Manual Fractals - this time with Legos!

NOTE: For the sake of easy reading, the MATH! is in the footnotes.

When I saw this task my first thoughts ran immediately to a 3D construction with Legos. I thought it would be possible and possibly not very difficult to build a fractal out of Legos. After all, a fractal is simply a geometric shape that is self-similar and constructed recursively¹.

The most basic fractal can be constructed with lines. Given a line of a certain length, say from 0 to 1 on the real line, copy this line below the original line, but divide the line into three segments and remove the middle portion. Divide each of the remaining two segments into thirds, remove the middle section, and copy the lines below. Repeat until you are down to dots and cannot split things any further.

This is called a Cantor ternary set². Here's an example of what it should look like:



Since the lines are proportionally similar to one another, and the topmost lines can be said to be made up of the smaller lines, it is self-similar. And it is constructed recursively by using the process in step two on the results of the process from step one, and so on. It's not very interesting to look at, but a simple example of how fractals can be constructed.

There are lots of different kinds of fractals but for the purpose of constructing one from Legos, I thought the Menger Sponge would be perfect. While not as eye-catching as some sets, it is square, easy to mentally calculate and visualize, and its boxy nature would lend itself perfectly to my building materials.

FRACTAL 1: The Menger Sponge

Attempt 1

The plan I had was to use as many of these



as I owned or could obtain, and stack them into cubes of 9 x 9 x 9, then remove the necessary pieces to create the first iteration of the sponge. Then, I would make 20 of those cubes³ to build a larger cube for the second iteration. The whole thing, I imagined, would be about a cubic foot or less – about the size of a breadbox. Totally manageable! Then if I got more Legos, like a LOT more, perhaps through collaborating with other FØEcakes, I hoped I could expand to a third iteration and that would be quite big enough; really as far as I would ever want to go.

This was, of course, a total pipe dream. For starters, while one iteration would only take 20 2x2 bricks, which seemed reasonable and doable, two iterations would take 400! I have a lot of Legos, but I knew I had little hope of making it that far.

Another reality I discovered that stood in my way is that Legos, individually, are quite expensive. Looking at the Lego website⁴, to get one hundred 2x2 bricks, it would cost me $110! Obviously ordering several hundreds of them was beyond reason.

I immediately put out a call to all FØEcakes for Legos, and though I was directed to a reliable Legomaniac resource, and still had doubts about getting that many 2x2 bricks. But before I could despair much over that problem, I realized another, more critical flaw in my plan:

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Menger sponge attempt 2

After figuring out a way to make this thing stay together (though I had to eliminate the idea of using all 2x2 bricks which made it not totally self-symmetric) I created this



but it was obviously, well, not a square.

Although outward symmetry wasn't as important as self-symmetry I longed for it anyway.

Menger sponge attempt 3

I decided I could perhaps construct a more fully square square, of many different kinds of pieces (hopefully cutting down in my need for many, many square bricks) and that would be my smallest unit. This also, turned out to be somewhat impractical. I tried a lot of stuff, but this was the first thing that worked used this square constructed of smaller 2x2 and 2x4 bricks:



It was not ideal, but I attempted to build the sponge again anyway.



This is indeed a fractal folks, just a really clunky ugly one.

This monster used a lot of pieces, though, and I was running out of pieces, fast. I would never be able to spell out FØES - or even just Ø - with the rate I was using pieces. I wasn't even going to make it to the second iteration. I had to rethink things again.

Menger sponge attempt 4

After playing around with a lot of different sizes for the sponge, I tried with something much smaller as my basic unit:



And finally came up with something that worked.



It still seemed off to me; it looked like a poorly constructed house, but I was at least certain it was a Menger sponge. Not feeling fully satisfied with the final result of my sponge, I decided to try building another fractal pattern.

FRACTAL 2: The Sierpiński Triangle

I know Koch snowflakes are fractals, but they seemed too complex to be able to build with Legos. My thinking was that, Koch snowflakes have a faster rate of expansion, I think, for the surface area/size of the perimeter, so the likelihood was greater that I would run out of pieces before creating anything useful or wondrous... or even recognizable. Also, there are more points on a Koch snowflake, and making pointy things would be difficult. I was concerned, too, that the aspect of making it 3D presented more of a problem of construction.

However, I was starting to think that something similar to the Menger sponge, the Sierpiński Triangle (or Sieve) might work.

A Sierpiński triangle is constructed by taking a triangle (usually equilateral) and shrinking it in size by one half, then copying it two times, and placing the copies at with one point at each vertice of the base of the original triangle. At the second iteration, you repeat this process for each triangle, and so on through successive iterations.

I especially thought it would be a good candidate for construction considering that the Sierpiński Triangle doesn’t have to be composed of triangles, it can also be made up of other shapes, like squares! (The instructions above still work, more or less, just start with a square instead of a triangle. See example below.)

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Sierpiński triangle attempt 1

I tried building one face, just to see if it would work, but I had hopes to build more.



This had definite promise.

I could easily extent this to 3D by making the face tilt in one direction



and then making subsequent faces connect top and bottom into a sort of hollow tetrahedron, and then I could stack four of them into a larger triangle. Worried I wouldn’t have enough pieces, I began to dig furiously into the Lego box.

Sierpiński triangle attempt 2

But in the end, it totally worked:



Then I realized that I am an idiot, and that I was completely wasting my time building something that looked like a Sierpiński Triangle, rather than building the thing itself. I mean, fractals are all about self-symmetry, and this thing I had built had the outer pattern on its surface, but my idea was to create 3D fractals, with my 3D materials.

Sierpiński triangle attempt 3

Beginning with a tiny pyramid of four 2x2 bricks, I began to build.



This is one iteration. From there, I build more.



Legos in the sizes and colors I needed were getting scarce. I wondered what my downstairs neighbors thought I was doing, as I scraped through my Lego bin over and over again. I had forgotten how much I loved that sound.

Eventually it came time for drastic measures, and I got to do something else I remember I secretly loved as a child -



Eventually, I found enough of everything I needed, and could finish the next iteration.



I went to sleep, fairly satisfied and having had a lot of fun.

But strange dreams plagued me, of my high school geometry teacher mocking me for my imprecision, and of every math test I was ever late or unprepared for... what could it mean?

When I awoke in the morning, I knew. I had made a grave error! Inspecting my fractal I realized it looked off, and the reason was because it was not entirely self-similar. I had made the layout of the base of the first iteration incorrectly, and it did not reflect my basic unit.

I was going to have to reconstruct the entire thing.

Sierpiński triangle attempt 4

After a lot of work, and a sad realization that I just did not have enough Legos to complete this differently constructed Sierpinski Triangle with the Legos I had and still maintain my easy to see color pattern, I was finally finished.



This is, indeed, a fractal. Though a bit unstable (I didn't have very many pieces of the correct size to try and connect some things differently in the second and third iterations so as to make it more stable) it holds its own.

Now, who wants to get together, bring their Legos, and build a really big one?⁵.

Other fun fractals

Oh! But, before I was done, when I was stewing over actual fractal Lego completion, while my backbrain was working on that problem, I also, created this dragon curve of Lego people heads:



I left a bloody trail of corpses in my wake



but here, you can better see their mathematically precise and smiling faces.

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Ridiculous footnotes:

¹ Wikipedia defines a fractal as "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," who references The Fractal Geometry of Nature by Mandelbrot, B.B. (1982).

Another imprecise definition, but one that might be easier to grasp for those with some knowledge of the terms from calculus, is that a fractal is a curve that is everywhere continuous but nowhere differentiable.

² The Cantor ternary set is defined as a perfect set that is nowhere dense (obviously analogous to the curvilinear definition I mentioned) and can be described mathematically as

(C_n-1)/3 U [2/3 + C_n-1)/3]

Also, does anyone know if LaTex or some equivalent works in praxes here? HTML math tags weren't working.

³ It only takes 20 2x2 bricks, rather than the 27 you might be thinking, intuitively (9²) because one brick is removed from each face of the cube as well as the one at the very center. So the second iteration would be... only 400 bricks!


⁴ The Lego website was disappointing and a bit sad. When I was a kid it was my greatest dream, next to going to Antarctica, to work as a Lego designer someday. Legos today are not what they used to be. So many sets are comprised of fewer, larger, and more specialized pieces. I imagine it's harder for kids today to reconstruct them into something else other than what the instructions intend.


⁵ I am totally serious about this! We could keep our Legos separate (because who wants to lose any of their Legos?) by color coordinating, but it could totally be done. I am imagining a giant SF0 monument with pieces from all over! Collaborate with me!