boxes and fractions

Last week at MathFest, Dusa McDuff gave an excellent series of lectures on symplectic geometry. I enjoyed the second one the most, because in it she described the solution to a concrete problem that had a beautiful and expected answer, and used several tools of varying levels of difficulty. The result was published in the Annals of Mathematics in 2012; you can get the paper here. I won’t be referring to it for the rest of the post, however. Instead I want to highlight one elementary construction she described during this lecture.

About halfway through the talk, McDuff made a comment along the lines of, “Here’s something I must have learned in elementary school, but I’ve been surprised by how many mathematicians don’t know it.” I certainly don’t recall having seen it, at least not in the generality she described, and I do think people of all ages and mathematical abilities could enjoy playing with it.

Start with any fraction—say, \(30/13\)—and make a rectangle whose side lengths are the numerator and the denominator of the fraction.

Now start marking off squares, as large as possible, inside the rectangle. In this example, we can cut off two \(13 \times 13\) squares at first.

When no more large squares fit, start marking off squares from the remaining rectangle along the side.

Continue until you run out of space. In this example, only one more step in the process is needed.

When you’re finished, you will have filled the rectangle with squares of various sizes.

Count how many squares you have of each size and put those numbers in sequence. In this example, we have two large squares, three medium squares, and four small squares, so the sequence is 2, 3, 4. (Yes, I chose the fraction \(30/13\) to get this sequence.) Now write these numbers into a continued fraction, as follows: \[ 2 + \cfrac{1}{3 + \cfrac{1}{4}}. \] To evaluate a continued fraction, we start at the bottom and work through the nested operations: \[ 2 + \cfrac{1}{3 + \cfrac{1}{4}} = 2 + \cfrac{1}{\cfrac{13}{4}} = 2 + \frac{4}{13} = \frac{30}{13}. \] The result is the fraction we started with!

Here’s another example. If we start with the fraction \(25/7\) and follow the same process, we get this picture.

This time there are four different sizes of boxes. Counting the number of boxes of each size gives the sequence 3, 1, 1, 3, and we can check that \[ 3 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{3}}} = 3 + \cfrac{1}{1 + \cfrac{3}{4}} = 3 + \frac{4}{7} = \frac{25}{7}. \]

I had fun figuring out why this works, so in the interest of keeping this post short, I’ll leave that as an exercise for the reader. A hint: the process of cutting off squares is essentially Euclidean division.

Although we started with rectangles whose side lengths are integers, there’s no reason to restrict the above process to that case. In fact, if this process seems familiar, you may have seen it before in the special case of a golden rectangle, in which only one square of each size can be included:

This is related to the fact that the (infinite) continued fraction of the golden ratio has all \(1\)s: \[ \frac{1 + \sqrt{5}}{2} = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}} \] which I explained in another way in my last post.

One more example worth considering. Here is the start of the process when the side lengths of the rectangle are \(\pi\) and \(1\):

So there are three of the largest boxes, seven of the next largest, then fifteen of the next size (although the resolution of this image doesn’t let you see that). The fact that the first two steps nearly fill the entire rectangle is why \(3 + 1/7 = 22/7\) is such a good approximation for \(\pi\).

P.S. Dusa McDuff is also married to John Milnor, who gave the Hedrick Lectures in 1965. What other husband-and-wife pairs (or other pairs of family members) have both given the same high-profile math lecture series?

geometry at the fair

Last month, West Springfield once again hosted the Eastern States Exposition (or “The Big E”), which brings together fair activities from six states: Maine, New Hampshire, Vermont, Massachusetts, Connecticut, and Rhode Island. It’s great fun to attend, and includes displays of the finest crafts to have competed in county and state fairs from all over the northeastern U.S. in the past year. This means, for instance, that there are a bunch of great quilts.

Symmetry naturally plays a large part in the design of these quilts. The interplay between large-scale and small-scale, and between shapes and colors, creates aesthetic interest. This quilt, for instance, presents squares laid out in a basic tiling pattern (a square lattice). Each square contains a star-shaped figure. The star itself has fourfold dihedral symmetry, which matches the symmetry of the lattice, but the choice of colors in the stars breaks the symmetry of the reflections, resulting in cyclic (i.e., pure rotational) symmetry.

This quilt also shows fourfold dihedral symmetry in the shapes, which is broken into cyclic symmetry by the colors. It hints at eightfold (octahedral) symmetry in some places, but this is broken into fourfold symmetry by the colors and by the relationship of these shapes to the surrounding stars.

This pattern shows fourfold cyclic symmetry at the corners, but that’s not what first caught my eye. The basic tile is a rectangle, which has the symmetry of the Klein four-group (no, not that Klein Four Group). For the two quilts above, I first noticed the large-scale symmetry that was broken at the small scale; here I first saw the limited small-scale symmetry that is arranged in such a way as to produce large-scale symmetry. (I think this is because I tend to notice shapes before colors.)

This quilt uses the square lattice on the large scale, but varies the type of small-scale symmetry. Each square contains the same shapes, but they are colored differently so that sometimes the symmetry is dihedral, sometimes cyclic.

This next quilt is geometrically clever in many ways. It has no reflection symmetries, even disregarding the colors, although the basic shapes that comprise it (squares and a shape with four curved edges, two concave and two convex, for which I have no name Edit 10/15: In an amusing exchange on Twitter, I learned that this shape is described among quilters as an “apple core”) do have reflection symmetries. (I am disregarding the straight lines that cut the curved shapes apple cores into smaller, non-symmetric pieces.) The centers of the squares lie on a lattice that matches the orientation of the sides of the quilt, but the sides of the squares are not parallel to the sides of the quilt. The introduction of curved shapes also acts in tension with the rectangular frame provided by the quilt medium.

Some of the quilt designs rejected fourfold symmetry altogether. Here is one based on a hexagonal lattice:

and another based on a triangular lattice:

(These two lattices have the same symmetries.)

Here is a quilt that stands out. It appears to simply be pixellated:

but if you look closely, you’ll see that the “pixels” are not squares, but miniature trapezoids.

It therefore has no points that display fourfold symmetry. All rotational symmetries are of order two.

All of the types of symmetries of the above quilts (except, perhaps, the one that used some tiles with dihedral symmetry, some with merely cyclic) can be described using wallpaper groups, which I leave as an exercise for the reader.

This next design seems more topological than geometric: it is full of knots and links.

This quilt has an underlying square lattice pattern, but the use of circles again evokes links, at least for me.

It was a surprise to come across a quilt with fivefold symmetry, but it makes perfect sense for a tablecloth.

Finally, this quilt was just gorgeous. The underlying pattern is simple—again a square lattice—but the diagonal translations are highlighted by the arrangement of the butterflies.

As you can see, it was decorated as “Best of Show”. We were particularly happy to see it receive this prize, because we had previously seen it in Northampton’s own 3 County Fair!

the best “real-life” use of geometry I saw this year

On May 3, 2003, the Old Man of the Mountain—a rock formation that had been known for at least two centuries as one of the natural wonders of New Hampshire—collapsed. No one saw it happen; that morning two park rangers looked up and realized he was gone. It had been expected that this day would arrive. The Old Man’s face was a remnant of ancient glacial movements, and it was not stable, thanks to erosion and freezing; it had already been repaired multiple times since the 1920s. In 2007, a project was begun to memorialize the Old Man, and in 2011 the “Profiler Plaza” was dedicated.

Over fall break this year, my wife and I made a trip to the western edge of the White Mountains, where the Old Man of the Mountain used to reside. We stopped by the memorial to the Old Man that is now located on the edge of “Profile Lake”, where I was astounded by the ingenuity of the project that had been created. Not content with photographs or descriptive plaques, the Old Man of the Mountain Legacy Fund sought to recreate the experience of viewing the famous visage.

This optical illusion is created by looking along any of several different steel structures, called “profilers”.

Each profiler has an array of raised features that, when viewed from an appropriate angle, line up to recreate the face on the mountain from the viewer’s perspective.

The distance from the Profiler Plaza to the Old Man’s former location is about half a mile, but for the profile effect to work requires careful placement of the viewer’s eyes. Thus each steel profiler comes equipped with three spots, marked according to the viewer’s height, so that they will be in the proper alignment. (Below is a picture of my wife looking at one of the profilers.)

I found this application of geometry to a memorial not only ingenious, but also quite stirring. The Old Man of the Mountain inspired several artistic works, including Nathaniel Hawthorne’s short story “The Great Stone Face”. When I was in high school, my mom directed a theatrical adaptation of this story, in which I played the role of the visiting poet who appears near the end of the tale. So I felt a special connection to this place as I visited it for the first time.

It seems this could make a useful cross-disciplinary lesson in school, say between English, geometry, and U.S. history. Students could study the stories of the Great Stone Face and the monument’s demise in 2003. Then they might be asked to choose a location and design the memorial, working out the necessary measurements. For instance, here is a link to a map with the face’s former location marked: 44.1606° N, 71.6834° W. The actual location of the profilers is on the north shore of Profile Lake. If anyone carries this out, I’d love to know how it goes!

Thanks for reading, and Happy New Year!