First and foremost, I am a professor of mathematics. Over the past decade, I have slowly realized that I am also an artist and a designer of jewelry and knitwear, but at the core of my practice is the desire to share the beauty of mathematics with others through my visual and tactile works.
Hand-knitted garments hold a particular fascination for me. There is something profoundly satisfying in knowing that every inch of yarn in an intricate shawl or sweater has flowed directly through my fingers.
I am currently Professor of Mathematics at St. Mary’s College of Maryland and an Associate Editor of the Journal of Mathematics and the Arts. Ellie Baker and I wrote the 2014 book Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist, which contains many of our jewelry patterns. My knit and crochet designs are listed on Ravelry, where my username is sgoldstine.
Collection: The Symmetry Completist
Each of the pieces in this collection catalogs all the symmetries of a particular mathematical type that are possible in a particular form of knitting color work. The three unifying design principles are as follows: the designs in each garment fall into one or several visually compatible families of patterns with shared motifs; each possible symmetry type appears exactly once in the garment; the resulting garment has a pattern that an experienced knitter can easily reproduce.
Seven Peaks:
Like the Fourteen Ciphers shawl, this cardigan grew out of my joint research with Carolyn Yackel into the mathematical structure of mosaic knitting. The seven two-color strip designs on the body of the cardigan exhibit the seven frieze groups (the possible symmetry types of patterns that repeat in one direction). These are echoed in miniature in the seven two-color strip designs at the cuffs. All of the patterns are built around the same simple triangular motif.
The direct inspiration for the garment was a wall hanging that I knit in four colors of yarn with the designs that became the lower body of the sweater. In the wall hanging, I used an I-cord edging to create polished side borders. It occurred to me that a variation of this technique could produce a sleek raised edge for a button band, and the seed for the cardigan was thereby planted.
Collection: The Symmetry Completist
Each of the pieces in this collection catalogs all the symmetries of a particular mathematical type that are possible in a particular form of knitting color work. The three unifying design principles are as follows: the designs in each garment fall into one or several visually compatible families of patterns with shared motifs; each possible symmetry type appears exactly once in the garment; the resulting garment has a pattern that an experienced knitter can easily reproduce.
Crystalline:
Double knitting is a form of two-color knitting that yields color-reversed versions of the same pattern on the front and the back. Its thick, reversible fabric is particularly suited to scarves.
Of the seventeen wallpaper groups (the possible symmetry types for designs in the plane that repeat in multiple directions), exactly nine are compatible with the regular grid of a double-knitting design. Crystalline contains each of these nine symmetry types. The nine designs fall into three families of three; a family with heart motifs with reflection symmetry, a family with scroll motifs with rotational symmetry, and a family with vine motifs with glide-reflection symmetry.
The pattern for Crystalline appears in the Deep Fall 2016 issue of the popular online magazine Knitty. Sneaking an account of symmetry groups into a general knitting magazine is one of my proudest feats of stealth mathematics.
(See my other submissions for the description of the collection, The Symmetry Completist.)
Fourteen Ciphers:
In the most common forms of two-color hand knitting, the knitter carries the two colors of yarn at the same time and stitches from each color as dictated by the pattern, which is physically challenging. In the 1970’s, Barbara Walker introduced mosaic knitting, a less intimidating two-color technique in which only one yarn is carried and worked at a time. The simplicity for the knitter comes at a cost for the designer: an unusual set of restrictions on color placement unique to mosaic knitting.
As Carolyn Yackel and I have discovered in our current research, the rules of mosaic knitting allow for all of the symmetry groups compatible with a square grid. However, if we consider two-color symmetry groups, in which there are both symmetries that preserve colors and symmetries that swap colors, some groups are excluded. Of the seventeen two-color frieze groups (the possible two-color symmetry types of a patterns that repeat in a single direction), only fourteen are possible in mosaic knitting. In this rectangular wrap, designs for twelve of those groups run along the length of the shawl, and designs for the other two appear at either end, an arrangement partially forced by the constraints of mosaic knitting.
