The Mathematics of Sewing!

I have been thinking about the mathematics of sewing for some time. It seems to me that one could develop a really interesting little memoir on the subject. Here's a mini table of contents for such a text :

1. Introduction
   
2. Metric spaces - from 1D to 2D
   
3. Metric spaces - from 2D to 3D
   
4. Fabric, manifolds and topology
   
5. Tilings and prints
   
6. Draping, gravity and constrained dynamic systems
   
7. Conclusion
   

I'm sure I haven't exhausted the possible topics, either. Here's a few notes about each of these topics.

Metric spaces - from 1D to 2D : The practice of drafting a sloper or block (or pattern) is one familiar to engineers and users of Computer-Aided Design (CAD) software - it consists of using interconnected lines of pre-determined length to lay out a set of two-dimensional shapes. In addition, the process of weaving threads to form two-dimensional sheets is another example of extruding one dimensional objects (threads under simplification) to form two dimension objects (sheets). The process of manipulating patterns (or fabric, for that matter) by cutting and then reattaching cut pieces at different locations exploits particular sets of the metric properties of 2D spaces. The relationship between the metric and its properties on the one hand, and the manipulation of sewing patterns on the other, could be rendered explicit.

Metric spaces - from 2D to 3D : The process of stitching fabric together in complex ways to form 3D objects constitutes a second class of transformations that could be explored.

Fabric, manifolds and topology : During the process of taking pieces of fabric that have been cut and sewing them to form 3D objects, topological properties of the sheets are also exploited. Indeed, whether sheets, tubes (thread) or 3D objects (garments), we are dealing with entities that are mathematically described as 'manifolds'. The topology of the objects concerned affects the fabrication process - for example, turning a jacket inside out is a manifestation of certain topoological properties of the garment.

Tilings and Prints : The procedure by which one generates a print pattern through repetitions of a motif is a form of tiling, for which there is a very interesting mathematics derived of both very old and very new ideas. Hence exploring tilings presents an interesting area of study, inlcuding repetitive tilings but also non repetitive tilings such as Penrose tilings.

Draping, gravity and constrained dynamic systems : Once garments are design, printed and constructed, they are worn. How they are worn depends upon the behavior of dynamic systems under gravity, especially in the presence of particular classes of constraints (e.g. adjacency or connectivity constraints).

A full exposition of these different aspects would make fascinating reading!

Footprints in the sand...

As part of my ongoing work on the fabric wiki, I have been researching so-called "eco-friendly" fabrics and the issues that surround them. This makes fascinating reading (see my wiki article on this for more details). It turns out there is no "perfect" eco-fabric - all of them have issues (even recycled fabrics), despite what some people on the net seem to believe. That said, it is possible to make intelligent choices that can put pressure on the system to change for the better.

What I found particularly interesting, though, is the whole notion of a "footprint". First of all, there seem to be several footprint concepts being used now, including the "ecological footprint", the "carbon footprint", even the "water footprint" or the "energy footprint", with a certain amount of confusion as to what each represents. Although we probably hear more about the carbon footprint, the most general concept is actually the ecological footprint. The ecological footprint is the amount of land that would be needed to produce and/or absorb the materials needed and/or generated by our activities. The reason for the "slashes" in this statement is that land is required to make things, but the pollution or garbage generated by our activities (and the nature of the world means that waste products are ALWAYS generated by activities) also requires land support. This notion of footprint is hence, truly a footprint - it is expressed in surface area that activity generates. The footprint per capita is then multiplied by the total population of the Earth to determine the total surface area needed to support all our current activities. Well, it turns out that we would need, roughly 1,5 planet Earth's to support current activity levels. In other terms, we are using up the planet's resources faster than they can be renewed. This is NOT a GOOD state of affairs for future generations.

Interesting, and perhaps surprisingly to many, our use of fabric is a primary component of our overly large ecological footprint. The textile industry is a heavy polluter as well as an energy gourmand, and the result is a direct contribution of something near 15 to 20% of our ecological footprint, due mainly, of course, to the use of fabrics in the developed world (Europe and North America). It should be noted that the ecological footprint of the developed world rings in at five (5!) planet Earths - we are using resources five times faster than the planet can renew itself. Therefore, the 15% to 20% contributed by the world of fabrics actually represents one whole Earth ON ITS OWN. This is a serious problem, and one that those of us interested in sewing and fashion need to be aware of and doing something about!

This means not just what and how we buy, but also how we talk about and promote fashion, how we engage with the community, and so forth.