it means that 6 months
later I find I have three warped 6x18 plates.
> > I worked by the assumption that the hinges/technic-pegs
>
> Don't forget turntables. Very useful, as I hope to show...
Whatever. Anything that easily changes stud direction, except connecting
regular bricks pivoted by their end studs (because that puts deformation
solely in the bricks making the sides, which I assume to be unacceptable).
>
> If indeed there is no stress in the wall itself, only the hinge, then it is
> neat.
*sigh* Well, this is NOT in any sense what I have been saying. I said I
assumed that the sides _permitted_ no stress, not that they _sustained_ no
stress. My error margin was designed to minimise the amount of stress that
they sustain.
> But I can tell you that the bricks appeared to be pulled apart a bit in
> the 5-5-7 wall that I built -- though it was hardly enough to pop the • studs. I
> didn't have a micrometer handy to measure the spacing. 8^)
Sure. Now, if you build various 5-5-7 walls built with different sized 1xN
bricks, I think you'll find that the one built with more, shorter 1xN bricks
will have visibly allowed more deformation because there are more gaps
pulled apart, i.e. the deformation varies with the the number of bricks
used. This was my whole point. Flexibility can vary widely with
construction. Allow for too much flexibility and builders are going to be
using inappropriate combinations of triangle and construction, and will find
that they can't build it, or worse, will damage their bricks.
> Also consider that one may build right triangles with only two angled • elements,
> and the right angle would be defined by the grid of studs. These would • actually
> be stronger than triangles made of three free-floating angles.
And these would allow even less deformation! Actually, you have brought up
an interesting point. I tested most of the triads using a right angle
defined by the stud grid, so I guess my constant error of 0.05 studs is that
allowed by _2_ joints.
>
> Not every LEGO fan is a mathematician. My initial table contains only the
> triangles formed by relatively-prime integers.
So you missed out the ones formed by integers that weren't prime, but had no
common factor? Bummer. Or is that what "relatively-prime" means?
> >
> > I suggest that you assume that the sides accept no strain, and limit the
> > strain to that accepted by the joints, as above.
>
> Now, in this case I expect that the strain allowable at the joints will be
> *larger*, if you use Technic pegs and beams. Unlike the horizontal hinge • bricks
> we were discussing above, the peg at the joint is free at both ends. If it
> bends a bit in its sockets, no big deal. But the hypotenuse wall will • likely be
> less forgiving. Consider that the greatest flexibility in the length of • this
> wall is obtained by allowing it to be constructed with an integral number • of *
> plates*. But this means that the studs run along the hypotenuse. If you • pull
> on a wall of stacked plates, they can come apart.
>
> Is there a mechanical engineer in the house? 8^)
Well, not every LEGO fan is a mechanical engineer.... ;-) Which is what
people would have to be if a table forced them to check whether their
particular construction method could accept the strain of a particular
triad.
I suggest that you assume that the sides accept no strain, and thus leave us
with a more universally applicable solution. Which, funnily enough, is what
I've already done.
>
> > It's rather ironic, when we were debating error margins on star maps, • _you_
> > were the one who was suggesting my error limits weren't strict enough. • Now,
> > the tables seem to have turned. Funny, eh? Still, we live and learn ;-)
>
> LEGO is plastic. It's deformable. The stars are where they are, • Heisenberg
> notwithstanding...
If I put a star at 5.5 parsecs in my Legoverse and it's actually at 5.8, it
doesn't matter at all. If I put a 0.08 stud strain on my LEGO bricks and
they actually only tolerate 0.04, then I might permanently wreck them!
Now, after saying that, if my attitudes towards error margins still puzzle
you, then I give up..... :-)
> > The next task for one of us is to tackle what is probably a greater
> > problem, where 3 (or more, but 3 is a good start) of ANY triangle each • share
> > a common side and all their sides must be integers (within error).
>
> This is hard to decipher. Are you talking about building a tetrahedron?
Hmmm, it was a bit ambiguous. What I mean is a figure made from three
triangles that tesselate, can butt together so one corner from each meets in
the middle, and those corner angles therefore add up to 360 degrees. It's
late here and I have to get to bed, so that explanation will have to do for
now ;-)
G'night all,
Paul
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Message is in Reply To:
 | | Re: Pythagorean Triads and Almost-Triads
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| (...) Oh great, another one of our geeky debates! 8^) (...) Do we know this for a fact? The LEGO hinge bricks are pretty snug. The joint is a Technic peg, but it is fixed to one brick. And you can't even disassemble the hinge plates. In spite of (...) (24 years ago, 21-Jan-00, to lugnet.build)
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