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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Sun, 16 Jan 2000 04:13:02 GMT
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Viewed:
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815 times
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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.)
Paul Baulch wrote:
> G'day all,
>
> Most of this will not come as a surprise to "real" expert builders, but this
> post is also, in a way, an invitation to confirm what I have found. I
> certainly haven't seen any sort of data like this posted on anyone's web
> pages. Hopefully various parts of this data will be useful to a few people
> planning creations, and maybe save them some trial-and-error.
>
> Ever wondered exactly what are ALL of the combinations of lengths you can
> use to make angled walls and beams in a model, that will give you an
> acceptable right-angle? A few days ago I was sitting making different
> triangles using three 2x4 brick-w/-hinge, and various lengths of 1xN plates.
> As most of us know, you can make a right-angled triangle using 1x3,1x4 and
> 1x5 stud sides respectively, and also using 1x5, 1x12, and 1x13. These
> length combinations are called Pythagorean Triads, or so my memory of
> schooling tells me.
>
> Anyway, I noticed that some triangles I made were "almost" right-angled, and
> I thought, "are they close enough for me to use? What other ones are there?"
Emerging from my LEGO Dark Ages last fall, we opened an Exploriens
Android Base #6958 that my wife had purchased for my son (who's three,
it was a little complex for him). Immediately upon seeing the 10 X 10
octagonal dome pieces, I asked myself the question, "Can I build a
45-degree wall underneath the diagonal portion of the dome?" I reasoned
that, since the triangular cutout was 5 X 5 and the hypotenuse was thus
the square root of 50 ( = 7.071), I might be able to make a seven-stud
wall that fit. With some fiddling, some 1 X 4 hinge bricks and some
tiles, I did just that.
I quickly learned that I wasn't the first person to build this
particular 45-degree wall. Ed Boxer used it in his magnificent
cathedral, and describes it quite thoroughly:
http://www.geocities.com/~edboxer/crown.html
However, like you, it occurred to me that other angles could be useful.
For example: the angles of a 5-12-13 Pythagorean right triangle are
nearly identical to the angles of a 4 X 8 Classic Space wing.
> So, being a computer programmer, I wrote a program that would searcn all
> combinations of triangles (of certain integral dimensions) and show me which
> ones were "close enough" to right-angled,
As a consequence of the rapid evolution of computer programming
languages, and five years in a doctoral program that kept me away from
computers, I have lost the ability to program(!). I had to do this in
Excel. I did this back in November, but wanted to construct and show off
a model with my new knowledge, before revealing my secrets. You've
forced my hand. :^)
> "close enough" being defined as
> (here I'll get technical, sorry to non-programmers):
>
> square_root( X*X + Y*Y ) - Z < error_margin
>
> Where error_margin was an amount, in studs, that I thought was acceptable. I
> chose 1/20th of a stud for the following results (see below). 1/20th of a
> stud seems to be an acceptable amount of "give" when I tested these results
> using 2x4 brick/w/hinge. It didn't seem to strain the bricks at all, I think
> that there's enough looseness in the hinges themselves to absorb the 0.05
> stud inaccuracy. Of course, some of the "almost-rtiads" have even less
> inaccuracy. The amount is also in the data below. I'd be interested to know
> whether 0.05 studs is an acceptable slack to take up when using Technic
> beams and pegs, I haven't tried this.
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.
The larger Pythagorean near-triples can certainly take more than 1/20
stud of deformation stress. My rule of thumb is that the error cannot
exceed 1.0% of the hypotenuse. The 5-5-7 triangle, which works,
actually exceeds this limit by a hair (0.071 studs/7.000 studs = 1.01%).
> So anyway, here are the results. Whenever I plan a new creation I use this
> table now, it comes in very handy as it shows all side lengths which work
> (to within 0.05 studs) up to 100 studs, and I can choose the one that's
> closest to the angle I want (if the sides are small enough). I plan to use
> the 8-9-12-stud triangle a lot as it's small and close to 45 degrees.
Here's my spreadsheet. All columns are separated by tabs, and rows by
carriage returns. You should be able to copy and paste this into your
favorite spreadsheet in order to straighten out the columns.
Description of the column contents as follows:
angle = narrow angle of the right triangle.
plate = names plate with listed angle if one exists, otherwise blank
x, y = dimensions of the legs of the right triangle, in studs
z = length of the hypotenuse of the right triangle, in studs
strain = percent difference between the hypotenuse (z) and the nearest
whole-number value
comp. angle = complementary angle, equals 90 degrees minus the narrow angle
My chart differs from Paul's in several ways:
1) All entries are ordered by the smallest angle.
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
and LEGO-related. First, as I mentioned, I was grinding these out in
Excel, not programming. Second, at least as I have envisioned using
this building technique, it will be necessary to make periodic contact
between the angled wall and the grid of studs, for strength and support.
An angled wall that "floats" much farther than 20 studs will probably
be too weak for my purposes. Other people may find these longer runs
useful. They're on Paul's table.
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
the 4 X 8 Classic Space wing. If you really cannot tolerate the
0.6-degree difference between the 5-12-13 triangle and this wing piece,
you have to go out to a 9-21-23 triangle to obtain a strain below 1.0%.
The 3-7 and the 6-14 triangles are too strained.
---------
Standard angle plates, Pythagorean triples, and near-triples all
measurements in studs
angle plate x y z strain comp. angle
15.9 3 10.5 10.92 0.73% 74.1
16.3 7 24 25.00 0.00% 73.7
17.5 3 9.5 9.96 0.38% 72.5
18.4 Snowspeeder 2 6 6.32 5.13% 71.6
18.4 Snowspeeder 3 9 9.49 5.13% 71.6
18.4 Snowspeeder 4 12 12.65 2.77% 71.6
18.4 Snowspeeder 5 15 15.81 1.19% 71.6
18.4 Snowspeeder 6 18 18.97 0.14% 71.6
19.4 3 8.5 9.01 0.15% 70.6
19.4 6 17 18.03 0.15% 70.6
21.8 3 7.5 8.08 0.96% 68.2
22.6 5 12 13.00 0.00% 67.4
23.2 Classic 4X8 3 7 7.62 5.05% 66.8
23.2 Classic 4X8 6 14 15.23 1.52% 66.8
23.2 Classic 4X8 9 21 22.85 0.67% 66.8
24.4 5 11 12.08 0.69% 65.6
26.6 Airplane 7X12 1 2 2.24 10.56% 63.4
26.6 Airplane 7X12 2 4 4.47 10.56% 63.4
26.6 Airplane 7X12 3 6 6.71 4.35% 63.4
26.6 Airplane 7X12 4 8 8.94 0.62% 63.4
27.6 6 11.5 12.97 0.22% 62.4
28.1 8 15 17.00 0.00% 61.9
29.7 4 7 8.06 0.77% 60.3
29.7 6 10.5 12.09 0.77% 60.3
32.5 7 11 13.04 0.29% 57.5
33.7 5 7.5 9.01 0.15% 56.3
36.9 3 4 5.00 0.00% 53.1
39.5 7 8.5 11.01 0.10% 50.5
41.6 4 4.5 6.02 0.35% 48.4
41.6 8 9 12.04 0.35% 48.4
45.0 various 45-deg. 5 5 7.07 1.01% 45.0
45.0 various 45-deg. 7 7 9.90 1.02% 45.0
----------
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.
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...
--
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 1 Reply:
 | | 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 (...) (25 years ago, 21-Jan-00, to lugnet.build)
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Message is in Reply To:
| | Pythagorean Triads and Almost-Triads [DAT]
|
| G'day all, Most of this will not come as a surprise to "real" expert builders, but this post is also, in a way, an invitation to confirm what I have found. I certainly haven't seen any sort of data like this posted on anyone's web pages. Hopefully (...) (25 years ago, 10-Jan-00, to lugnet.build)
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