Thank you thank you thank you! I've been wanting to do more 45 degree angles in my builds in bricklink studio but it doesn't like angles much when rotating things, I forgot all about 15706 because I almost never see it in use, this will REALLY help me!
√(2²+5²+14²)=15 is really neat in 3 dimensions. I discovered it while building my 30 stud diameter sphere. Now thanks to this - let's call it Euclidean quadruple - some of the pieces on the surface are placed perfectly accurate.
'Pythagorean quadruple' is a thing, according to Wikipedia. Some other simple ones include √(1²+2²+2²)=3, √(2²+3²+6²)=7, √(4²+4²+7²)=9, and √(1²+4²+8²)=9
Fascinating as always. Talking my language. Math, Lego and a little bit of one of my favorite carpentry disciplines: fence work. Happy fanksgibbing, Chris and family!
12:24 in this case the correct length would be 12.021 "fences": 0.021 fences are 0.167mm, while the gap between two bricks placed on a grid is 0.4mm. I call it legal.
Hey, I've actually used one of these! There's a spot in my city that has one of those 1x8x2 railings. It has two connections on the bottom that have 4 studs in between, so they're 6 apart, meaning I was able to do the 3-4-5 triple to put the fence at the perfect angle for my build! Mostly I'm glad to know that that's a mathematical principle and not going to damage my pieces or something. :P
Absolutely genius video man. Great job! I literally was thinking the other day about this exact thing and if there was math to prove that angled pieces placed like that were legal. Thanks for answering my question!
Something to think about. Diagonals of a plate are equal. Take two 2x4 plates. Place two 1x1 plates, one on each corner of the diagonal of one 2x4 plate and then attach the second 2x4 plate on top, but use the other diagonal. I hope that makes sense. You can use this with any plate, as long as the sides are of different lengths. Doing this in a square will just rotate the part 90 degrees. You don't even have to use the corners of the plate. Take a baseplate and put the 1x1 plates in such a way that they are the diagonal of a 7x13 rectangle. Now you can place a 8x16 plate on top at an angle, using the other diagonal of the 7x13 rectangle. Again, I hope this makes sense, it's easier to show than to describe.
@bricksculpt You're welcome. I accidentally discoverde that technique back in 1991, when I was building a starfighter. I didn't even realise what I had done until a few weeks later. Small addition to the original comment. If you place two 1x1 plates on the diagonal of a rectangle, you can rotate that rectangle (and the diagonal) 90 degrees around its center and you will have two more connection points. There should also be a connection point at the center of the diagonal. Where that connection point is would depend on the original rectangle used. Have fun discovering the possibilities.
I’ve done a mock with a mills plated section at 45 degrees in the middle. The other plates sit up to it and are tiled off using the 2x2 triangle tiles. Works really well
Hey, you're gonna like this. I'm putting up pegboard to help me organize my stuff, right. I've got a panel of it 8 inches wide and some of those 10 hole wide tool hangers with six sets of double rings. I was thinking about this video and how i could put a hanger at an angle for certain things and since there's some leeway with the holes in the pegboard, it totally works in a couple of configurations. And it's a Pythagorean triplet. I've got one set up with things like a pair of scissors, a telescoping back scratcher, a telescoping alligator clip and a screwdriver. Just thought I'd share this because you gotta be proud of yourself for the inspiration. Pat yourself on the back or something. And since it's still thanksgiving..... Thank you!👽
You can also half the size of a triple by using a jumper. So 3x4x5 triple becomes 1.5x2x2.5. with the "fencepost" stud, that is 2stud plus jumper, 3 stud, and then a 1x5 plate on the hypotenuse where the hollow stud of jumper connects between the last two studs.
@@bricksculpt I saw the trick you did with the underside-pole and hollow stud. That is a half length. You already figured it out. This is not above your level! I believe this would be a valuable addition to your pythagorian series. It is the same scaling that you had done. (3, 4, 5) can be reduced (halved) to (1.5, 2, 2.5). And then that can be multiplied by like 3.to give us (4.5, 6, 7.5). Anything with a .5 can be attached with a jumper or underside pole. Some of those unfortunately large triangles can be halved with the use of jumpers.
For studs on slopes, technic bricks can be a solution, if the slope's location allows for it. Of course you need to do the mathematics in plate heights, and divide the square root of the hypotenuse by 2.5 to get back to brick distances. In my experience, anything below 0.01 brick-width difference from a full brick is easily within LEGO tolerances (i.e. an axle will go through as if it was a single hole), and up to ~0.05 brick-width difference can be used without putting to much strain on the pieces.
Not sure if you've realized this, but those 45° solutions, at the end, should also be solutions for the square root of 2. The diagonal of a 1x1 square is root 2, & that cuts perfectly through the middle, meaning it cuts 90° angles in half, or into two 45° angles. You were missing those half stud offsets.
There are a few small near-triples which result in small gaps when you build them. I like this one: 4 - 7 - 8(.0623), which with jumpers you can make even smaller at 2 - 3.5 - 4-ish. This has an angle close to 30° as well, for those nice hexagons...
@haha no thank you! My head works better with visuals; and your 'demonstration' of the 'fence post problem' was legitimately the best example I've ever seen, simple but straight to the point and logical. Well done :)
Previous comments got automodded by adding a picture link, so I threw them in the discord, but you can get even more triangles using SNOT bricks and brackets (half plate thicknesses), even found the right angle to mate with 2x3 wedge plates though I'm not sure how useful it would actually be
If you only care about the angle and don’t need to attach it to a baseplate you can make the triangles 10x smaller if you use quarter plate offsets as your smallest unit :) I’ve used that to make cat eyes and ears where the head is a sloped surface and then ”eyebrows” extend to attach to a stud on the face. Looks kinda trippy and illegal, but it isn’t :)
This absolutely goes way beyond my understanding 🙈, but I'm proud to say I own that first retired piece! 😇 Two white ones. They come from the slide in the Paradisa playground. Maybe I'll find other use for them now!
1:55 the fencepost problem aka the off-by-one error. If you're connecting the studs, the corner is at the middle of the stud, so the last half stud length on either side of the plate is not counted in the side, meaning you need a piece 1 stud longer than you would think. Edit: I literally could've watched 5 seconds longer and he would've gotten to this, lol
Using the tubes on the bottom of tiles is a legal technique. It’s not particularly common but it is used. The Minecraft sets for example use it for fence posts.
I recently saw an article about that yellow piece because I'm trying to do a LEGO version of my university's smaller theatre space. The seats are on platforms, and there's a slight angle that I have no idea how to translate into LEGO. I just don't know how to make the floor smooth on top. (For the bottom, I'd do tiles and occasional jumpers where they line up, a la the triangles shown here.) Once I figure that out, it's just a matter of doing walls and a backstage area.
It would be an interesting demonstration / art piece to demonstrate as many diagonals coming from the same origin as you can, fanning out from 0 to 45 to 90
would there be a way to figure out where jumper plates can go underneath the hypotenuse similar to the way you can find spots for regular studs at 7:39?
I cannot understand why 6044 is retired. It's such an obviously useful piece and provides so much build potential. It's the kind of piece that I would have LOVED easy access to as a child. It's one of those pieces that frankly should never be retired. It's just too obviously useful.
Wouldn't you be able to scale them down 2 times tho? Because lego plates have this alternating pattern of convex and concave antistuds, right? So if the supporting studs have holes in them then a plate can be attached between 2 upper studs
If i have two 90 degree angles and i go 12 in one direction and 5 in the other on the fist angle and the other one mirrors it. then i connect the ends using two of the infamous 45 degree angle peices such that the longer side meet in a 45 and the shorter sides meet in a 135 is that legal. Edit: Google search seams to suggest the angles should be: 22.6 and 67.4 Which adds up to 45.2 and 134.8 when doubled not sure if thats a fatal amount to be off by.
Is there a way I can share some images of some relevant techniques with you? There are some ways to accomplish some non-Pythagorean angles that are also "legal". Very useful for attaching Winter Village buildings to a base plates at odd angles.
With the close-enough approximations that aren't quite exact triples - be careful with those because the mathematical errors add-up over distance, so they can't be extended indefinitely. Conversely, approximations that work well over longer distances can't handle mid-point attachments. Only exact Pythagorean triples are totally worry-free and compromise-free.
Idk how much this will help you guys build, but there is a formula to generate triples. As you know, a^2 + b^2 = c^2. But here's the neat thing. You can substitute (m^2 - n^2) for A, (2mn) for B, and (m^2 + n^2) for C. With this, any value of M or N applies as long as M is greater than N. With N=1 and M=2, you get the famous 3-4-5. If this is of use to you, you're welcome.
Does this only work when placingg down a 1-by-X (one wide piece), or is there a way to attach entire large plates at angles on top of other plates with connections spread across the entire plane, not just a line?
Very good question, yes you can! If you rotate a large plate by a certain amount around some stud which I’ll call (0,0), then the stud that used to be at coordinates (2,1) will now be at (1,2), and a whole bunch of other studs will line up too (they form a big square lattice, exactly one fifth of the studs will line up). This includes (1, -2) going to (2, -1), but more surprisingly, (5,0) will go to (4,3). In other words, this is exactly the 3,4,5 triple angle. Something similar happens with every triple a, b, c (in simplest form), if you rotate a big plate by that angle, then 1/c of the studs will line up
5, 5 and 7 are pretty close to a right triangle, because 5^2 + 5^2 is 50, and 7^2 is 49. By the way, did you use Pythagorean theorem to check your numbers, or just used what fits?
I want to see a contest for least illegal technique. The 12 12 17 triangle is only off by 0.17%. But I bet you could do better. From what I can find, legos are supposed to have a tolerance of 5 micrometers. So if you can find an illegal technique that's that close, is it legal?
The 45-degree triangle is 12-12-17 (or 13-13-18 in LEGO studs), not 12-12-18 as stated. sqrt(12^2+12^2) = 16.97056275. Also, counting the studs in your video, the plates are 13-13-18. Also, if the diagonal piece were 19 long, the middle would be directly under a stud, not half way in between.
Yeah I think the problem he's detailing is that the outer edge would be correct but the way he did it since the studs are a certain distance away from each other technically the bottom line is only two and the side is only three so it doesn't work he needs to go to 6:00 and 4:00 and then he can make a seven connect
Yes, and checking that this is a triple for any x and y is a straightforward algebra exercise. But more shockingly, you get EVERY primitive Pythagorean triple this way.
There's always the wedge A plate (15706) for the perfect 45 degree angle.
Yes yes yes
Thank you for bringing this to his attention! We would hate for a piece like that to slip under his radar!
Thank you thank you thank you! I've been wanting to do more 45 degree angles in my builds in bricklink studio but it doesn't like angles much when rotating things, I forgot all about 15706 because I almost never see it in use, this will REALLY help me!
√(2²+5²+14²)=15 is really neat in 3 dimensions. I discovered it while building my 30 stud diameter sphere. Now thanks to this - let's call it Euclidean quadruple - some of the pieces on the surface are placed perfectly accurate.
You are far beyond my brain lol
'Pythagorean quadruple' is a thing, according to Wikipedia. Some other simple ones include √(1²+2²+2²)=3, √(2²+3²+6²)=7, √(4²+4²+7²)=9, and √(1²+4²+8²)=9
The original pythagorean triples video was the first of your videos I watched. It's great to see the progress you've made since then.
Thanks! I felt I needed to share it with more subscribers now that the channel has grown so much.
Fascinating as always. Talking my language. Math, Lego and a little bit of one of my favorite carpentry disciplines: fence work.
Happy fanksgibbing, Chris and family!
Thank you! Happy Thanksgiving to you as well!
You can add more supports to your diagonals by using jumper plates. Remember that each triple can not only be doubled, but also halved.
I need to experiment with that
12:24 in this case the correct length would be 12.021 "fences": 0.021 fences are 0.167mm, while the gap between two bricks placed on a grid is 0.4mm. I call it legal.
Right on!
Hey, I've actually used one of these! There's a spot in my city that has one of those 1x8x2 railings. It has two connections on the bottom that have 4 studs in between, so they're 6 apart, meaning I was able to do the 3-4-5 triple to put the fence at the perfect angle for my build! Mostly I'm glad to know that that's a mathematical principle and not going to damage my pieces or something. :P
That's awesome! It's really handy to know when building!
Absolutely genius video man. Great job! I literally was thinking the other day about this exact thing and if there was math to prove that angled pieces placed like that were legal. Thanks for answering my question!
Awesome, thank you!
Very instructive video imo, thank you.
Glad it was helpful! Thank you
*I never knew 5-longs existed...*
Something to think about. Diagonals of a plate are equal.
Take two 2x4 plates. Place two 1x1 plates, one on each corner of the diagonal of one 2x4 plate and then attach the second 2x4 plate on top, but use the other diagonal. I hope that makes sense.
You can use this with any plate, as long as the sides are of different lengths. Doing this in a square will just rotate the part 90 degrees. You don't even have to use the corners of the plate.
Take a baseplate and put the 1x1 plates in such a way that they are the diagonal of a 7x13 rectangle. Now you can place a 8x16 plate on top at an angle, using the other diagonal of the 7x13 rectangle.
Again, I hope this makes sense, it's easier to show than to describe.
Omg that is genius! I'm going to play around with this and it will likely make a great video!!! Thanks!
@bricksculpt You're welcome.
I accidentally discoverde that technique back in 1991, when I was building a starfighter. I didn't even realise what I had done until a few weeks later.
Small addition to the original comment.
If you place two 1x1 plates on the diagonal of a rectangle, you can rotate that rectangle (and the diagonal) 90 degrees around its center and you will have two more connection points. There should also be a connection point at the center of the diagonal. Where that connection point is would depend on the original rectangle used.
Have fun discovering the possibilities.
This opened a huge habit hole for me. Big video about it coming tomorrow!!! Thank you so much!!!
I’ve done a mock with a mills plated section at 45 degrees in the middle. The other plates sit up to it and are tiled off using the 2x2 triangle tiles. Works really well
Well done and, yes, very useful. Thanks.
Glad it was helpful!
THAT PIECE NEEDS A COME BACK. Its SO much easier for kids to build roofs with it!
I agree!
I love that piece. I’ve used it for the leading and trailing face of an EMU (train) I designed.
Hey, you're gonna like this. I'm putting up pegboard to help me organize my stuff, right. I've got a panel of it 8 inches wide and some of those 10 hole wide tool hangers with six sets of double rings. I was thinking about this video and how i could put a hanger at an angle for certain things and since there's some leeway with the holes in the pegboard, it totally works in a couple of configurations. And it's a Pythagorean triplet. I've got one set up with things like a pair of scissors, a telescoping back scratcher, a telescoping alligator clip and a screwdriver. Just thought I'd share this because you gotta be proud of yourself for the inspiration. Pat yourself on the back or something. And since it's still thanksgiving.....
Thank you!👽
That's awesome. I never thought of it for Pegboard, but yes pegboard is basically the same as the Lego grid!
You can also half the size of a triple by using a jumper. So 3x4x5 triple becomes 1.5x2x2.5. with the "fencepost" stud, that is 2stud plus jumper, 3 stud, and then a 1x5 plate on the hypotenuse where the hollow stud of jumper connects between the last two studs.
We can use half lengths as well (1.5, 2, 2.5).
Yes, that I'd true but my math skills aren't on that level lol
@@bricksculpt
I saw the trick you did with the underside-pole and hollow stud. That is a half length. You already figured it out. This is not above your level! I believe this would be a valuable addition to your pythagorian series.
It is the same scaling that you had done. (3, 4, 5) can be reduced (halved) to (1.5, 2, 2.5). And then that can be multiplied by like 3.to give us (4.5, 6, 7.5). Anything with a .5 can be attached with a jumper or underside pole. Some of those unfortunately large triangles can be halved with the use of jumpers.
@@bricksculptJust take any pythagorean triple and half each side length. 3:4:5 becomes 1.5:2:2.5
I'm going to have to try this
For studs on slopes, technic bricks can be a solution, if the slope's location allows for it.
Of course you need to do the mathematics in plate heights, and divide the square root of the hypotenuse by 2.5 to get back to brick distances.
In my experience, anything below 0.01 brick-width difference from a full brick is easily within LEGO tolerances (i.e. an axle will go through as if it was a single hole), and up to ~0.05 brick-width difference can be used without putting to much strain on the pieces.
Yeah I hadn't thought of that
Not sure if you've realized this, but those 45° solutions, at the end, should also be solutions for the square root of 2. The diagonal of a 1x1 square is root 2, & that cuts perfectly through the middle, meaning it cuts 90° angles in half, or into two 45° angles. You were missing those half stud offsets.
Ah yes, BrickSculpt never fails to connect Lego the fundamentals of physics/engineering.
Much appreciated as always :)
Thank you!
There are a few small near-triples which result in small gaps when you build them. I like this one: 4 - 7 - 8(.0623), which with jumpers you can make even smaller at 2 - 3.5 - 4-ish.
This has an angle close to 30° as well, for those nice hexagons...
Bro just casually explained Pythagoras triples better than any math teacher could dream
Lol thanks!
@haha no thank you! My head works better with visuals; and your 'demonstration' of the 'fence post problem' was legitimately the best example I've ever seen, simple but straight to the point and logical. Well done :)
Previous comments got automodded by adding a picture link, so I threw them in the discord, but you can get even more triangles using SNOT bricks and brackets (half plate thicknesses), even found the right angle to mate with 2x3 wedge plates though I'm not sure how useful it would actually be
If you only care about the angle and don’t need to attach it to a baseplate you can make the triangles 10x smaller if you use quarter plate offsets as your smallest unit :)
I’ve used that to make cat eyes and ears where the head is a sloped surface and then ”eyebrows” extend to attach to a stud on the face. Looks kinda trippy and illegal, but it isn’t :)
This absolutely goes way beyond my understanding 🙈, but I'm proud to say I own that first retired piece! 😇 Two white ones. They come from the slide in the Paradisa playground. Maybe I'll find other use for them now!
1:55 the fencepost problem aka the off-by-one error. If you're connecting the studs, the corner is at the middle of the stud, so the last half stud length on either side of the plate is not counted in the side, meaning you need a piece 1 stud longer than you would think.
Edit: I literally could've watched 5 seconds longer and he would've gotten to this, lol
Lol love it!
Using the tubes on the bottom of tiles is a legal technique. It’s not particularly common but it is used. The Minecraft sets for example use it for fence posts.
I recently saw an article about that yellow piece because I'm trying to do a LEGO version of my university's smaller theatre space. The seats are on platforms, and there's a slight angle that I have no idea how to translate into LEGO. I just don't know how to make the floor smooth on top. (For the bottom, I'd do tiles and occasional jumpers where they line up, a la the triangles shown here.) Once I figure that out, it's just a matter of doing walls and a backstage area.
You learn to start counting with zero writing code! Cool video!
Thanks for watching!
the curve piece you use right in between the B and S of your big 1x2 brick is genius
Thanks to @lucahermann!
th-cam.com/video/ZRPGpweWg7E/w-d-xo.html
It would be an interesting demonstration / art piece to demonstrate as many diagonals coming from the same origin as you can, fanning out from 0 to 45 to 90
Stay tuned my next video is very similar to that.
7^2 + 7^2 = 49 + 49 = 98 ... Root 98 is almost 10. So 7-7-10 does also work for a 45 degree angle without stressing the pieces too much.
My first thought: A perfect 45/45/90 can be created by letting a 1x2 jumper plate be slightly offset on both ends.
Im glad it was mentioned.
would there be a way to figure out where jumper plates can go underneath the hypotenuse similar to the way you can find spots for regular studs at 7:39?
I need to experiment with that still
Fun fact: Part 79846 fits that 12x12x18 triangle at a right angle from all the bricks.
This is basically Numberphile "Impossible Squares" but in Lego
“I don’t think this is illegal enough to matter.” - I say that all the time!
Lol
I cannot understand why 6044 is retired. It's such an obviously useful piece and provides so much build potential. It's the kind of piece that I would have LOVED easy access to as a child. It's one of those pieces that frankly should never be retired. It's just too obviously useful.
I agree great piece!
Wouldn't you be able to scale them down 2 times tho? Because lego plates have this alternating pattern of convex and concave antistuds, right? So if the supporting studs have holes in them then a plate can be attached between 2 upper studs
I will have to test that
Could you do intermediate triples that work with jumper plates?
I need to try that next
Is it possible to use hinch pieces at the corners ?
Take a look at the instructions for the UCS TIE Interceptor. LEGO uses hinge plates to place plates on the sloped sides of the wedges for the wings.
Yes in certain angles but it's not easy
If i have two 90 degree angles and i go 12 in one direction and 5 in the other on the fist angle and the other one mirrors it.
then i connect the ends using two of the infamous 45 degree angle peices such that the longer side meet in a 45 and the shorter sides meet in a 135 is that legal.
Edit: Google search seams to suggest the angles should be:
22.6 and 67.4
Which adds up to 45.2 and 134.8 when doubled not sure if thats a fatal amount to be off by.
I would have to test it I'm not sure im following the layout.
3D print the missing pieces? Or is that "illegal"? Or build in Modulex?
If you don't mind building illegal, and you want 30° and 60°, you can use 4, 7, 8. Perfect for Hexagons
Is there a way I can share some images of some relevant techniques with you?
There are some ways to accomplish some non-Pythagorean angles that are also "legal".
Very useful for attaching Winter Village buildings to a base plates at odd angles.
Either Discord if you have an account or go to his channel info there's his mail.
Yes email me at bricksculpt.chris@gmail.com
Or post the on the Bricksculpt Discord Server.
With the close-enough approximations that aren't quite exact triples - be careful with those because the mathematical errors add-up over distance, so they can't be extended indefinitely. Conversely, approximations that work well over longer distances can't handle mid-point attachments. Only exact Pythagorean triples are totally worry-free and compromise-free.
Yes that's a good point!
Idk how much this will help you guys build, but there is a formula to generate triples. As you know, a^2 + b^2 = c^2. But here's the neat thing. You can substitute (m^2 - n^2) for A, (2mn) for B, and (m^2 + n^2) for C. With this, any value of M or N applies as long as M is greater than N. With N=1 and M=2, you get the famous 3-4-5. If this is of use to you, you're welcome.
It is easy to check that this is always a triple, the cool and shocking thing to me is that ALL (primitive) triples can be generated this way.
Does this only work when placingg down a 1-by-X (one wide piece), or is there a way to attach entire large plates at angles on top of other plates with connections spread across the entire plane, not just a line?
Not sure I haven't really tried that. Might be worth investigating.
Very good question, yes you can! If you rotate a large plate by a certain amount around some stud which I’ll call (0,0), then the stud that used to be at coordinates (2,1) will now be at (1,2), and a whole bunch of other studs will line up too (they form a big square lattice, exactly one fifth of the studs will line up). This includes (1, -2) going to (2, -1), but more surprisingly, (5,0) will go to (4,3). In other words, this is exactly the 3,4,5 triple angle.
Something similar happens with every triple a, b, c (in simplest form), if you rotate a big plate by that angle, then 1/c of the studs will line up
@@bricksculpt You absolutely can! If you build a plate at a Pythagorean angle, there are studs in every single row that line up with studs below.
Video on this coming tomorrow
@@bricksculpt wow! Thanks a bunch!
Another day of hating the irrationality of sqrt(2)
I agree
I've used a lot of 5, 12, 13
5, 5 and 7 are pretty close to a right triangle, because 5^2 + 5^2 is 50, and 7^2 is 49.
By the way, did you use Pythagorean theorem to check your numbers, or just used what fits?
I did in my original video and it's close. It's hard to get really accurate meslasurments when measuring from the center of studs.
I want to see a contest for least illegal technique. The 12 12 17 triangle is only off by 0.17%. But I bet you could do better.
From what I can find, legos are supposed to have a tolerance of 5 micrometers. So if you can find an illegal technique that's that close, is it legal?
That would be a good challenge
Sqrt(12^2+12^2) = 16.97
There's a good chance that's right on the edge of Lego tolerances.
especially as we're building with rectangles rather than straight lines, i would suspect that 12, 12, 17 is perfectly legal
The 45-degree triangle is 12-12-17 (or 13-13-18 in LEGO studs), not 12-12-18 as stated. sqrt(12^2+12^2) = 16.97056275. Also, counting the studs in your video, the plates are 13-13-18. Also, if the diagonal piece were 19 long, the middle would be directly under a stud, not half way in between.
I literally made a 3 4 5 triangle with legos a few hours ago… how
TH-cam is watching you :P
If you are just using the tube/jumber position on the underside of a plate, LEGO does not consider it illegal
I was calling the length illegal, not the attachment.
You forgot you live in 3D
Yeah I think the problem he's detailing is that the outer edge would be correct but the way he did it since the studs are a certain distance away from each other technically the bottom line is only two and the side is only three so it doesn't work he needs to go to 6:00 and 4:00 and then he can make a seven connect
345 method in building
2:43 > "though nobody actually says zero"
yes you do, talk to a programmer lmao
Yeah it's a funny concept
To generate a pythagorean triple, take two integer numbers x and y, and compute the three values
x²-y²
2xy
x²+y²
They will form a triple.
Yes, and checking that this is a triple for any x and y is a straightforward algebra exercise. But more shockingly, you get EVERY primitive Pythagorean triple this way.