There are two things I have done for as long as I can remember: explore mathematics, and create things with my hands. In my professional academic life, these impulses have merged into my specialization in mathematics and the arts. I am particularly devoted to exploring handcrafts, especially those in the fiber arts traditionally viewed as feminine. The interplay between mathematics and fiber arts is endlessly fascinating, both in the ways that mathematics allows for a deeper understanding of knitting, crochet, weaving, and so forth, and in the ways that these crafts can illuminate complex concepts in mathematics.
Linear Lace in Burgundy depicts the seven frieze symmetry types with knitted lace, in designs inspired by traditional Estonian knitting and the patterns of Nancy Bush. It is a refinement of an earlier work, Linear Lace Pendant.
Due to inherent asymmetries in the shaping stitches, representations of frieze groups with 180 degree rotations (the designs in the top row) merely approximate rotational symmetry and reflections with horizontal axes. If we discount these approximations, each of the three designs has the same frieze group as the design directly below it. The final frieze group, in the bottom edging, can only be realized horizontally.
Linear Lace in Burgundy was knit by hand on 2.75 mm knitting needles.
In our paper "Building a better bracelet: wallpaper patterns in bead crochet," Ellie Baker and I prove that 13 of the 17 wallpaper groups generate designs in bead crochet rope, but two of them only occur in fairly trivial patterns. This seamless necklace smoothly transitions between designs for all 13 groups. The necklace is fastened so that the two boring patterns sit at the back of the neck, the designs with no reflections or glide reflections sit in the inner loop, and the remaining designs sit in the outer loop. The earrings (inner loop), bracelet (outer loop), and clasp (back) are bead woven with planar wallpaper designs that generate the patterns on the necklace.
Serpentine Symmetries is hand made and contains over 7700 beads.