Knitting, Chinese Knots, Braid Theory

By Xah Lee. Date:

Am fascinated by the math aspects of knitting… ⁖ the pure math aspect of how those threads are inter-woven, such as in Braid theory.

knitting red courses stockinette garter
Knitting. image source
cable knitting
Cable kniting. image source

Besides the math aspect of which thread are intertwined with which other threads in what way, it is also possible to create a pattern on the surface. What such patterns are possible, how are they made, must by itself be a complex theory.

Serweta-na drutach2-s
Lace tablecloth made using knitting needles. 1536×2048 image source

Chinese Knots

Also related is Chinese knotting.

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Chinese knot. Source This one has the characteristics of fractal.
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Chinese knots. image source
chinese knot 7-luck
Chinese knot. image source
Chinese butterfly knot
Chinese knots. image source

Discovered this fantastic book, free: Encyclopedia Of Needlework , By Thérèse De Dillmont. @ Source.

It provides in-depth coverage, with hundreds of clear illustrations, many of which shows the pure weaving beauty of math.

Basket Weaving

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A weaving pattern (Wickerwork) commonly found in Asian chairs made of rattan.

Celtic Knots

Celtic knots pattern
Celtic knots strip
A traditional Celtic pattern (by Alastair Luke) The colorings are added on to make it easy to trace. In fact, the whole central cross with its round endings is of one single string. I just used different colors for different segments because otherwise it's one single color.

Decorative Weaving Patterns

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Left: From “Islamic Designs for artists and craftspeople”, by Eva Wilson, 1988. Above right, from a book.
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Computer generated weaving as decorative art. image source

This image shows the modern approach of possibilities on the subject of weaving. Note that many of the threads are closed circles, as links. A weaving made of entirely links is not explored in knitting, fabric, or any sort of weaving artifact, simply because it is not practical. However, as modern material science, this is a new vista. Or as mathematical classification of weaving, this have not been studied much, if at all.

Also, when we consider the math aspects of links, knots, weavings, we think of topology. That is, how each thread or inter-woven with other threads. However, weavings as a decorative art, the geometric positioning, is a critical part of the work.

Hyperbolic Surface Crochet

Unrelated to math aspects of weaving, but saw this the other day. Knitting a hyperbolic surface.

crochet hyperbolic surface
Knitting that forms a hyperbolic surface. image source

Actually, the person who knitted this is Daina Taimina, a mathematician, and she has written a book. Crocheting Adventures with Hyperbolic Planes , by Daina Taimina. amazon

Little Math Research

As far as i know there's not much serious mathematical study on knitting, Chinese knots, weaving, as a math subject. By that, i mean, definition, analysis, theories, of weaving from a professional mathematician, as a sub-field of combinatorial geometry, on what weaving are possible, their classification, etc. Not, for example, studies on how to create such work, or survey of abstract patterns on existing artworks, from artisans or artists or anthropologists, or algorithms about how to create such patterns by computer programers.

I've only seen some research papers by Branko Grünbaum and G C Shephard on the weaving studies. ⁖ A Catalogue of Isonemal Fabrics , Branko Grünbaum and G C Shephard, “Discrete Geometry and Covexity” vol 440.

Note that, among related folk arts, today there are relatively a lot math study of paper folding.