The Home of Mathematical Knitting (sarah-marie's mathematical knitting pages)

Also: Rachel's Cool Increase is a match for k2tog; two knit stitches flow out of one with no bumps.

Conferences and Books

Carolyn Yackel and sarah-marie belcastro co-organized two mathematics conference sessions, each of which included a sequence of talks and a mathematical fiber arts exhibit. We have co-edited two books on mathematics and fiber arts written for crafters, mathematicians, and mathematics educators.

Making Mathematics with Needlework is based on the based on the 2005 AMS Special Session in Mathematics and Mathematics Education in Fiber Arts (at the January 2005 Joint Mathematics Meetings in Atlanta, GA). A customizable pattern for the hyperbolic baby pants from Making Mathematics with Needlework is available for free at the Wolfram Demonstrations Project.

Crafting by Concepts is partially based on the 2009 AMS Special Session on Mathematics and Mathematics Education in Fiber Arts (at the January 2009 Joint Mathematics Meetings in Washington, DC).

sarah-marie's mathematical knitting

exhibits of sarah-marie's work:

YxI: Core and Cowl, Mathematical Art Exhibition (Juried), 2013 Joint Mathematics Meetings

Spring Forest (5,3) and Nonorientable Surfaces of Low Genus, ATMCS5 Art Exhibition (Juried), Algebraic Topology: Methods, Computation and Science, International Centre for Mathematics Sciences, Edinburgh, UK, July 2–6, 2012.

Negatively curved Mobius bands, Virtual and Physical Shape Exhibition (Juried), Shape Modeling International, College Station, TX, May 22–25, 2012.

Dusk or Dawn (K_6 on Mobius band) and 8-colored Hyperbolic Pants (Map on the Surface of a Two-Holed Torus Requiring 8 Colors), Sticks, Hooks, and the Mobius: Knit and Crochet Goes Cerebral, Lafayette College Art Gallery, January 8–February 5, 2012.

Negatively curved Mobius Band and K_7 embedded on a torus, Mathematical Art Exhibit (Juried), 2012 Joint Mathematics Meetings

A mirror pair of (3,2) torus knots embedded on tori, Exhibition of Mathematical Art (Juried), 2010 Joint Mathematics Meetings

Six of my pieces were included in Yarn Theory, curated by Martha Lewis. PS122 Gallery, New York City, Apr. 25–May 17, 2009.

Nonuniform Nonorientable Nonsurfaces, Mathematical Fiber Arts Exhibit (Juried), 2009 Joint Mathematics Meetings

Connected sum of four projective planes, Exhibition of Mathematical Art (Juried), 2008 Joint Mathematics Meetings

Dual Seven-Colored Tori, SIGMAA-ARTS Juried Art Exhibit, Mathfest 2007. images here (.pdf from the MAA, see p.14)

Dual Seven-Colored Tori, Gathering 4 Gardner 7 Exhibit, March 2006. images here (scroll down page)

media mentions of sarah-marie's work:

Mathematical Knitting, by Cory Doctorow, boingboing.net, February 15, 2013
Knitting reinvented: Mathematics, feminism and metal by Mark Ward, BBC News Technology, August 20, 2012
Interview by Lara Neel in Show 110 of the Math4Knitters podcast
All Things Considered piece by Ari Daniel Shapiro, January 10, 2012 (photos on his blog)
Math DongA (Korean high-school magazine) October 2010, p. 21.
Klein Bottle Hat, in 1000 Fabulous Knit Hats by Annie Modesitt, p. 133.
Crea-beta: Rechtlijnig, Mobiussjahl (.pdf), natuur-weenschap & techniek, jaargang 78, nummer 3, maart 2010, pp. 54-55.
The Bizarre and Brilliant World of Knitted Science, Discover Magazine Online Gallery by Eliza Strickland, September 2009.
Knitting by Numbers, Lucinda Mathews, Plus Magazine, Issue 51, June 2009 (runner up in the general public category of the Plus new writers award 2009).
New Math by Liz Worth, broken pencil blog, September 1, 2007.
Knitting a Mobius Band by Elise Gibson, Smith Alumnae Quarterly, Spring 2007.
Mathematical Knitting by Sarah E. White for about.com, Jan. 24, 2007
Knitting Network by Ivars Peterson in MathTrek, Science News Online, Jan. 27, 2007; Vol. 171, No. 4.
Crafting Geometry (.pdf from O'Reilly Media) by Arwen O’Reilly, CRAFT: magazine, Vol. 01, pp. 46–49.
Crafty Geometry (.pdf from fiberfolio) by Erica Klarreich, Science News, Dec. 23, 2006; Vol. 170, No. 26/27, p. 411.

Not on this page: sarah-marie's home page

talks on mathematical knitting

You know, just in case you want to invite someone to speak on mathematics and knitting...and if you've been to or given a talk not listed here, please let me know so I can add it.

• Mary Harris, Spring 2011, Some Mathematics Within? (YouTube recording of much of the talk, in four parts) Mary reviews some of her educational projects that expose the mathematical thinking embedded in practical knitting (and sewing and weaving and...) that is done around the world.
• Hugh Griffiths, Spring 2010, Geometric Knitting II . (slides) This describes how to knit Seifert surfaces, generate cellular automata lace (ruby code), and make a particular universal cover scarf (ruby code).
• Hugh Griffiths, Fall 2007, Geometric Knitting. (slides) Hugh describes how to use differential geometry to use the metric on an embedded surface to produce a knitting pattern.
• Amy Szczepanski, Spring 2006, Knitting Geometric Shapes: How do you knit a circle? Amy talks about how gauge and appropriate increasing combine to design a proper circle, and applies her methodology to discover and fix a mathematical flaw in Elizabeth Zimmerman's Pi Shawl.
• Amanda Redlich, Spring 2006, Knitting Knerdery. (abstract) Her survey for non-knitters included coordinatizing knitting, non-rectangular grids, different mobius strip constructions, and some mathematics which arises from knitting (knot theory, curvature/shaping, Turing machines).
• Nate Berglund, Spring 2006, Knitting as a means of Visualizing One-Sided Surfaces. (slides) His audience is mathematicians (beginning grad student level). Nate explains how knitting can model immersions of surfaces, and shows how he designed his knitted Boy's Surface using a polyhedral model of the immersion.
• and of course there were many talks at the AMS Special Sessions in January 2005 and January 2009.

A class on math and fiber arts: Jill Zarestky's seminar at TAMU.

Selected external links: Mathematical Knitting

Jeny Carden Staiman's double-knitted Möbius-band-into-torus experiment
Jana's Escher patterns: koi-1, koi-2, birds/rays/turtles cushions, birds/rays/turtles blanket, birds/rays/turtles symmetry, fish tessellation, fish tessellation geometry.
Staceyjoy Elkins' compact notation for texture patterns
Mark Shoulson's homemade topological shapes include various embeddings of nonorientable surfaces
Claire Irving's embeddings and immersions of the projective plane (look for a paper by Claire in the reference list)
Nate Berglund's wacky projective-plane-on-a-hat and other nonorientable knitting
Dan Isaksen's Möbius Knitting paper. I disagree with the conclusions of this paper, because (a) the special stitch introduced is not as symmetric as claimed (it still has a half-stitch glide reflection) and (b) plain seed/moss stitch is just as symmetric, but much simpler. But Dan is cool anyway, because he can use cohomology to do arithmetic.
Janet's Petersen Graph stitch pattern (subtle and brilliant)
Sarah Hauschka's knitted tessellation (it's 6 x 8 feet!)
Wooly Thoughts contains a variety of objects and patterns with mathematics ranging from elementary to advanced.
Knitting pattern generators:

• Brittany's Random Lace chart generator (constrained by having each increase immediately followed by a decrease)
• Brittany's Random Cables chart generator
• Susann Hajjar's Cellular Automaton chart generator
• Susann Hajjar's Random Squares chart generator (this gives a square motif with 90-degree rotational symmetry)
• Susann Hajjar's Maze chart generator
• Caitlin's very flexible random stripe generator
• Laura Kogler's Mosaic knitting pattern generator

generate cellular automaton knitting patterns with perl (for more on cellular automaton knitting, check out Debbie New's awesome Unexpected Knitting)
Brent Annable's perfect (yes! it is!) sphere and self-intersecting double-knitting and rotating ring of tetrahedra (youtube link)
Camille Fox's cellular automata tea cozies (Rule 30 and Rule 110)
Lisa Grossman's change-ringing socks (for more on the math, see here)
Pattern: Fibonacci sleeveless shirt
Lara Neel's Math4Knitters podcast/blog (where this website is mentioned, in Episode 12 (of the old incarnation of the podcast)! and again in Episode 33!(.mp3; text here))
Eleanor Kent hand knits mathematical images of the sort we mathematicians generally have to computer-generate.
Norah Gaughan's Celestine
Sondra Eklund's prime factorization sweater
Horst Schultz's tumbling blocks design (lots of other beautiful things on that site...)
Claire's tetrahedral dice bag
Rachel Bishop's Perfect Shuffle Scarves, which are cute. (Try to ignore her poor mathematical grammar and highly nonstandard terminology.)
Heather Taylor's article on knitting and mathematics in elementary school
Fuzzy Galore's use of probability and Fibonacci numbers in knitting design
fictional mathematical knitting...
...and who can resist Cliff Stoll's Acme Klein Bottle?
Marie-Christine Mahé has another klein bottle hat, in Knitty.

You can find more mathematical knitting projects if you are a ravelry member by flipping through my favorites.

I know there must be more out there: witness qB (Rachel Rawlins)'s comment from 1.12.2004 on a Making Light blog entry. "Damn! and I thought I was the only person to explore maths in knitting. One particular favourite was to start with the alphabet in morse code (x and blank), write an appropriate word or series of initials as the base and then run Xor or other similar function in a spreadsheet. Binary numbers would also do, I suppose. Then you have a unique pattern which can form the basis for work in either colour or texture."

(There are tons of technical textile sites and professional journals which deal with the engineering issues involved in the knitting industry. While there is plenty of mathematics involved, it all deals with the physical properties of yarn instead of examining the mathematics of knitting itself.)

Not just math, but other disciplines are covered at the very-cool thomasina's guide to geeky knitting and geeky knitting index.

Selected mathematical fiber arts (but not knitting, see above) external links

I haven't been able to make an exhaustive list of cool stuff on the web relating to mathematical fiber arts other than knitting, because, well, I don't know much about the other fiber arts, mostly. So I'm not quite sure how to effectively search. Thus, like just about everything else on my pages... if you know of something I've missed, please tell me.

Mathematical Quilting
Rebecca Chaky's Flow Snake
Sarah Mylchreest and Mark Newbold's A Piece of Hyperspace (another execution here)
Tanya Scharaschkin's Penrose Quilt
Sarah Mylchreest and Mark Newbold's Penrose Tiling quilt
Lisbeth G. Clemons' Penrose Quilt
Karen Meagher's quilts
Anabeth Dollins' mathematics quilt and quilted proof of the Pythagorean Theorem (gorgeous!)
Gwen Fisher's quilted Cayley tables
Elaine Ellison's mathematical quilts pages---check out pages 2, 4, 5 in particular.
Lorrie Kim's quilt that is also a GCD table
Lorrie Kim's quilt with Perl-generated blocks (generated in Perl by Mark Jason Dominus)
Irena Swanson's quilts
Travis's visit to a quilt show
PDF of slides for Jason Cantarella's talk on L-systems in quilt design (some L-systems create tilings which correspond to quilt pieces; those that don't suggest topstitching patterns)

Mathematical Crochet
Daina Taimina's home page and awesome geometry textbook page
Fluxx's logic statement
Ariel Barton's Seifert surface
Matthew Wright's Seifert surfaces and Alexander Horned Sphere
Patrick Rodriguez's Crochet Sphere Pattern Generator
Florine Meijer's Seifert surface, double torus, cube with holes, weird hyperbolic structure, and 3-4-5
Janet's self-orthogonal latin square of order 10

Mathematical Needlepoint
William Mitchell's Pascal's Triangle
John Young's QR Code Pillow and a different QR Code chart by Janet M. Perry

Mathematical Bead Weaving
Gwen Fisher's gallery

Defies Categorization
Anabeth Dollins' pieced polyhedral balls and page about stuffed rhombic hexecontahedra