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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..... :-)
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, because
their thickness and construction are unknown (i.e. dependent entirely on the
modeller's application). I worked by the assumption that the
hinges/technic-pegs 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? ;-)
>
> My chart differs from Paul's in several ways:
>
> 1) All entries are ordered by the smallest angle.
Who was it said something about "pasting these tables into a spreadsheet"?
Oh, that's right, it was both of us :-)
> 2) I've included possible HALF-stud entries, for those of you who use
> offset plates. I've been experimenting with this, and it looks
> promising.
>
> 3) My table only includes a few entries where either x or y exceeds 20
> studs. There were practical reasons for stopping here, both computer • [...]
>
> 4) For the standard angle plates (did I miss any?), I have listed some
> entries where the rise and run corresponding to the angled edge are both
> whole numbers, but for which the strain is too high. The reason for
> this is to emphasize the first entry that IS usable. For example, take
What kind of a reason is that? Put in the entries that ARE useable and they
emphasise themselves. Unusable ones just clutter up the table.
Oh, and another difference is:
5) It contains redundant multiples. I always assumed that people could
trivially extrapolate to find these, and would only want the fundamental
triads... silly me :-)
>
> 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. 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.
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 ;-)
> So, since you're a Space fan like me, I have to wonder whether I'll
> finish my wedge-shaped spacecraft before you do... are you married?
> Does your wife have the flu? How about your kid? This year's strain of
> influenza routinely incapacitates healthy adults for over seven days.
> Needless to say, I'm not getting much done this week...
Well, the next task for one of us is to tackle what is probably a greater pr
oblem, 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). 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).
Paul
http://www.geocities.com/Area51/Shuttle/5168/
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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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I wrote:
> In lugnet.build, Paul Baulch writes:
> >
> > John J. Ladasky Jr. wrote in message <38814540.ED2D820E@my-deja.com>...
> > > 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.
> >
> > 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.
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 plates. The two plates
were used as the bottom course of two eight-stud walls. These two walls were
perpendicular, and the closest ends of the two walls were separated by ten studs
in each direction. In between these two walls, I built a fourteen-stud wall at
45 degrees, which was suspended over the baseplate by the height of one plate.
The walls were six bricks plus one plate in height. The hinge bricks were
placed in the second and fifth rows. This way, I could see whether the strain
imparted to the bricks would prevent the attachment of additional components,
either above, below, or between the hinges.
Additional tension was noted when trying to add parts to the 45-degree wall, but
not enough to impede building. Once I reached the third course of bricks, I
could pile stuff on top of the 45-degree wall as if it was attached to the
baseplate. I was worried that bricks might pop off of the bottom course while I
was pushing on the top of the wall. Nothing moved at all. Tiles underneath the
45-degree wall are thus structually unnecessary. They are nice for decorative
purposes, however.
I did not observe any significant bending of the baseplate, nor any distortion
in the shape of the walls. I suspect that the strain is being distributed into
the cracks between the bricks of the 45-degree wall. I can't prove this without
the use of a micrometer, however.
So, the 45-degree wall here is 14.000 studs long, however it is meshing nicely
with the baseplate at coordinates separated by 14.142 studs. Let's hear it for
the power of experimentation, and the plasticity of ABS!
--
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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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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