The catalog photo of Fundamentals of Lace, a knitted wall hanging. The piece is a rectangle of blue knitted lace lashed to wooden dowels at the top and bottom. The fabric has white, yellow, and red beads knitted into the lace. The rectangle is divided into a 3 x 3 grid of different lace designs, and each section has its own pattern of colored beads that mark the symmetries of the design. The white beads, which are always in vertical or horizontal lines, mark the axes of reflection (mirror) symmetries. The yellow beads (also in vertical or horizontal lines) mark the axes of glide reflection symmetries. The red beads, which are slightly larger and more spaced out, mark centers of rotational symmetry.
Fundamentals of Lace after blocking and before attaching to dowels, lying flat on a wooden deck. Each of the nine lace designs is made up from smaller lace motifs arranged in four rows and six columns. Disregarding any beads, the motif in the upper left corner of each design is identical, a rectangle with eyelets along one diagonal and half of the other diagonal forming a lowercase lambda. The part of the motif under the lambda is an eyelet mesh, while the regions above and to the left are solid stockinette. The rest of each design consists of the same motif in various orientations, some like the original, some flipped vertically, some flipped horizontally, and some rotated by 180°. Mathematically, the motif is a “fundamental region” for the design; hence, the title of the work.
Fundamentals of Lace, pre-blocking. The lace is rumpled and uneven, and the lines of beads are much wobblier than in the finished work. Algebraically, we can prove that there are exactly nine possible symmetry structures for a design in a non-square rectangular grid repeating in two directions. The structures are expressed by the beaded markings: for instance, the middle panel gives an example of a design with no 180° rotational symmetries that has both reflections and glide reflections, and any repeating design that has evenly spaced, parallel symmetry axes that alternate between reflections and glide reflections and nothing else except translations (just sliding the design without twisting or flipping) has the same abstract symmetry structure.
#ShowMeYourKnits #knitterschoice
Fundamentals of Lace, 2025: the newest #mathart I’m exhibiting at the Joint Math Meetings next week. Details (like the allusion in the name) are through the link and in the alt text.
🧶 #knitsky #mathknitting #symmetries
gallery.bridgesmathart.org/exhibitions/...