# Twenty-One

As of 2005 I have drastically cut back on the number of LEGO projects and commissions I am taking on. Other interests are eclipsing LEGO. That is not to say that I have stopped completely, just that I need to really be intrigued by a project in order to undertake it.

One such mosaic is this: Twenty-One, done for the Mathematics Department at Auburn University. Being a graduate (twice over) of that department I was quite pleased when they approached me and asked if I would like to do a piece of LEGO artwork for them. Other than been mathematically themed, I had complete discretion as to what I wanted to do.

Mathematics and LEGO have always been closely tied together for me; I've often wondered which came first: my fascination with geometry or my fascination with LEGO bricks. Either way, the two definitely feed one another. I have done several mathematics-based LEGO projects before, on a smaller scale (for example, my LEGO pentominoes and my LEGO icosahedron), but never had I done a really large piece with a mathematical theme.

That's not to say I hadn't thought about it, though, and as soon as the Math Department came to me, I knew what I wanted to do: a large mosaic of squares.

Yes, squares.

Here's a geometry question for you: can a large square be partitioned into some number of smaller squares so that no two of the smaller squares are the same size?

Think about it some. It's easy to dissect a square into smaller squares if some of those squares are the same size (for example, just cut it into four quadrants -- four squares of the same size). But can the big square be dissected so that each of the smaller squares is a different size?

For some time (decades ago), it was believed this was impossible. But then, in 1939, the first example was found. It used 55 squares. Later, in 1978, it was mathematically proven that the smallest number of differently sized squares that could be used to form a larger square was 21. That 21 square solution is unique.

This was the inspiration for my LEGO mosaic (for those interested in learning more about "perfect square dissections", I encourage you to peruse the MathWorld website).

Coincidently, the integer dimension of the large square in that 21-example is: 112 units. Doubling that you get 224 which is divisible by 32 (32 * 7)... the stud dimension of a standard LEGO baseplate.

This was too good to be true. Without needing to cut any baseplates, or compensate with a border, I could build this 21-square mosaic so that it was 224x224 studs.

Next, I hoped to make each of the 21 squares a different LEGO color. Now, obviously some of the internal squares were much larger than others. The big ones I planned to make using common LEGO colors while the smaller squares would be built with rarer colors.

But would I have enough of the various colors. Over the past few years LEGO have greatly expanded their color availability, but twenty one different colors?

Amazingly, I did have enough bricks in enough different colors to complete it. Here's the breakdown (all of the colors are the "old" LEGO colors -- no new grays or brown were needed):

 Stud Dimensions LEGO Brick Color 100x100 Red 84x84 Blue 74x74 Yellow 70x70 White 66x66 Black 58x58 Green 54x54 Dark Red 50x50 Sand Red 48x48 Tan 38x38 Medium Gray 36x36 Brown 34x34 Teal 32x32 (Extra) Light Gray 30x30 Light Blue 22x22 Dark Gray 18x18 Orange 16x16 Dark Blue 14x14 Purple 12x12 Dark Orange 8x8 Very Light Blue 4x4 Dark Pink

All of this planning was done before I actually built the mosaic, of course. And, really, the planning was the most time consuming part. The only things I had to order were dozens of baseplates (I could have used 49 32x32 baseplates, but I incorporated some 48x48 ones because they are a better area-to-cost value).

 A plaque describing the mosaic is mounted next to the artwork. It reads: "TWENTY-ONE" Can a square be partitioned into a number of smaller squares such that no two of the smaller are the same size? Casual pondering might lead one to think this geometrically impossible. However, in 1939, the first such example was found (R. Sprague, 1939 -- using 55 squares). Improvements were made by others until, in 1978, it was proven that the "lowest possible order of a perfect square dissection" is 21 (A.J.W. Duijvestijn, 1978). Artist Eric Harshbarger (a graduate of the Auburn University Mathematics Department, 1992 BS., 1994 MS.) has recreated that unique solution in the adjacent mosaic. By using 21 different colors of LEGO bricks, a different color for each square, he has used the popular construction toy to illustrate the geometry of the problem. Asked to explain his motivation further, Harshbarger expounded: "The inherent geometry and squareness of the LEGO bricks very much reinforces the underlying theme of the square dissection problem; it is a natural fit. Furthermore, by employing a material more commonly thought of as a popular toy, I hope to draw the audience closer to the work. Too often Mathematics is not considered 'fun,' and yet here we see that it is so closely related to one of the most popular toys of all time."
Once all of the pieces were collected, the build went quite quickly (it is, after all, just a lot of squares... built in Studs-Out fashion, it was really just a matter of dealing with the seams between the square baseplates and where they fell in relation to the varying squares atop them).

My friend Sarah even got into the act and helped me build many of the squares (she let me do the big "boring" squares while she attacked the smaller "cooler colored" squares). Together we finished the mosaic over the course of about 10 days of very casual work.

The mosaic was completed in mid-May 2005, but it was not until a month later that I actually installed it in the Math building at Auburn University (Parker Hall).

I've installed plenty of mosaics at this point; I've learned to come prepared (I even made a checklist of tools and such), but each one is always an adventure. My biggest fright with this one came at the very beginning: the bricks into which I was drilling the anchor bolts were hard... VERY HARD. I was afraid the masonry bit on my drill was going to melt, break, or otherwise just not get the job done.

With determination, however, I did get the holes drilled (and amazingly, right in position -- the mosaic hung very flat to the wall). With the top row of baseplates hung the remainder of the plates went up very quickly. Within 80 minutes I had the whole thing mounted.

I then spent the next 40 minutes gluing the pieces around the edge of the mosaic (to keep overly curious hands from removing the LEGO bricks).

If any of you are ever on the Auburn University campus and find yourself entering Parker Hall, I doubt there's anyway you'll miss this one hanging close by. In a few years the Mathematics Department is supposed to have a new building constructed somewhere on campus. The mosaic will be relocated then.

Before I provide a list of links to a few more pictures, I should thank Professors Andras Bezdek and Michel Smith (Department Head) for helping the project become a reality.

1. Sarah works diligently on the dark red pieces. Note that those pieces are plates. The whole mosaic is one brick thick, so the dark red (as well as some of the other colors) were built atop two additional layers of plates.
2. The plates were arranged on my floor as we completed them
3. Here's a shot of most of the main colors done. You'll note the gaps where the seams between baseplates lie.
4. Sarah works on one of the final two baseplates. Tan is the current color.
5. I kneel next to the completed pieces. I did not attach the baseplates (to hide the seams) until I actually installed the mosaic on site.
6. Of course, my cat, Faux Pas, had to lay claim to the mosaic as soon as it was complete, (it's the cat's house after all... I'm only visiting).
7. I then trained the cat to snap pictures so that Sarah and I could be in a shot together.
8. The mosaic is located right near the stairway between floors of the Parker Hall. It is viewable from above and below.