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Pythagorean Triads and Almost-Triads
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lugnet.build
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Date:
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Mon, 10 Jan 2000 01:55:55 GMT
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899 times
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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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Message has 3 Replies:
 | | Re: Pythagorean Triads and Almost-Triads
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| (...) I believe Eric Brok has something like this on his site. (...) 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 (...) (25 years ago, 9-Jan-00, to lugnet.build)
| | | Re: Pythagorean Triads and Almost-Triads
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| (...) 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 (...) (25 years ago, 9-Jan-00, to lugnet.build)
| | | Re: Pythagorean Triads and Almost-Triads
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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.) (...) Emerging from my LEGO Dark Ages last fall, we opened an Exploriens (...) (25 years ago, 16-Jan-00, to lugnet.build)
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