[TIP] Almost a perfect fit

[aeh5040]

Here is an interesting little construction I stumbled upon.  I post it in case it is useful to someone.  The fit is not mathematically perfect, but very very close.  The angle produced by the frames is 180 - 2 arctan(4/6) = 112.62 degrees, whereas the connectors are 112.5!

What other constructions like this do you know?

Edited March 19, 2017 by aeh5040

[Didumos69]

Good to know! Something similar: sqrt(12^2 + 12^2) = 16.97. In the center you see another implementation of your find .

[Void_S]

aeh5040, it's so close, that even LDD allows it with a small exclusion: you won't add one of a black pins (only 3 of 4 could fit here) because of a small delay between the 2x4 L-Liftarm and 5x7 Frame:

 

[Lasse D]

Nice find aeh5040. With such a little gap it might even be legal for LEGO to use this construction. 0.12 degrees is very little considering that it is spread on 5 connections.

[Sprites]

Interesting. though i don't see any practical use for the moment. 0.1° can easily be handled by brick/joints torsion

[Didumos69]

On 19-3-2017 at 7:25 AM, aeh5040 said:

What other constructions like this do you know?

This construction follows the lines perpendicular to the sides of the Pythagorean triple (5, 12, 13). The hole of the connector coincides with the center of the triangle's in-circle with r = 2. The angle produced with this construction is 180 - 2 arctan(2/10) = 157.38 degrees, whereas the connector is 157.5 degrees. LXF-file can be found here.

Edited March 23, 2017 by Didumos69

[aeh5040]

Here is another consequence of the first observation above.  Stack five 5x7 frames on top of each other, each time using the twisted diagonal-diagonal connection.  Then the overall twist between the top and bottom frames is almost exactly 90 degrees.  So you can join another 5x7 frame vertically to both of them (with a liftarm to space it one stud away at one side).  Sorry I don't have access to a picture right now - it is an exercise for the reader!

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14 hours ago, Didumos69 said:

This construction follows the lines perpendicular to the sides of the Pythagorean triple (5, 12, 13). The hole of the connector coincides with the center of the triangle's in-circle with r = 2. The angle produced with this construction is 180 - 2 arctan(2/10) = 157.38 degrees, whereas the connector is 157.5 degrees.

That's very cool!

Edited March 24, 2017 by aeh5040

[Didumos69]

1 hour ago, aeh5040 said:

That's very cool!

Given your proof in the 'perfect fit' thread for the fact the sharp angle of the (5,12,13) triangle equals the sharp angle drawn by your stacked frames, it's no coincidence that the deviation of the 157.38 angle is the same as in the 112.62 angle: .12 degrees. Together they make exactly 270 degrees.

[aeh5040]

26 minutes ago, Didumos69 said:

Given your proof in the 'perfect fit' thread for the fact the sharp angle of the (5,12,13) triangle equals the sharp angle drawn by your stacked frames, it's no coincidence that the deviation of the 157.38 angle is the same as in the 112.62 angle: .12 degrees. Together they make exactly 270 degrees.

Awesome - I had not realized that!

[Didumos69]

6 hours ago, aeh5040 said:

Awesome - I had not realized that!

Looked into it a little deeper. The fact that the angles match has to do with the fact that the angles defined by the ratios 4 / 6 and 2 / 10 add up to the angle defined by the ratio 1 / 1 (45 degrees):

Arctan sum identity: arctan(x) + arctan(y) = arctan((x + y) / (1 - xy))

arctan(2/10) + arctan(4/6) = arctan(1)

 

Edited March 24, 2017 by Didumos69

[Didumos69]

Here is a (3,4,12,13)-quadruple. The angles made by the #3-connectors are 157.5 degrees, close to the 157.38 degrees needed. I used this in the back of my 4WD RC Buggy.

Edited May 14, 2018 by Didumos69

[Limga]

It maybe not so useful as it does involve half-stud offset, but still I'd like to share a new (for me) almost perfect fit that works flawlessly in reality. I need this for some suspension.
The hypotenuse is 4.47 (square root of 20) in 4 and 2 rectangular triangle. Given the +1 stud it gives 5.47 that is so close to 5.5 that LDD does allow it. It does snap with real parts withous sufficient bending or frictions, just like regular connection. 

[Erik Leppen]

Yeah, I use that one sometimes. Mostly in double-size, so as a 4-8-9 triangle. Especially as the 5x9 double-bent beam can act as the "4-8" part. This is very easy to test: put an axle in both axleholes of a 5x9 double bent beam and you can slide an 11L or longer beam over both axles.

It's basically the same triangle as the one used to mesh a 16t gear with a 20t gear. 2 studs aside, 1 stud up, hypotenuse 2.25, which is half the size of your triangle. (In general in Lego Technic, the distance between the centers of the axles of two meshing gears is 1/16 times the total number of teeth on those 2 gears. So 1/16 x (16 + 20), which is 2.25)

Similarly, 4-7-8 is also possible (or 2 - 3.5 - 4).

[astyanax]

When I saw that image from this article, I wondered if this is a perfect fit or almost perfect. So after a little bit of math I concluded that those holes in the gear rack (24121) are exactly 3 Sqrt(10) - Sqrt(30) modules apart, or approximately 4.0096 modules. So the 5M beam is 0.077mm too short. I guess that's within tolerance?

[brunojj1]

22 minutes ago, astyanax said:

I guess that's within tolerance?

No, you always shall keep it 100! This is illegal! Be purist and don´t ever dare to violate or bend the holy Lego pieces .

[astyanax]

1 hour ago, brunojj1 said:

No, you always shall keep it 100! This is illegal! Be purist and don´t ever dare to violate or bend the holy Lego pieces .

Careful now! Do you want me to point out all the illegal connections in your Ferrari 488 Pista model? Because I happen to know each and every one of them!

[brunojj1]

49 minutes ago, astyanax said:

Careful now! Do you want me to point out all the illegal connections in your Ferrari 488 Pista model? Because I happen to know each and every one of them!

I don´t know what you mean . 1. It´s not a Ferrari. 2. Everybody calls it illegal and buys it anyway. 3. The yellow gear rack you´ve shown above isn´t to serve in the ultimate MOD I suppose, so that will be fine .

Edited February 2, 2022 by brunojj1