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Mathematickal Arts

“To me the simple act of tying a knot is an adventure in unlimited space. A bit of string affords the dimensional latitude that is unique among the entities. For an uncomplicated strand is a palpable object that, for all practical purposes, possesses one dimension only. If we move a single strand out of the plane, interlacing at will, actual objects of beauty result in what is practically two dimensions; and if we choose to direct our strand out of this plane, another dimension is added which provides an opportunity that is limited only by the scope of our own imagery and the length of a ropemakers coil.” -The Book of Knots, Clifford W Ashley

Mathematickal arts workshop is organised by FoAM, as a part of Resilients (http://resilients.net) and Splinterfields (http://fo.am/splinterfields). The hypothesis of the workshop is that cultural resilience can be increased by (1) supporting a community of generalists able to connect disparate concepts and disciplines together, as well as (2) connecting traditional disciplines (such as mathematics and textile crafts) with contemporary and emerging technologies (such as computer programming or bioinformatics).

The workshop is designed and lead by Carole Collet, coming from textile education and sustainable design, together with Tim Boykett, a mathematician, artist and nautical enthusiast. The participants include a dozen people from a range of backgrounds: from textiles to free culture, graphic design to civil engineering, permaculture to lightweight structures, open source programming to cooking, with an even wider range of interests - origami to dough, fluffy dogs to algorithms.

Workshop leaders:

Day 1: 20110725

After the introductions to the workshop and its wider context of cultural resilience, the participants were invited to warm up by playing a human knot game - randomly joining all hands and trying to unknot to a loop or a twist without letting go (http://www.wikihow.com/Play-the-Human-Knot-Game). The theoretical session began with an overview of possible connections between mathematics and textile crafts. They talked about patterns and symmetries, knots and hyperbolic geometries, Jacquard looms and cellular automata. The presentation showed a broad range of possibilities that the workshop could unfold into, but its final direction depends on the participants and their interests. Slides can be found [[……]]

Dissecting a Moebius strip

The first practical exercise took one of the seemingly simple mathematical phenomena - the Moebius strip, and attempted to intuitively predict what will happen when one begins cutting it. What will be the lengths, twists, knots? How would you explain what happen to someone who'd like to try it themselves?

After proceeding to cut, count and get entangled in strips of paper, the group came together to look at each other's notational experiments and diverse attempts to make sense of what was happening to the Moebius strip through a simple process of cutting - in words, diagrams and algoritms. One of the ways to make sense is to use mathematical reasoning - from a conjecture, through evidence and observation to a proof. For example:

Conjecture: When the length is twice as long it has twice as many twists; odd number of twists - you go around the loop twice, so you double the length and you double the twists - you can never get more than double the original length.

  • odd number of twists –> 1/2 cut –> double length, double twist
  • odd number of twists –> 1/3 cut –> double length double twist + same length, same twist

Evidence:

  • 1 twist 1/2 cut = two pieces of twice the length, 2 twists
  • 1 twist, 1/4 cut = 1 same length, 1 twist + 1 2 lengths, 2 twists

Observations:

  • counting twists is hard
  • when you bring patterns in your cuts, it gives you more headaches, but becomes a creative exercise

Summary of experiments (i.e. “what did we do”):

  • We have taken a simple piece of material and done a series of experiments, taken notes and tried to make a summary of ideas in a working notation to help us remember what we've done, then sketched a proof to be able to repeat the experiment. The conclusion is that there are many different conjectures possible from a simple starting point.
  • We got into trouble quite quickly (intuition was contrary to empirical evidence). Mathematician Trevor Evans said that if maths contributed anything to the world it was to put common sense in a box on a top shelf, just next to a box with nonsense (paraphrasing Tim). Maths keeps messing with our intuitions (common sense), in a repeatable way.
  • We started with hands-on experiments, then moved on comparing and abstracting.

“Cutting a Möbius strip along the center line yields one long strip with two full twists in it, rather than two separate strips; the result is not a Möbius strip. This happens because the original strip only has one edge which is twice as long as the original strip. Cutting creates a second independent edge, half of which was on each side of the scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other, each with two full twists.” http://en.wikipedia.org/wiki/M%C3%B6bius_strip

On hyperbolic geometry

Example of non-euclidean geometry challenging Euclid's parallel postulate:

Euclid's geometry - flat surfaces, parallel lines never meet Circular geometry - there is always one point where two lines will meet Hyperbolic geometry - there are at least two parallel lines diverging (More: http://en.wikipedia.org/wiki/Hyperbolic_geometry)

After a short demonstration of FoAM's stitched hyperbolic surfaces (http://www.flickr.com/photos/foam/sets/72157626527782401/), the participants had a quick crocheting tutorial and made their first attempts at crocheting hyperbolic surfaces (http://www.math.cornell.edu/~dwh/papers/crochet/crochet.html) in wool and rope.

In addition to stitching and knotting, folding flat square pieces of paper (i.e. origami) was a third technique used at the workshop to explore hyperbolic geometry. The participants began with single hyperbolic paraboloids (http://erikdemaine.org/hypar/), and branched out into origami tesselations, such as the “water bomb” (http://www.youtube.com/watch?v=VXIVHjws15U)

Movie suggestion:

Software: Computational origami

References to practical exercises:

Ideas for day 2: Macramé to find out about different knot theories. Japanese bondage technique to avoid knots, installation with ropes. Remixing folds and knots, weaving the space, turning scribbles into weaves, weaves into knots…

Ideas:

  • museums in Brussels and Gent

http://lace.lacefairy.com/Lace/International/Brussells.html

  • Creating Sorting algorithms as per e.g. http://sortvis.org/ as textiles
  • macrame as large scale lacemaking (is this true?)http://www.wikihow.com/Macrame
  • 10mm braid rope - multiple colours, prefer nonpatterned rope so as to let the knot patters stand out
  • should it be a Research Workshop rather than a doing learning workshop?
    • Plan is that we do a series of smaller “exercises” over 1 (or 2?) days on various themes
    • 2D Symmetries in fabrics, relation to the planar symmetry groups
    • Use of symmetries, e.g. to be able to cut and still have appropriate pattern structures.
    • Hiding symmetries (esp repetitions) in textiles
    • Notational techniques in knitting and crochet, also weaving. relations to mathematical notations, procedural notations.
    • Splicing in 3 and 4 core rope. Symmetry (in end splices) and symmetry breaking (in eye splicing), the two splicing techniques
    • Parametrisation in textiles and sizes (related strongly to notation)
  • Biology and mathematics?
  • Knotting patterns that give emergent 3D structures (including hyperbolic_geometry): e.g. crochet→ Hyperbolic Reef (link needed).
  • Given a pattern, what is the simplest algorithm to make it → Kolmogorov complexity
  • Challenge: “How to learn some (nontrivial) mathematics through textiles”
  • large scale 3D weaving - space filling cords.
  • Museum visit at close of first day?
  • Material Budget: around 700
  • Dates: 3 days around the weekend of 23-25 July 2011
  • Bridges Conference examples:
    • Celtic knot patterns
    • self similar knots
    • weaving: see also html version here
    • Moebius Strip knitting
  • Question: are Celtic knot patterns related to Turk's Head knots?

Resources

misc:

  • Brunnian rings: Nils Baas networks that unravel with a single cut.
Notes from dave

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