[TIP] A perfect fit

[masterX244]

  On 3/23/2017 at 3:05 PM, Didumos69 said:

I had to study this one a little, but it's also a perfect fit. Can anyone tell me why? LXF-file here.

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Triangle works out as a pythagorean triangle which also perfectly fits the fitting pattern shown here:

 

[Didumos69]

  On 3/23/2017 at 4:44 PM, masterX244 said:

Triangle works out as a pythagorean triangle which also perfectly fits the fitting pattern shown here:

 

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I know. I should have been more clear. My question is actually: Why does the sharp angle of the Pythagorean triangle match the angle defined by the pattern with two frames from @aeh5040?

Edited March 23, 2017 by Didumos69

[aeh5040]

  On 3/23/2017 at 6:51 PM, Didumos69 said:

I know. I should have been more clear. My question is actually: Why does the sharp angle of the Pythagorean triangle match the angle defined by the pattern with two frames from @aeh5040?

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Nice find!  It can be seen as another example of the "Lego proof" of Pythagoras that I gave before, with the following parameters:

EG=9-4=5; GB=6+6=12; BE=9+4=13, giving a 5-12-13 triangle.

Edited March 24, 2017 by aeh5040

[Didumos69]

Great! Thanks @aeh5040!

[Didumos69]

I just noticed the red dot in the (5,12,13)-triple below, is also 'in-grid'. The 1L red lines make a right angle with the legs of the triangle.

Btw, the angle made by the red lines is practically equal to the angle made by a #3 connector: 157.5 degrees (in this triangle it is 157.38 degrees to be precise). See also the 'almost perfect fit thread'.

Edited November 20, 2017 by Didumos69

[schraubedrin]

Because i have a hard time seeing those perfect fits, i recently build tools to help me find them:

It's a collection of Pythagorean triples and multiples of those with axles in the corners to point at/insert in stud holes.

 

They make building stiff constructions a lot easier and faster.

 

[shadow_elenter]

  On 2/1/2019 at 10:23 AM, schraubedrin said:

Because i have a hard time seeing those perfect fits, i recently build tools to help me find them:

It's a collection of Pythagorean triples and multiples of those with axles in the corners to point at/insert in stud holes.

 

They make building stiff constructions a lot easier and faster.

 

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This is actually ridiculously useful, I'm going to make myself a set!
Thanks for sharing.

[technicmath]

Below are some more examples of a perfect fit:

More examples of such configurations can be found at https://bricksafe.com/pages/technicmath/liftarm_geometry_triangle.

[howitzer]

  On 9/18/2022 at 12:30 PM, technicmath said:

Below are some more examples of a perfect fit:

(snip)

More examples of such configurations can be found at https://bricksafe.com/pages/technicmath/liftarm_geometry_triangle.

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Wow, that's a lot of examples on your gallery, though many appear to be variants of a few arrangements. Did you use some kind of algorithm to find all these?

[technicmath]

  On 9/18/2022 at 1:08 PM, howitzer said:

Wow, that's a lot of examples on your gallery, though many appear to be variants of a few arrangements. Did you use some kind of algorithm to find all these?

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Thank you! First, the mathematical formula for such a configuration was derived. Then a java program was written to find solutions where all lengths are natural numbers. These solutions were given to a drawing program.

[howitzer]

  On 9/19/2022 at 6:53 PM, technicmath said:

Thank you! First, the mathematical formula for such a configuration was derived. Then a java program was written to find solutions where all lengths are natural numbers. These solutions were given to a drawing program.

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Yes, I thought it would be something like this.

[brickphisto]

Thank you very much for bumping up this very interesting thread. It is impressive to see how rather simple math can help you find and explain crazy looking perfect fits. Thanks also for the huge collection although I might never use any in my models.

[astyanax]

Another perfect fit: The inner angle of the bent liftarms is such that they can be stacked with half-module offset, maintaining the same alignment on the 'upward' side:

[Technic tango]

Just had alot of fun reading this, and thought I would share a tool I made for my grade 11 final project for making triangles that have whole number sides:

Triangle tool on github

[howitzer]

  On 9/19/2022 at 6:53 PM, technicmath said:

Thank you! First, the mathematical formula for such a configuration was derived. Then a java program was written to find solutions where all lengths are natural numbers. These solutions were given to a drawing program.

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Hmm, now that the thread was bumped I actually took a better look at your gallery, and it seems that there's a huge number of duplicates in there. In the current list it's quite hard to determine which designs are actually different and which are just repeated ones.

[technicmath]

  On 11/29/2023 at 8:29 AM, howitzer said:

Hmm, now that the thread was bumped I actually took a better look at your gallery, and it seems that there's a huge number of duplicates in there. In the current list it's quite hard to determine which designs are actually different and which are just repeated ones.

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Some configurations are indeed essentially duplicates. Some of these were automatically not considered by the java program. Others were retained on purpose to show different setups.