Ideas from Mathematics have been used as inspiration for fiber arts including quilt making, knitting, cross-stitch, crochet, embroidery and weaving. A wide range of mathematical concepts have been used as inspiration including topology, graph theory, number theory and algebra. Some techniques such as counted-thread embroidery are naturally geometrical; other kinds of textile provide a ready means for the colorful physical expression of mathematical concepts.
Counted-thread embroidery is any embroidery in which the fabric threads are counted by the embroiderer before inserting the needle into the fabric. Evenweave fabric is usually used; it produces a symmetrical image as both warp and weft fabric threads are evenly spaced. The opposite of counted-thread embroidery is free embroidery.
Stranded mathematical objects include Platonic solids, Klein bottles and the child’s face. Lorenz was created using manifold and hyperbolic plane claws. The work of hyperbolic plane crochet was embroidered by the decoration institute of the designs in the way that the people liked. Many wall patterns and frieze groups were used in cross-stitching.
The IEEE Spectrum has organized a number of competitions on quilt block design, and several books have been published on the subject. Notable quiltmakers include Diana Venters and Elaine Ellison, who have written a book on the subject Mathematical Quilts: No Sewing Required. Examples of mathematical ideas used in the book as the basis of a quilt include the golden rectangle, conic sections, Leonardo da Vinci’s Claw, the Koch curve, the Clifford torus, San Gaku, Mascheroni’s cardioid, Pythagorean triples, spidrons, and the six trigonometric functions.
Ada Dietz (1882 – 1950) was an American weaver, best known for his book Algebraic Expressions in Handwoven Textiles, which he described in 1949, which heavily based on the extensibility of polynomials.
Knitted mathematical objects include the Platonic solids, Klein bottles and Boy’s surface. The Lorenz manifold and the hyperbolic plane have been crafted using crochet. Knitted and crocheted tori have also been constructed depicting toroidal embeddings of the complete graph K7 and of the Heawood graph. The crocheting of hyperbolic planes has been popularized by the Institute For Figuring; a book by Daina Taimina on the subject, Crocheting Adventures with Hyperbolic Planes, won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year.
Embroidery techniques such as counted-thread embroidery including cross-stitch and some canvas work methods such as Bargello (needlework) make use of the natural pixels of the weave, lending themselves to geometric designs.
Ada Dietz (1882 – 1950) was an American weaver best known for her 1949 monograph Algebraic Expressions in Handwoven Textiles, which defines weaving patterns based on the expansion of multivariate polynomials.
Margaret Greig was a mathematician who articulated the mathematics of worsted spinning.
The short draw technique can be done from carded rolags, as well, but this does not produce a strictly worsted yarn. Yarns spun from a rolag will not have all the fibers parallel to the yarn though, with the short draw technique, many will be. Drum carded fiber, however, does have the fibers all parallel to each other, and thus can be used to create a strictly worsted yarn.
The original spinning machinery was based on the short draw technique. Instead of an active and passive hand, the drafting was done by two sets of rollers moving at different speeds. However, the short draw characteristics remain: the fibers in the resulting yarn are all parallel, and there is no twist in the drafting area. Even in the modern day, many spinning machines are based on this principle.
The silk scarves from DMCK Designs’ 2013 collection are all based on Douglas McKenna’s space-filling curve patterns. The designs are either generalized Peano curves, or based on a new space-filling construction technique.
The Issey Miyake Fall-Winter 2010–2011 ready-to-wear collection featured designs from a collaboration between fashion designer Dai Fujiwara and mathematician William Thurston. The designs were inspired by Thurston’s geometrization conjecture, the statement that every 3-manifold can be decomposed into pieces with one of eight different uniform geometries, a proof of which had been sketched in 2003 by Grigori Perelman as part of his proof of the Poincaré conjecture.
In 1890, Peano discovered a continuous curve, now called the Peano curve, that passes through every point of the unit square (Peano (1890)). His purpose was to construct a continuous mapping from the unit interval onto the unit square. Peano was motivated by Georg Cantor’s earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifold, such as the unit square. The problem Peano solved was whether such a mapping could be continuous; i.e., a curve that fills a space. Peano’s solution does not set up a continuous one-to-one correspondence between the unit interval and the unit square, and indeed such a correspondence does not exist (see below).
Peano’s ground-breaking article contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator. But the graphical construction was perfectly clear to him—he made an ornamental tiling showing a picture of the curve in his home in Turin. Peano’s article also ends by observing that the technique can be obviously extended to other odd bases besides base 3. His choice to avoid any appeal to graphical visualization was, no doubt, motivated by a desire for a well-founded, completely rigorous proof owing nothing to pictures. At that time (the beginning of the foundation of general topology), graphical arguments were still included in proofs, yet were becoming a hindrance to understanding often counterintuitive results.
A year later, David Hilbert published in the same journal a variation of Peano’s construction (Hilbert 1891). Hilbert’s article was the first to include a picture helping to visualize the construction technique, essentially the same as illustrated here. The analytic form of the Hilbert curve, however, is more complicated than Peano’s.
Source From Wikipedia