# Abstract legos

Legos are cool (I guess everyone who had legos in his/her childhood would agree…) ! Plus, they’re cool for a lot of different reasons.If you’re all grown up, you may not see your constructions from the same point of view as that of a child. Working on stuff involving group theory, algebra, etc., I can’t help but seeing the incredibly combinatoric aspect of legos. Take for example a set of six 2×4 bricks like the one below and try to combine them in different manners.

The number of possibilities with such a small set is already huge ! It was actually computed by the engineers in Lego, who came up with 102,981,500 for the number of possible constructions. But a researcher in Denmark, Søren Eilers, realized that this number was wrong since they only considered towers of maximal height (i.e height 6, where one and only one brick is stacked upon another one). By considering all possible heights, Søren Eilers came up with the number 915,103,765 ! Eilers has a number of papers about enumerating lego structures, which you can read on his web page. Here are some other interesting articles :

• “On the entropy of Lego” – B. Durhuus, S. Eilers – Available on Arxiv
• “Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length” – B. Durhuus, S. Eilers – Available on Arxiv
• “On the Asymptotic Enumeration of LEGO Structures” – M. Abrahamsen, S. Eilers – Experimental Mathematics, 20(2):145–152, 2011
• “Efficient Counting of Lego Structures” – M. Abrahamsen, S. Eilers – Available from Eilers web page.

In addition to the usual sets for kids, Lego released recently a series of sets about the most famous landmarks in architecture. Though expensive, they are quite interesting in their design which sometimes verges on the abstract. I wasn’t interested at first in playing with Legos again (plus, my old sets were being locked up in a cave), until I discovered that I had a Lego store not so far, which had a brick wall. So I came back one day with a cup full of $i$ x 1 ($\forall i \in \{1...4\}$) and $i$ x 2 ($\forall i \in \{2...4\}$) bricks !

Since there are already a lot of people designing fantastic structures in legos, whether it’s spaceships, Mars rovers, or new ideas for the Lego Architecture series (yes, the New York Apple Store), I decided to do something different.

Being attracted to conceptual art, and minimalism, both in art and music (if you don’t know the music of Steve Reich or Philip Glass, check it out on YouTube for example), I set out with a simple game: “With a given small set of basic bricks, try to come up with the greatest number of interesting abstract designs“. Now this game has a few more rules :

• Though the set of bricks is fixed, you can use as many flat tiles as you want for the finish.
• The structures have to be reasonably stable, meaning that they should resist a small applied pressure. This is to prevent some too theoretical structures from being built.
• Finally, they should be aesthetically interesting.

Ok, that last rule is very subjective ! But it will prevent the construction of bulky structures in which bricks are just stacked too regularly. To give you an idea of what I have in mind, let’s say that I like to see those small structures as micro-buildings (not unlike the Folies in the Parc de la Villette in Paris, designed by the architect Bernard Tschumi). I wouldn’t be surprised if it reminds you of a Sol LeWitt work.

So, to begin this Sol LeWitt-esque (I dare not say Sol LeWitt-y) project, here’s a first model, the 6-5 model, made of one 6 x 4 plate, six 2 x 2 bricks and five 2 x 4 bricks.

By the way, Lego has an amazing piece of software, the Lego Digital Designer, which allows you to build your lego structures virtually on your computer. This has sparked a community dedicated to the 3D rendering of lego designs. I’ll speak more about the tools used for the rendering in another post. For now, here are four designs in the 6-5 model.

More to come in the following posts, as well as other models too…