The Home of Mathematical Knitting

exhibits of sarah-marie's work:

Petersen-Embellished Non-orientable Blanket Square, Mathematical Art Exhibition (Juried), 2017 Joint Mathematics Meetings
Brunnian Link Cowls and Rainbow Brunnian Link Cowl, Mathematical Art Exhibition (Juried), 2016 Joint Mathematics Meetings
Spring Forest (5,3), MoSAIC (Mathematics of Science, Art, Industry, Culture Art) Traveling Exhibit (Curated), 2014--15
The smallest polyhedrally embedded snark on a small Klein bottle and Three (2k+2, 2k) links, Mathematical Art Exhibition (Juried), 2015 Joint Mathematics Meetings
Three two-component torus links and YxI: core, completion, and scarf, Mathematical Fiber Arts Exhibit (Juried), 2014 Joint Mathematics Meetings
Spring Forest (5,3): embedded, unembedded, and cowl and Modern Striped Klein, Mathematical Art Exhibition (Juried), 2014 Joint Mathematics Meetings
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.
Dual Seven-Colored Tori, Gathering 4 Gardner 7 Exhibit, March 2006. images here (scroll down page)

publicly available media mentions of sarah-marie's work:

Roots of Unity: Knotted Needles Make Knitted Knots by Evelyn Lamb, Scientific American blogs. February 26, 2014.
The Stunning Symbiosis between Math and Knitting by Evelyn Lamb, Scientific American online, and accompanying slideshow. February 24, 2014.
Math Digest piece by Claudia Clark, March 1, 2013
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
This website is mentioned in Lara Neel's Math4Knitters podcast/blog, in Episode 12 (show notes) and again in Episode 33 (show notes)!
All Things Considered piece by Ari Daniel Shapiro, January 10, 2012 (photos on his blog)
Klein Bottle Hat, in 1000 Fabulous Knit Hats by Annie Modesitt, p. 133.
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).
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.

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, and make a particular universal cover scarf.
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 that 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 and January 2014.

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

Selected links: Mathematical Knitting

Fiona Ellis describes mathematics used in shape/motif design.
Jeny Carden Staiman's double-knitted Möbius-band-into-torus experiment
Mark Shoulson's homemade topological shapes include various embeddings of nonorientable surfaces
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.
Lisa Grossman's change-ringing socks (for more on the math, see here)
Asher Kaboth's python script that generates cable pattern charts for change-ringing permutation sequences
- Syntax: placechart.py -n [number of bells] -t [title of method] [place notation for method]
- Place notation is described in Asher's pattern A Scarf of Cambridge Major (.pdf)
- By convention, change ringing only lists the first half of the permutations in a sequence, and ringers then reverse the first half to get the second half. The script doesn't incorporate this, so the user must enter the whole place notation sequence to get the entire chart.
Staceyjoy Elkins' compact notation for texture patterns
Janet's Petersen Graph stitch pattern (subtle and brilliant)
Sarah Hauschka's knitted tessellation (it's 6 x 8 feet!)
Woolly Thoughts contains a variety of objects and patterns with mathematics ranging from elementary to advanced.
Knitted items at the Joint Mathematics Meetings Art Exhibits:
- Susan Goldstine's double-knitted double knitting group sampler
- Heather Lewis's Norwegian Frieze
- Susan Goldstine's Fibonacci Downpour and Knit Bifurcation
- Heather Lewis's Flexagon Illusion
- Robin Endelman's Yang-Baxter equation scarf
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
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)
Eleanor Kent hand knitted 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 neat. (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.
Finally, some publicly-accessible Ravelry projects:
- Emily Peters's Fractal Cowl
- CountablyInfiknit's Sierinski Hatsket
- fbz's wolfram 1d cellular automata 01001001 scarf
- knitvana's kGeometry: Cube Tea Cozy with Circles
- chronographia's fib blanket

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

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

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 that 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.)

Selected links: Mathematical Fiber Arts (but not knitting)

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
Chawne Kimber / Ethan Berkove's origami 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 (and commentary)
Lorrie Kim's quilt with Perl-generated blocks (generated in Perl by Mark Jason Dominus)
Lorrie Kim's e quilt
Irena Swanson's quilts
Heather Dye's catenary quilt
Nina Paley and Theodore Gray's quiltimations and pi 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 that correspond to quilt pieces; those that don't suggest topstitching patterns)

Mathematical Crochet
Daina Taimina's home page and photo blog and and text blog
Hinke Osinga's crocheted Lorenz manifold
Sorrel Forbes's irrational ribbons
Ariel Barton's Seifert surface
Matthew Wright's Seifert surfaces and Alexander Horned Sphere
Emily Premise-Conclusion's ideal crocheted sphere
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
Lana Holden's space-filling curve
Moira Chas's hexagonal torus tiling
Susan Goldstine's bead-crocheted jewelry and hyperbolic surface
Heidi Burgiel's Hyperbolic {3,7} Afghan
Andrea Heald's Collatz cornsnake and Penrose doily and snood
Gabriele Meyer's many crocheted hyperbolic surfaces
Ron Asherov's crocheted spirals

Mathematical Needlepoint
Wing Mui's simple symmetric square sampler
Larry Riddle's space-filling curves and dragon curves and dragon curve tilings
Maggi Harriss's tiled cushions
William Mitchell's Pascal's Triangle

Mathematical Bead Weaving and Other Weaving
Gwen Fisher's gallery
Mary Klotz's triaxial weaving
Sally Eyring's ASCII weaving
Donna Contractor's Cantor dust weaving

Other
Veronika Irvine's periodic bobbin lace and reflection symmetry bobbin lace
Ellie Baker's invertible infinity scarf
Ayelet Lindenstrauss Larsen's Mobius color wheel
Louise Gould's multiplied snub cube
Anabeth Dollins' pieced polyhedral balls and page about stuffed rhombic hexecontahedra