Mathematics and fiber arts
Encyclopedia
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
ideas have been used as inspiration for a number of fiber art
Fiber art
Fiber art is a style of fine art which uses textiles such as fabric, yarn, and natural and synthetic fibers. It focuses on the materials and on the manual labour involved as part of its significance.-Fiber:...
s including quilt
Quilt
A quilt is a type of bed cover, traditionally composed of three layers of fiber: a woven cloth top, a layer of batting or wadding and a woven back, combined using the technique of quilting. “Quilting” refers to the technique of joining at least two fabric layers by stitches or ties...
making
Quilting
Quilting is a sewing method done to join two or more layers of material together to make a thicker padded material. A quilter is the name given to someone who works at quilting. Quilting can be done by hand, by sewing machine, or by a specialist longarm quilting system.The process of quilting uses...
, knitting
Knitting
Knitting is a method by which thread or yarn may be turned into cloth or other fine crafts. Knitted fabric consists of consecutive rows of loops, called stitches. As each row progresses, a new loop is pulled through an existing loop. The active stitches are held on a needle until another loop can...
, cross-stitch
Cross-stitch
Cross-stitch is a popular form of counted-thread embroidery in which X-shaped stitches in a tiled, raster-like pattern are used to form a picture. Cross-stitch is often executed on easily countable evenweave fabric called aida cloth. The stitcher counts the threads in each direction so that the...
, crochet
Crochet
Crochet is a process of creating fabric from yarn, thread, or other material strands using a crochet hook. The word is derived from the French word "crochet", meaning hook. Hooks can be made of materials such as metals, woods or plastic and are commercially manufactured as well as produced by...
, embroidery
Embroidery
Embroidery is the art or handicraft of decorating fabric or other materials with needle and thread or yarn. Embroidery may also incorporate other materials such as metal strips, pearls, beads, quills, and sequins....
and weaving
Weaving
Weaving is a method of fabric production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth. The other methods are knitting, lace making and felting. The longitudinal threads are called the warp and the lateral threads are the weft or filling...
. A wide range of mathematical concepts have been used as inspiration including topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
.
Quilting
The IEEE SpectrumIEEE Spectrum
IEEE Spectrum is a magazine edited by the Institute of Electrical and Electronics Engineers. The IEEE's description of it is:IEEE Spectrum began publishing in January 1964 as a successor to Electrical Engineering...
has organized a number of competitions on Quilt Block Design, and several books have been published on the subject. Notable quilt makers 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 section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s, Leonardo da Vinci
Leonardo da Vinci
Leonardo di ser Piero da Vinci was an Italian Renaissance polymath: painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer whose genius, perhaps more than that of any other figure, epitomized the Renaissance...
's Claw, the Koch curve, the Clifford torus
Clifford torus
In geometric topology, the Clifford torus is a special kind of torus sitting inside R4. Alternatively, it can be seen as a torus sitting inside C2 since C2 is topologically the same space as R4...
, San Gaku, Mascheroni
Lorenzo Mascheroni
Lorenzo Mascheroni was an Italian mathematician.He was born near Bergamo, Lombardy. At first mainly interested in the humanities , he eventually became professor of mathematics at Pavia....
's cardioid
Cardioid
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 is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp...
, Pythagorean triple
Pythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, 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 k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...
s, spidron
Spidron
In geometry, a spidron is a continuous flat geometric figure composed entirely of triangles, where, for every pair of joining triangles, each has a leg of the other as one of its legs, and neither has any point inside the interior of the other...
s, and the six trigonometric functions.
Knitting and crochet
Knitted mathematical objects include the Platonic solidPlatonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s, Klein bottle
Klein bottle
In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
s and Boy's surface
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...
.
The Lorenz manifold
Lorenz attractor
The Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape...
and the hyperbolic plane
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
have been crafted using crochet. Knitted and crochetted tori
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
have also been constructed depicting toroidal embeddings
Toroidal graph
In mathematics, a graph G is toroidal if it can be embedded on the torus. In other words, the graph's vertices can be placed on a torus such that no edges cross...
of the complete graph
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.-Properties:...
K7 and of the Heawood graph
Heawood graph
In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.-Combinatorial properties:...
. The crocheting of hyperbolic planes has been popularized by the Institute For Figuring
Institute For Figuring
The Institute For Figuring is an organization based in Los Angeles, California that promotes the public understanding of the poetic and aesthetic dimensions of science, mathematics and the technical arts...
; a book by Daina Taimina
Daina Taimina
Daina Taimina is a Latvian mathematician, currently Adjunct Associate Professor at Cornell University, known for crocheting objects to illustrate hyperbolic space. She received all her formal education in Riga, Latvia, where in 1977 she graduated summa cum laude from the University of Latvia and...
on the subject, Crocheting Adventures with Hyperbolic Planes, won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year
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 at the Frankfurt Book Fair, commonly known as the Diagram Prize for short, is a humorous literary award that is given annually to the book with the oddest title...
.
Cross-stitch
Many of the wallpaper patterns and frieze groupFrieze group
A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
s have been used in cross-stitch.
Weaving
Ada DietzAda Dietz
Ada K. Dietz was an American weaver best known for her 1949 monograph Algebraic Expressions in Handwoven Textiles, which defines a novel method for generating weaving patterns based on algebraic patterns. Her method employs the expansion of multivariate polynomials to devise a weaving scheme...
(1882 – 1950) was an American weaver
Weaving
Weaving is a method of fabric production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth. The other methods are knitting, lace making and felting. The longitudinal threads are called the warp and the lateral threads are the weft or filling...
best known for her 1949 monograph Algebraic Expressions in Handwoven Textiles, which defines weaving patterns based on the expansion of multivariate polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s.
used the Rule 90
Rule 90
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 a 1 value; in each time step all values are simultaneously replaced by the exclusive or of the two neighboring values...
cellular automaton
Cellular automaton
A cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...
to design tapestries
Tapestry
Tapestry is a form of textile art, traditionally woven on a vertical loom, however it can also be woven on a floor loom as well. It is composed of two sets of interlaced threads, those running parallel to the length and those parallel to the width ; the warp threads are set up under tension on a...
depicting both trees and abstract patterns of triangles.
Fashion design
The Issey MiyakeIssey Miyake
is a Japanese fashion designer. He is known for his technology-driven clothing designs, exhibitions and fragrances.-Life and career:Miyake was born 22 April 1938 in Hiroshima, Japan. As a seven year-old, he witnessed and survived the atomic bomb dropped on Hiroshima on August 6, 1945. He studied...
Fall-Winter 2010–2011 ready-to-wear collection featured designs from a collaboration between fashion designer Dai Fujiwara and mathematician William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
. The designs were inspired by Thurston's geometrization conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
, the statement that every 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
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 Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
as part of his proof of the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
.