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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Fri, 21 Jan 2000 07:29:19 GMT
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Viewed:
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706 times
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In lugnet.build, Paul Baulch writes:
>
> John J. Ladasky Jr. wrote in message <38814540.ED2D820E@my-deja.com>...
> > Geez, Paul,
> >
> > This is the second time I've bumped into you ,doing a project identical
> > to one of mine! (The other one was the 3-D map of nearby stars, in case
> > you've forgotten.)
>
>
> John, how could I possibly have forgotten that? ;-)
>
> > I think that you want to measure your error margin as a fraction of the
> > total hypotenuse length, rather than as the absolute number of studs.
>
> Here we go again..... :-)
Oh great, another one of our geeky debates! 8^)
> 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.
Tell you what. When I go home tonight, I will try to build a 45-degree wall
that is 14 studs long, defining the hypotenuse of a 10-10-14 triangle. I'll
report back and tell everyone whether it works. If it does work, the error will
be 0.142 studs. We'll know which way of measuring error is more realistic.
> 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...
> I worked by the assumption that the hinges/technic-pegs
Don't forget turntables. Very useful, as I hope to show...
> will be absorbing the deformation stress, and last time
> I looked, that was a constant for all triangles I examined, that is, three
> ("3") :-) Therefore, my error margin turns out to be constant too. Neat,
> huh? ;-)
If indeed there is no stress in the wall itself, only the hinge, then it is
neat. 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^)
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.
[snip]
> It contains redundant multiples. I always assumed that people could
> trivially extrapolate to find these, and would only want the fundamental
> triads... silly me :-)
Not every LEGO fan is a mathematician. My initial table contains only the
triangles formed by relatively-prime integers.
> > Now, consider that we've only talked about angles in the plane defined
> > by studs. But what about tilting a wall up in the vertical direction?
> > You can build such walls using Technic pegs, or with 1 X 2 hinge bricks.
> > What dimensions are permissible? I'm still working on the practical
> > aspects of this, though I have a table of theoretically-acceptable
> > triangles. I suspect that the strain permitted in the vertical-plane
> > triangles will not be as high as in the horizontal-plane triangles.
>
>
> 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^)
> Otherwise people are going
> to spend a lot of time having to build stuff that they're not sure will
> work, simply because their sides are built slightly differently. This
> greatly limits the usefulness of the table, which was meant to save time
> (IMO). Then again, your table already contains a lot of data that is either
> redundant or not actually useable. I also, therefore, suggest that any
> triads for which the strain is outside limits but still possibly useable
> (depending on construction) be put in a separate table for people to resort
> to if there are no acceptable "strict" solutions.
I've thought of this. Two tables might be useful, prior to knwoing just what is
acceptable. The model I'm working on uses the 5-12-13 triangle, rather than
attempting to match the Classic Space wing exactly. But, for the impatient, you
*can* just look down the table for those entries with zero stress...
> 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...
> 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?
> If people
> are building with hinges/pegs they need not merely be incorporating single
> right-angled triangles into conventional orthogonal constructions. I, for
> one, would like to know what arbitrary triangles I should choose if I want
> them to each share a side with each other (and connect up properly with
> integral stud lengths).
I am having trouble imagining a LEGO model that would actually need to join
triangles in this way. If you need a specific angle, why concatenate two
triangles to get there? Why not just build a single right triangle that will
give you that angle?
But, you haven't thought through the math on this last one. It's really easy.
You did say *arbitrary* triangles, right? You didn't constrain the angles in
any way.
Suppose that we have A, B, and C. If A + B > C, and A + C > B, and B + C > A,
then we can make a triangle with sides of length A, B, and C. Try it. You will
see that finding integers that meet these simple conditions is trivial. Now, to
join a second triangle to your first, select a side -- say A. Then choose D and
E such that A + D > E, A + E > D, and D + E > A.
Here are five values that work, though I am sure that you can come up with many
more: A = 3, B = 4, C = 5, D = 6, E = 7.
--
John J. Ladasky Jr., Ph.D.
Department of Structural Biology
Stanford University Medical Center
Stanford, CA 94305
Secretary, Californians for Renewable Energy <http://www.calfree.com>
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Message has 2 Replies:
 | | Re: Pythagorean Triads and Almost-Triads
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| (...) O.K., I did it, and it WORKS! Here are the details (I wish I had a digital camera, or LDRAW right now): On a 32 X 32 baseplate, I constructed three walls using several 1 X 4, 1 X 6, and 1 X 8 bricks, four 1 X 4 hinge bricks, and two 1 X 8 (...) (24 years ago, 22-Jan-00, to lugnet.build)
| | | Re: Pythagorean Triads and Almost-Triads
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| John J. Ladasky Jr. wrote in message ... (...) the (...) joint (...) disassemble (...) 45-degree, (...) 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 (...) (24 years ago, 24-Jan-00, to lugnet.build)
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Message is in Reply To:
| | Re: Pythagorean Triads and Almost-Triads
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| John J. Ladasky Jr. wrote in message <38814540.ED2D820E@m...ja.com>... (...) John, how could I possibly have forgotten that? ;-) (...) Here we go again..... :-) I think I don't. You see, my rule of thumb is that the walls that form the sides of the (...) (24 years ago, 21-Jan-00, to lugnet.build)
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