Pythagorean Theorem

The good ol' Pythagorean theorem. Probably the most widely known theorem in mathematics.
As you may know, the sides of a right angled triangle (a, b, c) satisfy:

a ² + b² = c²

(1)

Vice-versa, if the sides of a triangle satisfy (1), then it is a right angled triangle. But what have this theorem ever done for us?

This latter form of the theorem allows us to build cross beams with lego. The Lego has a unit, in both Lego and Technic, the dimensions of any parts are the multiple of this unit (except some half length parts).

 

This unit allows only positive integer solutions of the Pythagorean equation, which is exactly why it is an important equation.

Let us see some manifestations of the famous (and smallest) Pythagorean triple: 3-4-5. LDD files here.

On the first example you can see that I used a 5 beam and a 2×4 L-shaped beam and a 7 beam (actually only 6 holes of it).
You may think that the sides of these triangles are acutally 4-5-6, bigger than the theoretical 3-4-5 triple.
This is because the length of the beam is one unit bigger than the actual centers of rotation.

Like this fellow:

It is 4 units long, but the center of the two holes are 3 units wide from each other. See the cardan joint, the calculation is a bit different for that. The Pythagorean triple should appear as the distance of the rotation centers, not the total length of the beam or axis.

Now you can design your own Pythagorean constellations in various places. Let us see an examples in an official Lego Technic set.

• The windshield of the 42000-B set, the racer truck.
  
• The wing door of the 8448 street sensation car
  
  
  

There are other Pythagorean triples then 3-4-5. An other small triple is 5-12-13.

An example for that is in the 5_12_13.lxf file.

Some bricks incorporate the 3-4-5 or the 5-12-13 triple, further examples in the second LDD file.

Since the 3-4-5 and the 5-12-13 are both Pythagorean triples, the cuboid with sides 3-4-12 has an interesting property. It has an integer side-diagonal, and its space-diagonal is integer too.

Anyway, it is useless in Lego, because you should have two sphere joints to build this cuboid with its space-diagonal.

However, you may use other Pythagorean triples in 2D, here are some:

• 3-4-5
• 5-12-13
• 7-24-25
• 9-40-41

There is an overall formula to get every triple. For any positive integers z and x, for which z is greater than x:

z² - x², 2 x z , x² + z²

gives a Pythagorean triple.

Good luck!

PANZ3R

Hi everyone, as I promised in my former post, here is the brand new, custom made and revolutionary Lego military device. Conquer the world with Mindstorm EV3 PANZ3R tank.

See the building instructions in my Google Drive folder.
See the demonstration video:

Binary Transmission

So the problem is that you have a limited number of motors but you want to motorize a lot of functions.
I ask, how can you power the most axes with the least possible number of motors?

The usual parts of a gear box are given:

This setup is able to transmit the force from one axis to two switchable axes. There are three states of this little fellow:

• power to first axis
• neutral
• power to second axis

If you want an automatic gadget, then an additional motor is required, to operate the shift lever.

As a naive example, suppose that you have a car and you want to drive it, and steer it as well. This requires two motors, one for drive, one for steering. However, if you use a transmission box, like above, then the same motor can drive and steer, but the shift is up to you. The operation of the shift lever allows no spare of your motors. See the LDD file 00 for demonstration.

The minimum number of motors to power two separate axes is two, this is an information theoretical limit (with this setup).

The key idea is to use binary tree to extend the number of powered axes. Two gear boxes can be set to 2×2 = 4 states. Three of these can carry out 2×2×2 = 8 states and so on. In the LDD file 01 you can see a setup with 3 motors and 4 powered axes.

Note that the shift levers on the second stage are bound together, this is the key part why you can power more axes than the number of your motors.

With this binary tree stuff you can power 2^n axes with n+1 motors: one motor for the power, and n motors for the shifts. For example 4 motors can power 8 axes.

So, 3 motors can power 4 axes, and 4 motors can power 8 axes, it means that, in theory, 3 motors can power 8 axes as well. Moreover, this calculation is recursive.

I tried to construct such a 3 to 8 transmission, see the LDD file 02.
I would firmly say, that three motors can separately power any number of axes, however the structure is uneffectively large and complex. A better thing to do is to buy more motors.

PANZ3R coming soon

MINDSTORMS EV3, AWESOM3 PROGRAMMABL3 BRICK, AWESOM3 FIR3POW3R.

Since I have never seen such before, I decided to make a Lego tank. It is under construction, see here.

The creation uses bricks from the

• 42000 Grand Prrix racer
• 42006 Excavator
• Mindstorms EV3 (31313)
• and some additional gears, like this one 
  

The caterpillar body is quite compact and finished, although it does not drive well on solid floor, it slips on it, I recommend a carpet.

------- NEWS ---------
the PANZ3R is ready, see here.

Start

Hi everyone, whoever reads these thoughts carved into bricks!

This ought to be a blog to submit my lego creations, and ideas. I'am a 25 years old mathematician working as a developer.

I used to play a lot with Legos when I was a child, and nowadays this passion seems to come back.
A lot of my submittions contain Lego Digital Designer files, I advise you to download and use it extensively.
I have a google drive folder, containing the fun stuff.

Have Fun!