LEGO Theory Pythagorean Triples. Methods For Building At Angles.
Vložit
- čas přidán 17. 03. 2024
- Hello and welcome to Brick Sculpt! This exciting video is going to show you a few different methods (and the math behind them) for attaching LEGO pieces at angles. I hope you enjoy this video and if you have any questions, comments, or suggestions please fell free to ask in the comments. Thanks for Watching!!!
#lego #legomoc #legocity
It's really cool to see a video like this not from a mathematician but from someone who just discovered Pythagorean Triangles.
That's the best comment I have gotten yet. I like this so much because it really shows what I am trying to do with this channel. I'm just a AFOL who wants to learn more and more about this stuff by showing what I know and getting feedback. I am not the expert, but rather I am just starting a conversation with the community, hoping that we can all learn from each other. Thanks for sharing this!
12x12+12x12=16.97x16.97. The 18 stud long plate is 17 units in length. So, it’s really close. So, technically illegal, but 0.17% off, so probably close enough that you won’t damage the bricks.
So you vote illegal!
It's worth considering the tolerances of LEGO parts. I can't find an official statement from LEGO on it, but it looks to be around 0.01mm, with most blocks hitting 0.001mm or so in practice. Unfortunately, a lego-unit error of 0.03 translates to 0.24mm of drift, which is quite far outside the approved range. This likely will result in increased wear on the parts, but isn't enough to cause immediate failure.
29x29x41 might be the safer option, and would fit on a standard MILs plate
@@acshephard That has an error of 0.097mm. Since half the error will show at each end, that comes out to about 0.05mm on each end, still outside the spec but only by about 5x instead of 12x.
This is the measurement that makes an official home plate in baseball a mathematical impossibility!
The square root of two is an irrational number, so it's literally impossible for there to be a perfect 90-45-45 triangle if you're constrained to integer length sides. However, if you scale it up enough, you can get close enough to be within the engineering tolerance of the Lego bricks, so it will "work". That's what's happening with your triangle: 12 x 12 + 12 x 12 = 288, and the square of 17 is 289, so it's off by a bit more than a third of a percent, and that's "close enough" to fit.
Yeah that's the conclusion I have come to after reading everyone's input. Thanks for watching!
The reason you need to "add 1" to each side is because each of the triangle you are trying to create meets at the center of each stud. There's a triangle on the inside edge, another on the outer edge, but triangle you are making is from the centerline.
👍
While it is impossible to construct a 45-45-90 triangle with all integer side lengths, it is possible to turn any pythagorian triple into a type of isosceles triangle by reflecting the triangle around one of its legs and adding the reflection to the original triangle.
I like that, would be interesting to see how it would be applied to a moc in a practical sense.
The LEGO Creator Expert Corner Garage 10264 uses a (12, 12, 17) triple for its wall that is at a 45° angle!
Nice find! So maybe it is legal!
12,12,17 is close.... it's 0.24mm short of perfect. But 0.24mm is an awfully small variation across the 136 mm of your 17 side.
@@bricksculptin 10264 LEGO makes it legal by using 2-stud bricks with round corners to link a 16-stud wall to the corners.
The round corners allow an angel to shorten the distance between the corner studs of the wall (by 0.12mm) to 16.97 studs c-c, and not 17 studs c-c as if they used a straight 18 stud wall.
If you look up the instructions online, available from LEGO as pdf, you can clearly see that the round 2-stud pieces linking the walls at 45 degree angles are not at 45 degrees themselves.
@@mattadulting I get that the diagonal is just 0.1177 mm shorter, but it's still illegal.
But LEGO solved it in 10264 by linking the corners using 2-stud round bricks.
@@bricksculpt after looking at the 10264 instructions, I kind of see them as proof that building a 45 degree wall requires a technique to compensate for the difference, to make it legal.
These are some useful tips. I have some more triangles that may be useful.
Regarding Pythagorean triples;
A 3, 4, 5 triangle has angles of 53.38 and 36.62. The Lego equivalent lengths are 4, 5, and 6 studs.
A 5, 12, 13 triangle has angles of 25.36 and 64.64. The Lego equivalent lengths are 6, 13, and 14 studs.
A 8, 15, 17 triangle has angles of 33.83 and 56.17. The Lego equivalent lengths are 9, 16, and 18 studs.
A 9, 12, 15 triangle has angles of 53.38 and 36.62. The Lego equivalent lengths are 10, 13, and 16 studs.
A 7, 24, 25 triangle has angles of 17.20 and 72.80. The Lego equivalent lengths are 8, 25, and 26 studs.
A 9, 40, 41 triangle has angles of 89.23 and 0.77. The Lego equivalent lengths are 13, 13, and 42 studs.
A 12, 35, 37 triangle has angles of 20.45 and 69.55. The Lego equivalent lengths are 13, 36, and 38 studs.
A 15, 36, 39 triangle has angles of 25.36 and 64.64. The Lego equivalent lengths are 16, 37, and 40 studs.
Regarding 90 degree isosceles triangles;
The long side of a 5, 5, 7 triangle is off by 0.07 units. This means a triangle of 6, 6, 8 counting studs in Lego.
The long side of a 12, 12, 17 triangle is off by 0.03 units. This means a triangle of 13, 13, 18 counting studs in Lego.
The long side of a 17, 17, 24 triangle is off by 0.04 units. This means a triangle of 18, 18, 25 counting studs in Lego.
The long side of a 24, 24, 34 triangle is off by 0.06 units. This means a triangle of 25, 25, 35 counting studs in Lego.
The long side of a 29, 29, 41 triangle is off by 0.01 units. This means a triangle of 30, 30, 42 counting studs in Lego.
The long side of a 41 , 41 , 58 triangle is off by 0.02 units. This means a triangle of 42, 42, 59 counting studs in Lego.
Dang lots of good stuff there!
I reckon one could still make use of the other Pythagorean triplets, but it wiuld require use of jumpers since this let's you access half units. take for example [35,12,37] and turning it into [17.5,6,18.5] to get an angle of about 18.9°. Obviously this would require making plates of non unit length, but that can easily be created with the use of jumpers, although the construction might not be super stable
Yeah it might be something to look into for a future video. Thanks for watching
9:24 an isosceles triangle is two sides of equal length it doesn’t matter if it has a right angle or not but if it does then it is a right angled isosceles triangle ❤❤❤❤
Thanks for the correction
Wow this is maybe one of the most important MOC videos I’ve ever seen.
Thank you so.much for the great compliment you're the best!
Note, another brilliant one was found in Thors house build
You have a 2x4 plate. If you put two studs on opposing corners, you can put another 2x4 at an angle
I will have to check that out
@@bricksculpt Pretty much works for any plate
Yes because you are matching the diagonals on both bricks. You are putting the diagonal from up left to down right on top of down left to top right.
the general formula for the angle of 2 A×B plates (A is the longer side) is 2arctan(B-1/A-1)
angles with small plates:
- 6×2: 22.62°
- 5×2: 28.07°
- 4×2: 36.87°
- 6×3: 43.6°
- 3×2 and 5×3: 53.13°
- 6×4: 61.93°
- 4×3: 67.38°
- 5×4: 73.74°
- 6×5: 77.32°
a 13×6 setup is very close to 45°, only about 0.24° off
The diagonal of a 45-45-90 triangle is square root of 2, or about 1.41 given the length of a leg is 1. So one module is Lego speak for a 8 mm centre-to-centre distance. The diagonal must then be 11.31 mm (rounded number, of course). Thank you for making this video, I find it highly instructive.
Thank you. And thanks for the input!
I was just working out a lot of the same math this weekend! The new Hagrid's Hut set actually uses a slight variant on the 3-4-5 triangle you showed, just using hinges to create the angles between plates instead of stacking a plate on the studs. It makes for a very nice way of having stable octagonal walls, and I'm using some similar techniques for a windmill MOC for my medieval village. I ended up doing a bit of extra math and found some neat ways to further expand the footprint with wedge plates too. 2x2 and 3x2 wedge plates perfectly meet up to expand out the 53/37 degree angles from the 3-4-5 triangles.
Awesome input. I will likely be doing a similar video with hinges soon!
You are trying to make a 45-45-90 triangle, whose sides have ratio 1,1,root 2. The square root of 2 is irrational, so there is not perfect fit.
Any number can be expressed as a continued fraction, and these give the best estimate for numbers less than a given denominator.
For the square root of 2 we get that if p/q is an estimate (p+2q)/(p+q) is a better estimate.
Starting from 1/1 we get 3/2 then 7/5 then 17/12 then 41/29 then 99/70 and so on.
This corresponds to Lego pieces with studs 2/2, 4/3, 8/6, 18/13, 42/30 and 100/71 respectively.
The first three are not close enough for the fit to even be forced. You showed that 18/13 is close enough to force, but may not be within the tolerances of Lego. 42/30 would work better but is getting big. 100/71 would almost certainly be within official tolerances.
There is also the 20,21,29 Pythagorean triple that is nearly a 45 degree angle, or a 21,22,30 stud length triangle.
Given any n and m there is a 2mn, (n^2-m^2), (n^2+m^2) Pythagorean triple.
You know math better than I do. Thanks for the input.
Geometry nerd + Lego nerd = Amazing video. Thanks!!!
Thank you! Unfortunately math isn't my strong suit but I'm glad it worked out.
you can actually think of the stud grid as being twice as dense and use twice as large of numbers by using jumper plates
I'm twice as dense as most people doing LEGO math 😂
You might be able to use the half stud trick to scale down larger Pythagorean triplets to a more manageable size
Possible I would have to try it
You can also do it with hinged pieces, you just measure by the actual stud length of the piece, and you don’t have to account for the sharp corner getting the way of the adjacent studs. The outer corners will have a gap though, but I feel like it’s generally more practical building that way.
I'm going to be doing a hinge video soon. Thanks!
Great video, Ive resorted to using the specialized 45 degree brackets, moc builders have used hinges and lego has used 2x2 turntables to achieve the 45 angle in the past. Lego makes so many angled wedge plates but there are only a few ways to make it flush with anything
Yeah 45 degree angles always cause trouble.
As for the 45° angle: There are no Pythagorean triples a^2+b^2=c^2 where a=b. The proof basically boils down to a^2+a^2=c^2 => 2*a^2 = c^2 => sqrt(2)*a = c. Sqrt(2) is irrational - meaning there is no fraction that can represent it - thus there is no scaling factor that will make a and c integers at the same time. You found one that is close enough so that within the Lego tolerances it is a fit. 🙂
If you want to build at a 45 degree angle, you can also compromise on angle instead of exact stud matching. The 5-12-13 triangle has a corner that’s only slightly larger than half of 45 degrees, so if you build another 5-12-13 triangle on top of it, the hypotenuse of that will be less than a quarter of a degree in excess of 45 degrees.
That's a cool idea. I would be interested in seeing that technique built out.
I think you should consider the stud pitch to be 8mm, saves allot of accumulated error from conversion(not which is better metric or standard, just that this is how lego was designed)
Fair point
Wow thanks so much for this Lego Math lesson. The fact you were albe to use actual geometry with Lego blew my mind. Freaking brilliant!
You're very welcome!
this would be great for a third grade math class on angles.
I remember in third grade memorizing math tables, but kids will remember this because it's LEGO bricks.
I would have listened in school if it was all LEGO related.
I love videos about techniques exactly like this. I hope you make more ❤
Thank you! Will do!
@@bricksculpt That's wonderful news. Thanks for the reply!!
Amazing video, this is what I've been looking for for a long time
Glad you liked it. I have been looking for this info also so I figured I would just work it out myself and make a video.
The reason the isosceles triangle works is because 12 root 2 is approx 16.97. Apparently 0.03 lego units is within the tolerance of lego brick connections.
👍
Very nice video. Thanks for the explanation
Thanks for watching
Looks legal to me and the math works, too. I like it.
A vote for legal!
a^2+a^2 = (sqrt(2)*a) -> a isosceles right triangle has a longer side, which has a lengh of square root of 2 times the lengh of the shorter side. That means that both the long side and short side can‘t have a hole number of units in lengh at the same time. But it gets pretty close, when the short side has a lengh of 12. then the long side is 12*sqrt(2) which is close to 16,97.
👍
With Jumper-plates you can get it to half the size as you can get half-a-stud of distance.
I need to experiment with this!
I believe these are the different options you've presented (the final stud length in each example is the hypotenuse, or diagonal):
- 3,4,5 Pythagorean triple = 4x5x6 studs
- 5,12,13 Pythagorean triple = 6x13x14 studs
- isosceles #1 = 12x12x18 studs
- isosceles #2 = 8.5x8.5x13 studs
- isosceles #3 = 6x6x8.5 studs
I think you summarized it nicely.
So great wish i had class like this for teaching maths.....GREAT CONTENT ! Thanx
Glad you liked it, thanks!
You have hit upon something very close to a triple with 12, 12, 17. 12X12 + 12X12 = 288, and 17 X 17 = 289. So, we are close, but miss it slightly as A^2 + B^2 misses C^2 by 1.
In effect: For a right triangle, the sides are either going to be 12, 12, 16.97056275 or 12.02081528, 12.02081528, 17.
And, for a triangle of sides 12, 12, 17 the angles are not 45, 45, 90. They are 44.900528, 44.900528, 90.198944.
It's close, but it will never be perfect because a right isocoles triangle will always have one a*square root of 2 side
Yeah I have come to that conclusion
@@midtisel4222 Correct, but we can get extremely close. The 12, 12, 17 triangle is basically the 4th such near triple. Using the definition A^2 + B^2 = C^2 +/- 1. So, 12^2 + 12^2 is 288. 17^2 is 289 (either 1 more or less than the sum of squares of the other 2). So not a perfect A^2 + B^2 = C^2, but the next best thing. Consider the seventh such triangle of 169, 169, 239. A true right triangle would be either 169, 169, 239.002092 or 168.9985207, 168.9985207, 239. The angles of a 169, 169, 239 triangle are 45.00050152, 45.00050152, and 89.99899696.
No, the 45° triangle is not legal.
45° triangles have a side length ratio of 1:1:sqrt(2), and sqrt(2) is an irrational number, meaning you can not use whole number ratios to get it. 12*sqrt(2)=16.9705, which is really close, but not perfect. The error is about 0,18%, so you will notice if you make a really large triangle, it won't fit.
Yes I have come to the same conclusion after making this video.
5-5-7 is a fairly close enough small right triangle. 5^2 + 5^2 ≈ 7^2, which is 25 + 25 = 50 ≈ 49. Still, with all the 45 degree examples, since it will be slightly off, at some point in greater scales, the lego will have to bend or stretch past its limits, but they should be fine at smaller scales.
I will have to experiment with this
@@bricksculpt This is actually backward. 5-7-5 is off by about 1%, 12-17-12 is off by about 0.2%. At larger scales you're more likely to find something that's "close enough" to work.
There is no way to make a fully legal 45 degree connection with this technique, at least not when using standard pieces.
This is because if you have an isosceles right angle triangle, the length of the hypothenuse is always going to be the length of individual leg multiplied by square root of two, which is an irrational number - i.e. it cannot be represented as fraction of whole numbers.
And since this connection is entirely based on rational numbers, the 45 degree connection cannot be legal. You can get close enough that the tolerances of the pieces accept it, but it's not legal.
For example, your proposed 12 by 12 triangle connected by diagonal line of length 17 gives a rational fraction of 17/12 = 1.41666... while the square root of two is approximately 1.41421356... which means the diagonal piece is slightly longer than the actual length it's supposed to fit in, and therefore illegal.
That doesn't really mean it can't be used, because it's possible to possible to find numbers which form a very close approximation of square root of two. With a very close approximation of square root of two, there will be minimal stress (tension or compression) on the pieces.
Here are the closest matches for leg lengths between 1-50, with difference from ideal representing the "stress" of the illegal connection (lower is better):
29 and 41 (differs by 0.02973%)
41 and 58 (differs by 0.02974%)
46 and 65 (differs by 0.08274%)
17 and 24 (differs by 0.1731603%) (same as 24 and 34)
12 and 17 (differs by 0.1734607%)
If we want to get closer ratios than this, the lengths of the legs become really rather impractical.
If we allow half-lengths, then these lengths can be divided by two, and that can translate into the following values:
14.5 and 20.5
20.5 and 29
23 and 32.5
8.5 and 12
6 and 8.5
Of these ratios I would say the 17/12 (or 8.5/6) is the most practical due to the small lengths it allows. If you have the space, you can significantly reduce the stress of the connection further by moving to 41/29 (or 20.5/14.5).
Thank you for adding this.
The trip-triangle at its normal proportions is on the inside measurement for anyone wondering (as the bricks are one unit thick)
Yes good point!
I’d pay good money for a book which covers all the various LEGO math techniques so I don’t have to muddle for an age.
You and me both!
Both of these allow the hypotenuse in one of 8 orientations.
👍
I think anything is a legitimate building technique if it works and looks good. Purism is for the people that have more money than sense.
I like your approach and agree!
There are certain angles that will never be possible, an Isosceles triangle will never be possible because you'll always be dealing with a square root of 2.
👍
"Lego is all in right angles" **Laughs in part # 6054852** Srsly though, great video man, definitely going to save this to come back to next time I build at an angle
Thank you so much! 6054852 might deserve it's own video!
The reason a triangle with the two sides connected to right angle being equal cant work is because the length of the third side will be a multiple of the square root of 2, which is, as everyone knows, is an irrational number, similat to pi.
👍
Next level: use dots on diagonals which of grid as new atationary, to make its more complicative
👍
(1.5, 2, 2.5) is an even smaller version of the (3,4,5) triangle, that could still be useful, because you can put hollow studs in the half position at the bottom: (1 stud + jumper plate = 1.5; 3 studs = 2; 3.5 studs on the bottom of a plate = 2.5).
This is also the explanation for the 18 plate, that is 17 in length or 2*8.5. Because 12*sqrt(2) = 16.97 which is apparently close enough to 17.
Thanks for your input!
That whole explanation about why the 12×12×17 worked about the 17 being in a different measurement unit is wrong. It's because 17/12 = 1.416666... which is very close to √2 (1.414...) which is the ratio the two sides should be for Pythagoras theorem to hold.
a²+a² = c²
2a² = c²
√(2a²) = √c²
√(2)|a| = |c|
√2 = |c|/|a| (since c and a are lengths, we can ignore the negative option)
√2 = c/a
Now, if you choose a c and and a that are close enough to their ratio being √2, it'll work. 12 and 17 work very well because it's only got an error of 0.18%, probably smaller than Lego's own error when fabricating a piece.
Yes I gathered that after making the video 😁 thanks for watching!
How would the stability change, if you also use 1x2 and 2x2 jumper plates? Would there be more attachments?
Not sure. I haven't explored that deep into it.
To being with we have the fence post problem.
I looked up the fence post problem and you are correct! That is exactly what I did wrong.
I'm pretty sure the 45 degree one doesn't work. I tried doing something very similar. Not with the math but recognizing that a an off-sett stud, like when you use the jumper plates, could seemingly work for a 45 degree angle I was wanting. I used a plate that was 2 studs wide so maybe that was somehow a factor but it definitely was being stressed.
I have co.e to the conclusion it us illegal but when pitting it on it was not difficult and felt very little if any stress.
For a right traingle with angles of 45 degress, that means A=1 B=1 C=sqrt(2). If you do not properly compensate for the irrationality of root-2, the building technique is illegal. Proper compensation of the irrationality of root-2, can be done, but your method is not the way to do it.
You are correct its illegal. What method would you recommend instead? Just curious.
@@bricksculptuse 1x2 on a jumper plate, so you can rotate the connection points just a smidge closer to one another. It will no longer be a perfect triangle, like the triples, but it's the only way to get 45 degree.
That's how I've always done angles like that myself.
I’m gonna be that guy in a second…but I want to repeat all the positive comments first (great video)
Now, I should mention, for education sake, that you did give an inaccurate definition of an isosceles triangle. It does not need to be a right triangle. I’m only saying this because kids will be learning their Euclidean geometry from this video years in the future…
Thank you!
I keep meaning to experiment with angles. Presumably the different jumper plates placing studs in halfway points opens up other options, and then there's one technique I saw using hinges that give you an even shallower option.
Hinge angle video will likely be coming soon. Also possibly a jumper plate angle one (if I can figure it out).
I bet ya it’s legal. :D thanks!
Thanks for watching!
Yes, the difference between cardinal numbers and ordinal numbers. Counting studs is ordinal numbers, counting distances between studs is cardinal numbers.
Thanks for adding this, I didn't have that terminology in my brain!
This is not quite accurate. Cardinal numbers measure the size or total count of something. Ordinal numbers are used for describing sequential ordering (think 1st, 2nd, 3rd, etc.). Only cardinal numbers are relevant here.
This issue is often referred to as the fencepost problem, or simply as an off-by-one error. A 4x1 brick is 4 brick units long in total, but the distance between the centers of the first and last studs is only 3 brick units.
And we were saying geometry won't be useful in the future...
I know right!
Who the hell said something stupid like that? Geometry is probably the most useful field of math.
ok I was board and wonder how many of these triangles you can make within a 100 by 100 search area and within a 0.01 unit tolerance you get 197 different triangles!
Interesting. Nice work!
With Hinges, I created a 45 degree angle that was 13 studs long. Was my creation off?
Not sure I would need to experiment with it. I might do a hinge angle video soon.
Sqrt(2) is irrational so a 45° Pythagorien Triple can not exsist
You are correct
True, but there are some triangles that are very close. In a triangle of 12, 12, 17, the large angle is only 0.198944 degrees more than a true right angle. There are also 29, 29, 41 triangles and 70, 70, 99 triangles that are remarkably close. The 70, 70, 99 triangle has angles 44.99707675, 44.99707675, and 90.00584651.
These types of isosceles almost triples are very rare, for A^2 + B^2 = C^2 (+/-) 1, the next one after 70, 70, 99 is 169, 169, 239. The one after that is 408, 408, 577.
So I’m just gonna add to the complication, but what if you also started to add jumper plates in?
Jumpers would add a whole other level of complication. Maybe a future video. (Assuming I could figure it out)
You'd just count in half studs instead of full studs. For example, the 3;4;5 Pythagorean triangle in half studs would be 1.5;2;2.5 full studs and 5;12;13 half studs would be 2.5;6;6.5 full studs.
@@bricksculptjumper plates just means you can use triplets in 0.5 intervals. Such as 1.5, 2 ,2.5 or 4.5, 6, 7.5
It’s not that much more complicated
3blue1brown made a video about finding pythagorean triples a couple of years back: czcams.com/video/QJYmyhnaaek/video.html
So some quick excel later these are the only possible triples on a 32*32 stud board (my biggest board)
in studs
a b c smallest angle
4 5 6 38,66
6 13 14 24,78
7 9 11 37,87
8 25 26 17,74
9 16 18 29,36
10 13 16 37,57
11 25 27 23,75
13 17 21 37,41
16 21 26 37,30
19 25 31 37,23
21 22 30 43,67
I will need to watch that
Isosceles triangles aren’t necessarily right
For each length of base side on an isosceles there's an infinite number that don't have a right angle and exactly one that does.
Yea I figured that out after the video.
I’m not sure what the tolerances are for LEGO, and where things become ‘illegal’, but the 18 stud diagonal that you use has 8.5 studs from its centre to either of the end anti-studs, whereas the right-angled isosceles triangle with 12-stud sides has a hypotenuse of √(2 × 12²)/2 ≈ 8.485 studs. The difference is 0.0147 studs, which is the mismatch error at each of the end connection points, if the central connection is spot on. Since the diameter of a stud is 3/5 of the stud-to-stud spacing, this is about a 2.45% error in alignment of the stud with the anti-stud.
Just adding to that, since I hadn’t watched to the end. The calculation you do at the end will inevitably work out, no matter what right-angled triangle you mark out on a baseplate. To show that the 18-stud piece is a good fit, you should have measured *that* and compared it with the diagonal on the ground. Useful insights nonetheless.
You're math is better than mine! Lol thanks for adding that. Do you think that would make the isocelese triangle illegal?
@@bricksculpt I’m really not sure whether it’s illegal or not. It’s not an exact fit, but it could possibly be within the tolerances LEGO has decreed ‘legal’. And even if it’s slightly ‘illegal’, it could still be useful. (My own slight purist tendencies would probably lead me not to use it though, but each to their own!)
Incidentally, if you move away from requiring right angles at all, there are plenty of triangular buildings you could make, and even isosceles triangles - for example, doubling up the 3-stud side or the 4-stud side of a 3, 4, 5 triangle to give a 4, 5, 6 or 3, 5, 8 triangle. (This is basically just putting two Pythagorean right-angled triangles together to make a bigger triangle.)
I agree it is likely illegal, but I would use it anyway.
I think lego only allow exact geometry. And I would probably do the same.
There is a mistake in your définition of isoceles. Such a triangle just need to have 2 sides of the same length. No need of a 45º angle.
For other angles there are other solutions than Pythagore. For example I've seen something about reciprocal diagonals that uses hinges bricks or plates
Yeah there are definitely other ways to accomplish angles. Thanks for watching!
1:14 I think that’s like 7th grade school stuff.
3:54 do you know about a² + b² = c²
Covered it later in the video.
@@bricksculpt ok. In school we learned the formula directly so we could do the stuff better. It’s also really useful to find the shortest path between top objects. Or calculate its distance.
Just for reference, as long as |round(n*sqrt(2)) - n*sqrt(2)| < p, where n is the smaller side of the isosceles triangle and p is a given allowable precision, this should work. n=12 gives a precision of 0.029, so if we allow the max precision allowable to be 0.03 we get a few triples.
[12, 12, 17], [29, 29, 41], [41, 41, 58], [58, 58, 82], [70, 70, 99], [99, 99, 140] for n less than 100.
You could use this same method for half blocks, as well. Instead of counting full blocks, you would just need to count the half blocks as your length given for n.
Further, you could use this method for other right triangles, as long as you knew their ratios. For instance, a 3,4,5 triangle has ratios 1:sqrt(3):2, so by taking n to be your shortest side we can get a new inequality |round(n*sqrt(3)) - n*sqrt(3)| < p. Setting p = 0.03 again, we get:
[15, 26, 30], [41, 71, 82], [56, 97, 112], [71, 123, 142], [82, 142, 164], [97, 168, 194] for n less than 100.
For more complicated ratios, such as 5,12,13, you would start needing trig to figure out the ratios, and you would need to worry about two inequalities (one for each side in relation to your smallest side).
A bit more interesting, suppose you wanted a given angle, for example 40 deg of of a straight line. you would need to satisfy both |round(csc(angle)) - csc(angle)| < p and |round(cot(angle)) - cot(angle)| < p. For our example of 40 degrees and p = 0.03, we only get [99, 118, 154].
For most integer degree angles, there won't be any (where the smallest side is under 100), but just for completeness sake, here are the rest.
[1, 19, 19], [1, 28, 28], [1, 29, 29], [8, 15, 17], [8, 32, 33], [9, 39, 40], [13, 82, 83], [15, 56, 58], [17, 21, 27], [18, 30, 35], [28, 30, 41], [35, 50, 61], [37, 706, 707], [39, 80, 89], [40, 46, 61], [42, 122, 129], [43, 112, 120], [51, 322, 326], [57, 59, 82], [59, 139, 151], [61, 137, 150], [77, 106, 131], [78, 272, 283], [81, 211, 226], [83, 427, 435], [85, 126, 152], [90, 361, 372]
Following this method, I see a few issues, like the [1, n, n] triples that pop up. I assume these won't actually work, but I don't have any legos to test. I also tried to remove any multiples of standard triples, but I did this part by hand. Hope someone finds this useful!
Thanks for the input. You are far more inclined in math than I!
@@bricksculpt Always happy to work on something like this! The things people are able to build with LEGOs are crazy and I feel a little out of my depth, so if anyone who works on them finds this at least interesting than I'm happy. Keep up the awesome videos!
I can say 100% it is NOT a perfect fit no matter what lengths you pick. The diagonal length on a 45 degree isosceles right triangle is root 2 times the leg length. Root 2 is an irrational number, so no multiple of it will give an integer. Some values (like 17) are very close to a multiple though (12 x root 2 is roughly 16.971) so works given the tolerances of Lego. It may be "legal" if the construction rules allow for non-integer lengths that are within a close tolerance.
👍
Your 45 degree isn't just not a whole number in actual side lengths, it's not quite in lego either, and never will be at any size - that longer length is the square root of two times the shorter one, and that is an irrational number, and irrational numbers can never be made whole by multiplying by anything but another irrational number (or 0, technically, but that's not very helpful) - even fractional values like you get with jumpers and such are still rational numbers with no hope of making an irrational whole like that. The reason it works in practice is because while you'll never find lengths that add up there perfectly, you can find ones that get arbitrarily *close*, and in the real world, things aren't perfectly rigid and so you don't need to match perfectly and can get away with just really close. And with some achievable values the difference from the true correct length will likely even be less than the engineering tolerances, so at the very least *those* options should be considered legal, since if you didn't you'd have to count basically all building illegal due to the miniscule differences between otherwise identical bricks - though I'd have to find numbers on engineering tolerances to see if anything that falls within those is practical to achieve
Well put. You are far better with math than I.
I thought you said you were bad at math? 😂
Trust me I am! High school algebra and geometry are my limit lol. I struggled through college algebra calculus and stats but C's get degrees baby!
There is no way to make a "legal" triangle with 45, 45, 90 degree angles. The long side would have to be root(2) times the side of the short sides and root(2) is an irrational number therefore no matter what the short side is you will never get a whole number on the long side. In your 12 example root(2)*12 = 16.97 is simply close enough to 17 that you don't really notice the difference.
Also your explanation with the diagonal tiles is nonsense ;). You still use the same measurements for both the straight and the diagonal lines because you look at the studs on the lines not the grid.
Yes you are correct. I kinda was just figuring this all out as I went and got a ton of good feedback from the video. I quickly realized that my logic at that part was nonsense and realized that if it were legal it would be a pythagorean triple which it couldn't be. Thanks for watching and thanks for the feedback!