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.
An outgrowth of current research by the artist and Carolyn Yackel, this scroll shows the two-color frieze groups attainable in mosaic knitting. A popular form of two-color knitting, mosaic knitting imposes unusual constraints on color placement that limit the types of symmetry compatible with the form.
There are seventeen two-color frieze groups, which are the possible symmetries of a strip pattern that has at least one color-swapping symmetry, However, three of them are provably impossible in mosaic knitting, and nine more are impossible to knit horizontally. These nine appear side-by-side at the bottom of the scroll, and the remaining five run across the top.
Float Free, Bumblebee was knit by hand on 1.75 mm knitting needles.
This scroll depicts the seven frieze groups in knitted lace. Each frieze design has the same fundamental region, a rectangular pattern with no internal symmetries, and each design uses only enough copies of this rectangle to generate three translations of its primitive cell.
Glass beads incorporated during the hand-knitting process mark the symmetries in the scroll. White beads mark reflection axes, yellow beads mark glide reflection axes, and blue beads mark centers of rotation. The red beads along the edges of the designs mark their translational symmetries, with the minimal duplication needed to preserve reflections, glide reflections and rotations.
Fundamental Frieze Scroll II was knit by hand on on 2.75 mm knitting needles.