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!

circles, tangents, and conceptual art

The math department at Smith College recently acquired a new art installation: Sol LeWitt’s Wall Drawing #139 (Grid and arcs from the midpoints of four sides). This piece was a gift to the Smith museum, and was first installed there in 2008. It is an example of “conceptual art,” of which LeWitt was a major exponent during the 20th century. While conceptual art was/is a large movement, of which I am almost completely ignorant, in this case (like many others of LeWitt’s wall drawings) it means that the art resides in a concept—more precisely, a set of instructions—which is created by the artist and carried out by a team in each physical location. This is analogous to the creation of music, with the artist playing the role of the composer and the installation team acting like the musicians, who must take the artist’s instructions and interpret them in their particular setting.

(You can click on each image below for a full-sized version.)

 In this case, the directions (paraphrased) are as follows:

• Draw a grid of lines evenly spaced 1 inch apart over the dedicated wall space.
• Draw circles centered at the midpoint of each of the four sides, with radii increasing by 1 inch, all the way across the wall.

Here are the four midpoints:



You can learn more about the original installation at the museum from a video. I just wanted to make these pictures available and to highlight the possibility of asking innumerable mathematical questions about this piece. For instance, the grid and circles produce varying patterns and densities throughout the space:


Can you tell where each of these pictures was taken? In the center of the piece, many coincidences appear and tangencies among the circles and the grid lines become evident:


The installation was done by three Smith students in art and math, directed by a professional installer from the LeWitt studio over the course of nine days in January. At a presentation last week, the students described the exactness and concentration that this project required, as well as certain accommodations that had to be made—for example, not all of the wall edges are perfectly straight, and so they had to determine how to adjust the grid, and what points to use as the midpoints. Apparently one circle has a radius that is slightly too large, because of slackness in the compass they were using. (I haven’t yet found where this circle is.) Clearly there is an interesting interplay between form and accident (in the Aristotelian sense), leading to all sorts of philosophical questions that I’m not up to expounding at the moment.

This is the first of LeWitt’s works that I have encountered. I’m sure others have plenty of mathematical material to explore, as well.