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 depicts the seven types of frieze symmetries in knitted lace, with designs inspired by traditional Estonian knitting and the patterns of Nancy Bush. It delves more deeply than the Double Knitting Groups into the shape of the stitches themselves.
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 Pendant was hand knit on 2.75 mm knitting needles.
The Double-Knitting Groups exhibits all of the wallpaper symmetry types possible in double knitting. For a harmonious overall design, I grouped the nine structures into three pattern families: scrolls, hearts, and vines. Roughly speaking, the symmetry groups get more complex from top left to bottom right: the upper left pattern has only translations, the three adjacent to it have only translations and one other type of plane symmetry (clockwise: reflections, rotations, and glide reflections), and the remaining patterns have at least three symmetry types.
This work is based on my scarf design, Crystalline, whose pattern is published in the Deep Fall 2016 issue of Knitty. The square wall hanging was knit by hand on 1.75 mm needles.