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| | | Author: | chetzler  | | Posted: | Feb 5, 2023 11:45 | | Subject: | LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 155 times | | Topic: | General | |
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| I finally got around to building my copy of
and was delighted to see LEGO utilize a 3-4-5 Pythagorean triple in the geometry
of the foundation. Unlike some other sets containing angled walls that simply
float over a bed of tiles, this angled wall actually connects on every sixth
stud (every 5 stud lengths).
The only other official set where I recall seeing this satisfying piece of geometry
put to use was on the bow of this boat
using this part
Surely there have to be more, right? Are there any other math geeks out there
that are aware of any other sets where LEGO has used a Pythagorean triple?
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| | | | | Author: | Leftoverbricks | | Posted: | Feb 5, 2023 12:05 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 46 times | | Topic: | General | |
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| In General, chetzler writes:
| | I finally got around to building my copy of
and was delighted to see LEGO utilize a 3-4-5 Pythagorean triple in the geometry
of the foundation. Unlike some other sets containing angled walls that simply
float over a bed of tiles, this angled wall actually connects on every sixth
stud (every 5 stud lengths).
The only other official set where I recall seeing this satisfying piece of geometry
put to use was on the bow of this boat
using this part
Surely there have to be more, right? Are there any other math geeks out there
that are aware of any other sets where LEGO has used a Pythagorean triple?
|
Very nice question! I'm not a math expert but there are many sets around
where the LEGO geometry is applied in its best form.
You could ask Brickarchitect https://brickarchitect.com/ to chime in but I don't
know if he's active here.
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| | | | | Author: | SylvainLS | | Posted: | Feb 5, 2023 12:13 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 53 times | | Topic: | General | |
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| In General, chetzler writes:
| | I finally got around to building my copy of
and was delighted to see LEGO utilize a 3-4-5 Pythagorean triple in the geometry
of the foundation. Unlike some other sets containing angled walls that simply
float over a bed of tiles, this angled wall actually connects on every sixth
stud (every 5 stud lengths).
The only other official set where I recall seeing this satisfying piece of geometry
put to use was on the bow of this boat
using this part
Surely there have to be more, right? Are there any other math geeks out there
that are aware of any other sets where LEGO has used a Pythagorean triple?
|
Look at
There should also be a few of 3-4-5 in Technic sets as its the bas for
and other such parts.
I think what LEGO designers use more are what I would call “crossed diagonals”:
take rectangle ABCD, AC is a diagonal, the other diagonal BD has the same length,
it can therefore be placed where AC was. You get an angle which is twice the
angle of the diagonal.
Hmm, I’m not sure it’s clear… see pic.
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| | | | | | | Author: | SylvainLS | | Posted: | Feb 5, 2023 12:15 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 42 times | | Topic: | General | |
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| In General, SylvainLS writes:
| | […]
There should also be a few of 3-4-5 in Technic sets as
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*it’s the basis for
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| | | | | | | Author: | chetzler | | Posted: | Feb 5, 2023 21:54 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 46 times | | Topic: | General | |
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| In General, SylvainLS writes:
| | In General, chetzler writes:
| | I finally got around to building my copy of
and was delighted to see LEGO utilize a 3-4-5 Pythagorean triple in the geometry
of the foundation. Unlike some other sets containing angled walls that simply
float over a bed of tiles, this angled wall actually connects on every sixth
stud (every 5 stud lengths).
The only other official set where I recall seeing this satisfying piece of geometry
put to use was on the bow of this boat
using this part
Surely there have to be more, right? Are there any other math geeks out there
that are aware of any other sets where LEGO has used a Pythagorean triple?
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Look at
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Ugh, I even built that set and I remember that the area around the stairs was
really clever, but i didn't make the connection that it is indeed based around
a 3-4-5 triangle.
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There should also be a few of 3-4-5 in Technic sets as its the bas for
and other such parts.
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Huh, that's one I would never have guessed. It took me a bit to work out
because I had always (wrongly) assumed it was a 135-degree angle. I think it's
hard to see because the angle is correct but because the hypotenuse is only 3
units (center-to-center on the holes), the legs don't end up being integers.
After playing with it a bit though, I'm convinced (see pic).
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I think what LEGO designers use more are what I would call “crossed diagonals”:
take rectangle ABCD, AC is a diagonal, the other diagonal BD has the same length,
it can therefore be placed where AC was. You get an angle which is twice the
angle of the diagonal.
Hmm, I’m not sure it’s clear… see pic.
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I have stumbled across the crossed diagonals before when experimenting with off-the-grid
building. I don't know that I've ever see it in an actual set, but those
guys are pros and it may have been so naturally incorporated that I just didn't
notice it.
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| | | | | Author: | peregrinator | | Posted: | Feb 5, 2023 14:26 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 43 times | | Topic: | General | |
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| In General, chetzler writes:
| | Surely there have to be more, right? Are there any other math geeks out there
that are aware of any other sets where LEGO has used a Pythagorean triple?
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I don't think there are any right triangles in it, but the new Galaxy Explorer
had some pretty satisfying geometry
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| | | | | | | Author: | SylvainLS | | Posted: | Feb 5, 2023 15:04 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 42 times | | Topic: | General | |
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| In General, peregrinator writes:
| | […]
I don't think there are any right triangles in it, but the new Galaxy Explorer
had some pretty satisfying geometry
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Well, the build hinges (haha) on the 3-4-5 Pythagorean triangle because the wings
are 45° and 18.43° (1:3 wedges)¹, but the attachments aren’t in-system (bars
in holes) and are rather complicated.
¹ 3-4-5’s angles are 90°, 53.13° and 36.87°. 1:1 wedge = 90°/2, 1:2 wedge =
53.13°/2 and 1:3 wedges = 36.87°/2. So you can make a 3-4-5 triangle with wedges
(see pic).
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| | | | | Author: | BricksOfFaith | | Posted: | Feb 5, 2023 14:30 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 41 times | | Topic: | General | |
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| | This was a fairly difficult build, but I for real enjoyed every second of it!! |
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| | | | | Author: | rylie_aitch | | Posted: | Feb 5, 2023 15:07 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 51 times | | Topic: | General | |
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| In General, chetzler writes:
| | I finally got around to building my copy of
and was delighted to see LEGO utilize a 3-4-5 Pythagorean triple in the geometry
of the foundation. Unlike some other sets containing angled walls that simply
float over a bed of tiles, this angled wall actually connects on every sixth
stud (every 5 stud lengths).
The only other official set where I recall seeing this satisfying piece of geometry
put to use was on the bow of this boat
using this part
Surely there have to be more, right? Are there any other math geeks out there
that are aware of any other sets where LEGO has used a Pythagorean triple?
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The wing attachments on this set, if I remember right:
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| | | | | Author: | yorbrick | | Posted: | Feb 6, 2023 08:17 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 35 times | | Topic: | General | |
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| In General, chetzler writes:
| | I finally got around to building my copy of
and was delighted to see LEGO utilize a 3-4-5 Pythagorean triple in the geometry
of the foundation. Unlike some other sets containing angled walls that simply
float over a bed of tiles, this angled wall actually connects on every sixth
stud (every 5 stud lengths).
The only other official set where I recall seeing this satisfying piece of geometry
put to use was on the bow of this boat
using this part
Surely there have to be more, right? Are there any other math geeks out there
that are aware of any other sets where LEGO has used a Pythagorean triple?
|
I doubt there are (many) other Pythagorean triples, since the next one is 6-8-10,
which is the same thing (and they might as well strengthen with a stud after
5 instead of 10), then the next is 5-12-13, so a hypotenuse would be 14 studs,
so only one extra connection point every 13th stud. That is quite a long span
before being secured.
But of course there are quite a few examples where they use non-integer Pythagorean
lengths to change angles of layers with respect to each other using the crossed
diagonals idea. I think the first time I saw it in an official set was in a Christmas
tree a few years back but I have used it for some time, after seeing someone
else using it on flickr.
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| | | | | | | Author: | SylvainLS | | Posted: | Feb 6, 2023 08:42 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 41 times | | Topic: | General | |
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| In General, yorbrick writes:
| | […]
I doubt there are (many) other Pythagorean triples, since the next one is 6-8-10,
which is the same thing (and they might as well strengthen with a stud after
5 instead of 10), then the next is 5-12-13, so a hypotenuse would be 14 studs,
so only one extra connection point every 13th stud. That is quite a long span
before being secured.
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Yes, 3-4-5 is the simplest.
| | But of course there are quite a few examples where they use non-integer Pythagorean
lengths to change angles of layers with respect to each other using the crossed
diagonals idea. I think the first time I saw it in an official set was in a Christmas
tree a few years back but I have used it for some time, after seeing someone
else using it on flickr.
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Hmm, I think I saw it in Le Guide non officiel des techniques avancées de
construction LEGO, the French and augmented¹ version of The Unofficial
LEGO Advanced Building Techniques Guide.
And I quickly found it used by LEGO in X-Wings.
———
¹ Well, at least, the French version has chapters that the English I found earlier
doesn’t. Maybe I’ve the wrong English version….
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| | | | | | | | | Author: | yorbrick | | Posted: | Feb 6, 2023 09:21 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 27 times | | Topic: | General | |
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| In General, SylvainLS writes:
| | In General, yorbrick writes:
| | […]
I doubt there are (many) other Pythagorean triples, since the next one is 6-8-10,
which is the same thing (and they might as well strengthen with a stud after
5 instead of 10), then the next is 5-12-13, so a hypotenuse would be 14 studs,
so only one extra connection point every 13th stud. That is quite a long span
before being secured.
|
Yes, 3-4-5 is the simplest.
| | But of course there are quite a few examples where they use non-integer Pythagorean
lengths to change angles of layers with respect to each other using the crossed
diagonals idea. I think the first time I saw it in an official set was in a Christmas
tree a few years back but I have used it for some time, after seeing someone
else using it on flickr.
|
Hmm, I think I saw it in Le Guide non officiel des techniques avancées de
construction LEGO, the French and augmented¹ version of The Unofficial
LEGO Advanced Building Techniques Guide.
And I quickly found it used by LEGO in X-Wings.
———
¹ Well, at least, the French version has chapters that the English I found earlier
doesn’t. Maybe I’ve the wrong English version….
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I just checked - Winter Toy Shop 10199 was the official one where I remember
first seeing it. I cannot believe that was 13 years ago now. But it might have
been used before that and I didn't notice (even if I built it)!
I wonder if it was also used in a set that I built 12 years ago, and another
from 5 years ago ...
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| | | | | | | | | | | Author: | SylvainLS | | Posted: | Feb 6, 2023 09:34 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 30 times | | Topic: | General | |
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| In General, yorbrick writes:
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I just checked - Winter Toy Shop 10199 was the official one where I remember
first seeing it. I cannot believe that was 13 years ago now. But it might have
been used before that and I didn't notice (even if I built it)!
I wonder if it was also used in a set that I built 12 years ago, and another
from 5 years ago ...
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I can’t help much find when it was first used officially. I only think it wasn’t
used before 1990 (my Dark Ages) and I know it’s in
that I built in LDD.
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| | | | | | | | | Author: | peregrinator | | Posted: | Feb 6, 2023 10:20 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 25 times | | Topic: | General | |
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| In General, SylvainLS writes:
| | Yes, 3-4-5 is the simplest.
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Only if you don't have any bricks that are √2 in length
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| | | | | | | | | | | Author: | yorbrick | | Posted: | Feb 6, 2023 10:38 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 25 times | | Topic: | General | |
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| In General, peregrinator writes:
| | In General, SylvainLS writes:
| | Yes, 3-4-5 is the simplest.
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Only if you don't have any bricks that are √2 in length
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That wouldn't be a Pythagorean triple though, as a triple requires three
integers.
Many sets use the diagonal method for a 1-1-√2, which rotates the top plate by
90 degrees compared to the bottom one! You don't even need spacer 1x1 plates
for it to work.
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| | | | | Author: | chetzler | | Posted: | Feb 6, 2023 11:55 | | Subject: | Re: LEGO Pythagorean 3-4-5 Triangle | | Viewed: | 74 times | | Topic: | General | |
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| Another member just pointed this set out to me:
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USA, Minnesota
Netherlands, Overijssel
France, Nouvelle-Aquitaine
United Kingdom, England
