# Symmetry groups of viral doilies

Are you looking for interesting examples to use in your abstract algebra course to illustrate planar objects with rotational and reflective symmetries? Tired of the usual regular polygons and corporate logos. Already shown your class ambigrams? Feeling pressured to inject biology into your mathematics courses? Look no farther.

My colleague sent me a link to the website of the artist Laura Splan. Her work is a juxtaposition of the biological with the ordinary—such as pillows onto which images of skin are inkjet printed and block-printed wallpaper with her blood as ink. She writes

I use anatomical and medical imagery as a point of departure to explore these dualities and our ambivalence towards the human body. Viruses, blood, and x-rays of bones and viscera can be at once unsetting and enticing. I often combine scientific images and materials with more domestic or familiar ones. The ornamentation of wallpaper or the design of a doily lends a sort of relief in its familiarity and pleasing pattern.

It is one of her more tame works, Doilies (2004), that exhibits interesting symmetries. She describes it as follows.

The design of each doily is based on the structure of a different virus. I begin with a digital image of the virus, which I then base a design on in a graphics editor. The design is then imported into computerized embroidery software and the stitches are laid out and manipulated. Finally, the designs are output from a computerized sewing machine.

Here are the doilies. They exhibit several different symmetry groups; I’ve given the first two.

• SARS (symmetry group: $D_6$)
• HIV (symmetry group: $\mathbb{Z}_2\oplus\mathbb{Z}_2$)
• Herpes
• Influenza