Brutkey

Jens Finkhäuser
@jens@social.finkhaeuser.de

Tired: I'm playing with Lego.

Wired: I'm exploring the geometry of Pythagorean lattices.

Inspired: I'm playing with Lego.

Render of a Lego Technic grid of right triangles, specifically ones made of Pythagorean triples (7, 9, 11). The "lattice" is a backdrop of Lego studs. The triangles are made of Technic beams, which have holes of the same size and spacing as the studs. Therefore it's possible to lay out beams along the lattice in horizontal and vertical direction, and have their holes align perfectly with the lattice studs. It's also possible to lay beams diagonally and have some holes align with the lattice - but this is not possible at all angles. However, triangles where the side lengths consist of Pythagorean triples are right triangles that can be aligned on their corners. More specifically, for a Pythagorean triangle with the legs aligned horizontally and vertically along the lattice, the corners of the hypotenuse will also be aligned. The triangle with sides 7, 9 and 11 in length just so happens to be useful, because there exist Technic beams with those lengths. In the render, 3 triangles of this size are laid out so that they together form a triangle with double side length. Well, nearly. The only way this can be made to work with a Lego lattice is to re-use corners for two triangles. So for two stacked on top of each other, each of which is 7 holes high, the top hole of the bottom triangle's leg is also the bottom hole of the top triangle's leg. So you get 7 + (7 - 1) in length. The same applies for the other triangle sides.
Jens Finkhäuser
@jens@social.finkhaeuser.de

You might have noted that the triangles in the previous post have side lengths of 7, 9 and 11 (in holes). The alt text goes into some details on that.

But I didn't pick this particular triangle at random. You could easily form others.

I picked this because of the piece below. This is said to be bent at 53°, though that's imprecise. This also just happens to be related to the previous Pythagorean lattice.

A blue Technic beam bent at an angle of approximately 53 degrees.
Jens Finkhäuser
@jens@social.finkhaeuser.de

It helps to know that the 53 degrees are not measured between the two arms in the beam, but that they describe how much the beam is "bent". That is, it's measured from the imaginary line formed if the beam was straight.

The 53 degrees are also the angle at which the hypotenuse is inclined in the triangles.

In other words, you can plonk this piece onto e.g. the middle of the red points on the bottom of the grid, and rotate it to align with the beams.

Lego did not pick this angle by accident.