Biology in the News Explained

Mathematics and quilting

My recent dearth of posts has only to do with a temporary priorities shift. I have roughly two discretionary hours per day (and that assumes that housework counts as “discretionary”) and I elected for the past week to spend it finishing up a quilt top I began almost a year ago, which I decided to display here to prove my excuse. (Although the quilt is neither about biology nor music, it is about math and art, which I decided are close enough.)
 
Quilt demonstrating regular and semi-regular tilings in the Euclidean plane
The quilt (above) was designed to showcase the eleven regular and semi-regular Archimedean tilings of the Euclidean plane known to those who have studied geometry. Tilings are patterns of polygons which fit together on a flat surface, with no gaps. (There are also multiple tilings of polyhedra in three-dimensional space, for example a soccer ball, which consists of hexagons and pentagons. These can be fit together to form a (nearly round) polyhedron, but cannot fit together in a plane – if a soccer ball were flattened out, some gaps would appear between the shapes. ) Here are all of the tilings in the Euclidean plane, with color coding.

A regular tiling is one in which one type of regular polygon (a polygon with equal sides and angles) can be fit together repeatedly in a plane with no gaps. The three regular tilings are certainly familiar to most people, who see them used in floors all the time: squares (block 2 above), hexagons (8), and triangles (10). The eight semi-regular tilings use regular polygons of mixed shape to cover a flat surface. These are: octagons and squares (block 1 – also often seen on floors); hexagons and triangles (two ways, block 3 above and the entire quilt); squares and triangles (also two ways, blocks 4 and 7); hexagons, squares, and triangles (9); dodecagons and triangles (5); and dodecagons, hexagons and squares (6). These can be easily derived by figuring out which combinations of angles in the polygons add up exactly to 360°, which is necessary for the tiling to be flat. For example, the angles in a regular hexagon are 120°, so three hexagons can meet at a flat vertex. Squares have 90° angles, and you can either fit four of them together, or combine two squares (180°) with three triangles of 60° each to form a flat 360° vertex. And so on.

Of course, several of these are well known to quilters the world over. Blocks 2, 4, and 10 are most often seen in quilts because they involve squares and triangles, which are relatively easy shapes to cut out, and do not involve sewing into corners. The reason the construction of the blocks took me nearly a year is because the majority of them do involve sewing into corners, which means you cannot assemble the block by sewing straight lines only. Sewing into corners is especially hard on a machine, which I used, and when the angles are acute, which they get to be in the blocks with four or five shapes meeting at a vertex. Blocks 3 and 7 were the real killers for this, and each took me months because I didn’t have the patience to focus on them for any length of time. Plus I’m not particularly experienced with this in the first place, so I did a lot of seam ripping.

Because I am a hand quilter, it will probably be another year (at best) before the quilt is finished – it is a very large quilt. So, when more gaps appear in the blog, well, that’s just one of the other things I’m doing with my time.

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3 Responses to “Mathematics and quilting”

  1. Rob Swanson says:

    I finished half a quilt (using a machine).

    What I find it attractive are patterns with chromatic shifts (spectra) as well as some kind of design. This can be simple (a quilt my great grandmother did and I have inherited); or slightly more complex….perhaps I will see if I can take a picture of what I did – I do not have a digital camera since I am not a fan of digital camera (and like many things I think it speeds up life more than we already need to have had it speeded up).

    Do you teach any group theory or commutative algebra (rings/fields) to students? These are very interest subjects to be exposed to.

  2. Tammie says:

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