Ideas from

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

have been used as inspiration for

fiber art Fiber art (fibre art in British spelling) refers to fine art whose material consists of natural or synthetic fiber and other components, such as fabric or yarn. It focuses on the materials and on the manual labor on the part of the artist as ...

s including

quilt A quilt is a multi-layered textile, traditionally composed of two or more layers of fabric or fiber. Commonly three layers are used with a filler material. These layers traditionally include a woven cloth top, a layer of batting or wadding, a ...

making,

knitting Knitting is a method by which yarn is manipulated to create a textile, or fabric. It is used to create many types of garments. Knitting may be done by hand or by machine. Knitting creates stitches: loops of yarn in a row, either flat or i ...

cross-stitch Cross-stitch is a form of sewing and a popular form of counted-thread embroidery in which X-shaped stitches in a tiled, raster-like pattern are used to form a picture. The stitcher counts the threads on a piece of evenweave fabric (such as line ...

crochet Crochet (; ) is a process of creating textiles by using a crochet hook to interlock loops of yarn, thread (yarn), thread, or strands of other materials. The name is derived from the French term ''crochet'', meaning 'hook'. Hooks can be made from ...

embroidery Embroidery is the craft of decorating fabric or other materials using a needle to apply thread or yarn. Embroidery may also incorporate other materials such as pearls, beads, quills, and sequins. In modern days, embroidery is usually seen on c ...

and

weaving Weaving is a method of textile production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth. Other methods are knitting, crocheting, felting, and braiding or plaiting. The longitudinal th ...

. A wide range of mathematical concepts have been used as inspiration including

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

and

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

. Some techniques such as

counted-thread embroidery Counted-thread embroidery is any embroidery in which the number of warp and weft yarns in a fabric are methodically counted out for each stitch, resulting in uniform-length stitches and a precise, uniform embroidery pattern. Even-weave fabric is ...

are naturally

geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

; other kinds of

textile Textile is an umbrella term that includes various fiber-based materials, including fibers, yarns, filaments, threads, different fabric types, etc. At first, the word "textiles" only referred to woven fabrics. However, weaving is not the ...

provide a ready means for the colorful physical expression of mathematical concepts.

Quilting

The

IEEE Spectrum ''IEEE Spectrum'' is a magazine edited by the Institute of Electrical and Electronics Engineers. The first issue of ''IEEE Spectrum'' was published in January 1964 as a successor to ''Electrical Engineering''. The magazine contains peer-revie ...

has organized a number of competitions on

quilt block In the textile arts, a motif (also called a block or square) is a smaller element in a much larger work. In knitting and crochet, motifs are made one at a time and joined together to create larger works such as afghan blankets or shawls. A ...

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 In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618. Golden rectangles exhibit a special form of self-similarity ...

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...

's Claw, the

Koch curve The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...

, the

Clifford torus In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdo ...

San Gaku Sangaku or San Gaku ( ja, 算額, lit=calculation tablet) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classe ...

, Mascheroni's

cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spi ...

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...

spidron ''This article discusses the geometric figure; for the science-fiction character see Spidron (character).'' In geometry, a spidron is a continuous flat geometric figure composed entirely of triangles, where, for every pair of joining triangles, ea ...

s, and the six

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

Knitting and crochet

Knitted mathematical objects include the

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...

s and

Boy's surface In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' be immersed in 3-space ...

. The Lorenz manifold and the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

have been crafted using crochet. Knitted and crocheted tori have also been constructed depicting toroidal embeddings of the

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is c ...

''K''₇ and of the

Heawood graph Heawood is a surname. Notable people with the surname include: *Jonathan Heawood, British journalist *Percy John Heawood (1861–1955), British mathematician **Heawood conjecture **Heawood graph **Heawood number In mathematics, the Heawood number ...

. 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 ''Crocheting Adventures with Hyperbolic Planes'' is a book on crochet and hyperbolic geometry by Daina Taimiņa. It was published in 2009 by A K Peters, with a 2018 second edition by CRC Press. Topics The book is on the use of crochet to make ...

'', won the 2009

Bookseller/Diagram Prize for Oddest Title of the Year The ''Bookseller''/Diagram Prize for Oddest Title of the Year, originally known as the Diagram Group Prize for the Oddest Title and commonly known as the Diagram Prize, is a humorous literary award that is given annually to a book with an unusua ...

Embroidery

Embroidery techniques such as

including

and some

canvas work Canvas is an extremely durable plain-woven fabric used for making sails, tents, marquees, backpacks, shelters, as a support for oil painting and for other items for which sturdiness is required, as well as in such fashion objects as handba ...

methods such as

Bargello The Bargello, also known as the Palazzo del Bargello, Museo Nazionale del Bargello, or Palazzo del Popolo (Palace of the People), was a former barracks and prison, now an art museum, in Florence, Italy. Terminology The word ''bargello'' appears ...

make use of the natural

pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the smal ...

s of the weave, lending themselves to geometric designs.

Weaving

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

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

s. used the

Rule 90 In the mathematics, mathematical study of cellular automaton, cellular automata, Rule 90 is an elementary cellular automaton based on the exclusive or function. It consists of a one-dimensional array of cells, each of which can hold either a 0 or ...

cellular automaton A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...

to design

tapestries Tapestry is a form of textile art, traditionally woven by hand on a loom. Tapestry is weft-faced weaving, in which all the warp threads are hidden in the completed work, unlike most woven textiles, where both the warp and the weft threads may ...

depicting both trees and abstract patterns of triangles.

Spinning

Margaret Greig was a mathematician who articulated the mathematics of worsted spinning.

Fashion design

The silk scarves from DMCK Designs' 2013 collection are all based on Douglas McKenna's

space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space ...

patterns. The designs are either generalized Peano curves, or based on a new space-filling construction technique. The

Issey Miyake was a Japanese fashion designer. He was known for his technology-driven clothing designs, exhibitions and fragrances, such as '' L'eau d'Issey'', which became his best-known product. Life and career Miyake was born on 22 April 1938 in Hirosh ...

Fall-Winter 2010–2011 ready-to-wear collection designs from a collaboration between fashion designer Dai Fujiwara and mathematician

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...

. The designs were inspired by Thurston's

geometrization conjecture In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimens ...

, the statement that every

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

can be decomposed into pieces with one of eight different uniform geometries, a proof of which had been sketched in 2003 by

Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...

as part of his proof of the

Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...

References

External links

Mathematical quiltsMathematical craft projectsCrocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina
(2005) {{Textile arts Mathematics and culture Textile arts Recreational mathematics Mathematics and art