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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?"
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, "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.
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.
The columns are, left to right: shortest side of the triangle, next
shortest side, longest side, the deviation of the longest side (in studs)
from right-angled, and finally the largest "non-right-angled" angle made
inside the triangle (approximately). For example the first one is a 3-4-5
triangle with a devation 0 studs making an angle of about 53.1 degrees.
Hmmm, and I suggest copying the text and pasting it into a spreadsheet
program, otherwise it just looks like so much gibberish :-)
Cheers,
Paul
3 4 5 0 53.1
8 9 12 0.04159458 48.4
5 12 13 0 67.4
7 11 13 0.03840481 57.5
8 15 17 0 61.9
11 13 17 0.02938637 49.8
6 17 18 0.02775638 70.6
9 19 21 0.02379604 64.7
11 19 22 0.0455016 59.9
14 17 22 0.02271555 50.5
13 19 23 0.02172887 55.6
7 23 24 0.04163056 73.1
7 24 25 0 73.7
7 25 26 0.03849003 74.4
17 21 27 0.01851217 51
16 23 28 0.01785145 55.2
20 21 29 0 46.4
13 27 30 0.03335187 64.3
15 26 30 0.01666204 60
11 29 31 0.01612484 69.2
8 31 32 0.01562119 75.5
11 30 32 0.04690938 69.9
19 27 33 0.01514804 54.9
8 33 34 0.04414631 76.4
14 31 34 0.0147027 65.7
23 25 34 0.0294245 47.4
12 35 37 0 71.1
23 29 37 0.01351105 51.6
22 31 38 0.01315562 54.6
21 34 40 0.03751759 58.3
9 40 41 0 77.3
23 34 41 0.0487515 55.9
9 41 42 0.02381628 77.6
26 33 42 0.01190308 51.8
13 41 43 0.01162634 72.4
22 37 43 0.0464865 59.3
13 42 44 0.03410413 72.8
16 41 44 0.01136217 68.7
31 34 46 0.01086828 47.6
32 33 46 0.03262026 45.9
19 43 47 0.01063709 66.2
23 41 47 0.01063709 60.7
29 37 47 0.01063709 51.9
25 41 48 0.02082881 58.6
28 39 48 0.01041554 54.3
14 47 49 0.04079934 73.4
17 46 49 0.04079934 69.7
31 38 49 0.04079934 50.8
10 49 50 0.009999 78.5
17 47 50 0.020004 70.1
29 42 51 0.03920062 55.4
10 51 52 0.02885416 78.9
26 45 52 0.02885416 60
32 41 52 0.0096145 52
28 45 53 0 58.1
31 43 53 0.00943312 54.2
37 38 53 0.03772242 45.8
23 50 55 0.03635162 65.3
25 49 55 0.00909016 63
18 53 56 0.02679212 71.2
37 42 56 0.02679212 48.6
21 53 57 0.00877125 68.4
15 56 58 0.02586784 75
31 49 58 0.01724394 57.7
34 47 58 0.00862005 54.1
11 58 59 0.03388857 79.3
26 53 59 0.03388857 63.9
37 46 59 0.03388857 51.2
11 59 60 0.01666435 79.4
24 55 60 0.00833275 66.4
11 60 61 0 79.6
19 58 61 0.03277808 71.9
11 61 62 0.01613113 79.8
19 59 62 0.01613113 72.1
38 49 62 0.00806399 52.2
11 62 63 0.03175403 79.9
22 59 63 0.03175403 69.6
34 53 63 0.03175403 57.3
37 51 63 0.00793601 54
42 47 63 0.03173804 48.2
43 46 63 0.03175403 46.9
11 63 64 0.04689218 80.1
27 58 64 0.02344179 65
29 57 64 0.04689218 63
31 56 64 0.00781202 61
16 63 65 0 75.7
33 56 65 0 59.5
41 53 67 0.00746227 52.3
23 64 68 0.00735254 70.2
30 61 68 0.0220624 63.8
37 57 68 0.04413197 57
43 54 69 0.02897942 51.5
49 50 70 0.00714249 45.6
41 58 71 0.02816343 54.7
12 71 72 0.00694411 80.4
17 70 72 0.03471385 76.3
31 65 72 0.01388755 64.5
33 64 72 0.00694411 62.7
44 57 72 0.00694411 52.3
17 71 73 0.00684899 76.5
29 67 73 0.00684899 66.6
43 59 73 0.00684899 53.9
48 55 73 0 48.9
12 73 74 0.02027305 80.7
17 72 74 0.02027305 76.7
21 71 74 0.04052944 73.5
17 73 75 0.04668119 76.9
27 70 75 0.02666193 68.9
51 55 75 0.00666637 47.2
21 73 76 0.03948394 74
27 71 76 0.03948394 69.2
41 64 76 0.00657866 57.4
38 67 77 0.02596965 60.4
47 61 77 0.00649323 52.4
40 67 78 0.0320447 59.2
46 63 78 0.00640999 53.9
51 59 78 0.01282157 49.2
49 62 79 0.0253124 51.7
25 76 80 0.00624976 71.8
54 59 80 0.0187522 47.5
13 80 81 0.04936767 80.8
18 79 81 0.0246876 77.2
25 77 81 0.04322141 72
33 74 81 0.0246876 66
35 73 81 0.04322141 64.4
37 72 81 0.04939778 62.8
39 71 81 0.0061726 61.2
47 66 81 0.0246876 54.5
13 81 82 0.03657721 80.9
22 79 82 0.00609733 74.4
41 71 82 0.01219603 60
57 59 82 0.03657721 46
13 82 83 0.02409289 81
31 77 83 0.00602388 68.1
43 71 83 0.00602388 58.8
49 67 83 0.00602388 53.8
13 83 84 0.01190392 81.1
13 84 85 0 81.2
36 77 85 0 64.9
13 85 86 0.01162869 81.3
29 81 86 0.03487665 70.3
34 79 86 0.00581376 66.7
47 72 86 0.01744363 56.9
13 86 87 0.02299154 81.4
26 83 87 0.02299154 72.6
29 82 87 0.02299154 70.5
44 75 87 0.04598917 59.6
53 69 87 0.00574694 52.5
59 64 87 0.04596487 47.3
61 62 87 0.02299154 45.5
13 87 88 0.03409752 81.5
46 75 88 0.01704711 58.5
52 71 88 0.00568163 53.8
57 67 88 0.03409752 49.6
13 88 89 0.04495517 81.6
23 86 89 0.02246907 75
32 83 89 0.04495517 68.9
39 80 89 0 64
41 79 89 0.0056178 62.6
19 88 90 0.02777349 77.8
23 87 90 0.0111118 75.2
37 82 90 0.03889729 65.7
59 68 90 0.02777349 49.1
19 89 91 0.00549434 77.9
23 88 91 0.04396667 75.4
53 74 91 0.02197537 54.4
57 71 91 0.04943712 51.2
19 90 92 0.01630579 78.1
27 88 92 0.04890005 72.9
56 73 92 0.00543462 52.5
63 67 92 0.03261448 46.8
19 91 93 0.03764203 78.2
27 89 93 0.00537619 73.1
42 83 93 0.02150289 63.2
49 79 93 0.03764203 58.2
60 71 93 0.0430207 49.8
14 93 94 0.04786016 81.4
51 79 94 0.03190948 57.2
54 77 94 0.04786016 55
66 67 94 0.04786016 45.4
14 95 96 0.02603814 81.6
36 89 96 0.00520819 68
53 80 96 0.03646526 56.5
43 87 97 0.04638066 63.7
59 77 97 0.0051545 52.5
65 72 97 0 47.9
14 97 98 0.00510191 81.8
24 95 98 0.01530732 75.8
41 89 98 0.01020461 65.3
47 86 98 0.00510191 61.3
58 79 98 0.00510191 53.7
20 97 99 0.0403958 78.3
28 95 99 0.0403958 73.6
31 94 99 0.02020408 71.7
34 93 99 0.02019996 69.9
49 86 99 0.02020408 60.3
54 83 99 0.02019996 57
61 78 99 0.02019996 52
69 71 99 0.00505038 45.8
14 99 100 0.01500113 82
51 86 100 0.01500113 59.3
65 76 100 0.00499988 49.5
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"Paul Baulch" <paul@vic.bigpond.net.au> writes:
> I certainly haven't seen any sort of data like this posted on anyone's
> web pages.
I believe Eric Brok has something like this on his site.
> A few days ago I was sitting making different triangles using three 2x4
> brick-w/-hinge, and various lengths of 1xN plates.
Part of the difficulty of creating these triangles is how you make the
angles...for example, with Technic beams, you have to measure from the
center of one end hole to the center of the other. If you're using 1x2
plate hinges, the lengths of the sides are increased, etc.
> These length combinations are called Pythagorean Triads, or so my
> memory of schooling tells me.
I was taught Pythagorean Triples--same difference.
> I wrote a program that would search all combinations of triangles
> (of certain integral dimensions) and show me which ones were "close
> enough" to right-angled
Did you add an algorith to remove multiples? (I saw that 6,8,10 10,24,26
and others are missing.)
> 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'm pretty sure it would be. After all, LEGO parts are slightly
flexible, so you could probably have a greater error margin.
Thanks for your efforts!
--Bram
Bram Lambrecht / o o \ BramL@juno.com
-------------------oooo-----(_)-----oooo-------------------
WWW: http://www.chuh.org/Students/Bram-Lambrecht/
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Bram Lambrecht wrote in message <20000109.104425.5095.2.braml@juno.com>...
>
> I believe Eric Brok has something like this on his site.
Ah, cool. I hadn't seen it before.... actually I can't see it now. What is
the URL? :-)
Wow, I'm almost embarassed to ask, I've been to Eric's site so many times,
just not recently.
>
> Part of the difficulty of creating these triangles is how you make the
> angles...for example, with Technic beams, you have to measure from the
> center of one end hole to the center of the other. If you're using 1x2
> plate hinges, the lengths of the sides are increased, etc.
Sure, but the side lengths are all just integer units, so when one uses
Technic beams one treats side lengths as the number of holes along the
Technic beams.
>
> Did you add an algorith to remove multiples? (I saw that 6,8,10 10,24,26
> and others are missing.)
I sure did. Any triangle whose two shortest sides had a common factor (e.g.
common factor of 6 & 8 is 2) was eliminated. After all, we can just obtain
those from the other, "fundamental" ones.
> > 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'm pretty sure it would be. After all, LEGO parts are slightly
> flexible, so you could probably have a greater error margin.
Yeah, but I was thinking more along the lines of when you start stacking
more and more bricks on top of the triangle in your model, and they're all
applying a force towards right-angled, and then later on you find your
hinges/pins are buckled :-) then again, I doubt that being out by 0.05
studs would cause this.
> How difficult do you think it would be to edit your program give one (or
> more) sides in integral plate heights instead of stud widths? That might
> give some useful results, too.
Well, given that 3 plates == 1 brick, and 5 plates == two studs, I thought
that people could just interconvert in their head :-)
Silly me! ;-)
Cheers,
Paul
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"Paul Baulch" <paul@vic.bigpond.net.au> writes:
> Bram Lambrecht wrote:
> > I believe Eric Brok has something like this on his site.
>
> Ah, cool. I hadn't seen it before.... actually I can't see it now.
> What is the URL? :-)
Whenever you know someone's name, but not their URL, check
http://www.lugnet.com/links/
I know Eric is there.
> > If you're using 1x2 plate hinges, the lengths of the sides are
> > increased, etc.
> Sure, but the side lengths are all just integer units
Not necessarily--especially if you use different hinges on either end.
> > How difficult do you think it would be to edit your program give one
> > (or more) sides in integral plate heights instead of stud widths?
> > That might give some useful results, too.
>
> Well, given that 3 plates == 1 brick, and 5 plates == two studs, I
> thought that people could just interconvert in their head :-)
Not all integer stacks of plates are integer stud widths tall. It's the
ones that aren't that I'm interested in. I can convert the other ones.
--Bram
Bram Lambrecht / o o \ BramL@juno.com
-------------------oooo-----(_)-----oooo-------------------
WWW: http://www.chuh.org/Students/Bram-Lambrecht/
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"Paul Baulch" <paul@vic.bigpond.net.au> writes:
> So, being a computer programmer, I wrote a program that would search
> all combinations of triangles (of certain integral dimensions)
How difficult do you think it would be to edit your program give one (or
more) sides in integral plate heights instead of stud widths? That might
give some useful results, too.
--Bram
Bram Lambrecht / o o \ BramL@juno.com
-------------------oooo-----(_)-----oooo-------------------
WWW: http://www.chuh.org/Students/Bram-Lambrecht/
-----------------------------------------------------------
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I wrote:
> How difficult do you think it would be to edit your program give one
should be "...edit your program to give..."
I really should proofread.
> (or more) sides in integral plate heights instead of stud widths? That
> might give some useful results, too.
> --Bram
>
>
> Bram Lambrecht / o o \ BramL@juno.com
> -------------------oooo-----(_)-----oooo-------------------
> WWW: http://www.chuh.org/Students/Bram-Lambrecht/
> -----------------------------------------------------------
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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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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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