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!