Geometrical Dissections II: Four to One

This week, I’d like to discuss a piece of artwork which began as a geometrical dissection — I call it Four to One.

I thought it would be interesting to discuss the process of creating such a piece from beginning to end.  The creative process is not really mystical, but because we so often only see the finished product, it may seem that way at times.

It all began about 15 years ago, when I was teaching the Honors Geometry course I mentioned in last week’s post.  In Greg’s book Dissections Plane & Fancy, he takes a few chapters to discuss dissections from squares to other squares — frequently two different squares to one, or three to one.  But there was very little about four-to-one dissections, so I thought I’d explore this avenue a bit more.

I can’t recall precisely how I arrived at the identity

I might have written some for loops — but computers were not as fast back then….  Likely I used something like Lebesgue’s formula on p. 80 of Greg’s book, which gives a formula for creating three-to-one square dissections.  Then if one of those squares could be written as a sum of two others, I’d have a four-to-one dissection.  In particular, once I (might have!) found out that

I could use the fact that (which is just a multiple of the Pythagorean triple ) to obtain the possibility of a four-to-one dissection described above.

Now this only suggested the puzzle, not the actual dissection itself.  And there certainly is a dissection — at the very least, we can cut up all the squares into unit squares and reassemble!

This is hardly an elegant solution; but I did come up with the following one:

I liked it because each square was cut into just two or three pieces, which was particularly nice.  Moreover, only one piece needed to be rotated.  But even though the number of pieces is relatively small, there is still the possibility that a dissection may exist using fewer pieces.

Of course my original solution was sketched on a yellowing piece of graph paper — but what to do with it now?

My first attempt looked like this:

I was thinking of creating various pathways through the dissected squares so that when they were rearranged, the pathways would still line up.  I abandoned this approach, however.  I can’t remember exactly why, but the results didn’t appeal to me — and besides, the paths themselves actually had nothing to do with the dissection puzzle itself.

But then I had the thought — which was in fact a real challenge — can I communicate what’s happening with the dissection using only one square?  In other words, could I depict the geometrical dissection by just showing the largest square without giving the viewer the four smaller squares?  I think what might have moved me in this direction is that there was just no elegant way of putting all five squares together in a composition.  There were just too many corners.

So I though of overlapping the smaller squares onto the largest square, as shown below (note:  you’ll notice an error in the geometry, but as it was a draft I discarded, I didn’t bother to fix it):

Now if you look very carefully, you can find all the pieces of the dissected squares in the largest square.  There is some overlap, of course — but smaller circles were overlaid on larger ones so colors from both circles could be seen.  (I copied the original dissection again so it’s easier to compare.  I used different colors as the images were created at different times, so watch out! )

I liked the idea — I felt I was getting somewhere.  But I wasn’t happy with the colors.  Now creating mathematical art makes you hungry — I can clearly recall driving to lunch while I was in the middle of this project, and I can even remember the road.  It was Fall in Princeton, NJ, and the leaves had already turned color.  No more oranges and reds — but lots of greens and yellows, as well as browns from the tree trunks.  My color palette!

What intrigued me about the idea was the fact that I was working with a very abstract, almost purely mathematical problem — and here I was, thinking about using organic colors from nature, from my life experience.

Now I had already been working with the ideas from Evaporation, and realized if I was using an organic palette, I couldn’t have the circles be regular, precise — and the colors couldn’t be pure either, just like you might find hundreds of shades of yellow in a Fall forest.

So, as shown in this close-up of Four to One, the colors were varied by using random numbers just as was done for Evaporation, but there were no extremes — each piece of each square had to be clearly recognizable if the dissection was to be clearly seen.

The sizes of the circles varied as well, helping to contribute to a natural texture.  Here, you can clearly see how smaller circles were overlaid on larger circles for the two-color effect.  The smaller circles, however, had only about one-fourth the area of the larger ones, so it was clear which color was dominant.

And there it is!  The creative process is not magical, not mystical — in fact, much of the time it seems to consist of failed inspiration….  Consider yourself lucky if your first attempt turns out to be your last as well — but more often than not, creativity is an iterative process involving constant revision.

So my advice is to stick to it!  Don’t worry if the first attempt isn’t what you imagine.  Now I used Mathematica to create this image — and I’ve been programming in Mathematica for over twenty years.  So I’m pretty good at taking an idea and implementing it fairly quickly.  But if you’re relatively new to programming, you’ve got to be patient with your programming skills as well.  I can tell you though — it’s worth it.  Don’t let anyone else tell you any different….