WSJ: The Geometry of Dresses and Bodies

 

When a mathematician tries to sew a dress, the concept of positive and negative curvature can help—up to a point.

 

ILLUSTRATION: TOMASZ WALENTA

By 

Eugenia Cheng

May 7, 2020 11:02 am ET

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After sewing myself a mask, I decided to make myself a dress while sheltering in place at home. I had never made a dress before, but I sew well, and I’ve tried on many dresses in my life, and I am a mathematician. I told myself that it’s all just applied differential geometry.

Geometry is the study of shape, and differential geometry studies shapes that can be smoothly pieced together from small flat portions. In dressmaking, the aim is to take flat pieces of fabric and piece them together to fit around the curves of a body, which is the art of draping. 

 

It’s a difficult task. All bodies are curved, but some are more curved than others, requiring extra skill—which perhaps explains why some designers prefer making clothes for unnervingly thin people who are more or less cylindrical. Moreover, some clothes aim to accentuate curves, while others, like men’s suits, aim to hide them.

 

The late British mathematician Sir Christopher Zeeman wrote an article in 1994 about his attempt to make a dress for his wife using the principles of differential geometry—in particular, positive and negative curvature. An object with positive curvature, like a bowl, has more surface in the middle than at the edges. With negative curvature, it is the other way around: a saddle, for example, has more surface at the edges than in the middle.

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Without getting too explicit, let’s just say that the curves of a woman’s body have both of those features in different places. Understanding curvature helped me to be aware of areas where the dress needed to have “darts”—short folds sewed into the material, essentially removing some surface area. Darts at the edges create positive curvature, while darts in the middle accommodate negative curvature. 

 

Unfortunately, Zeeman thought that the mathematical theory would help him rather more than it did. He admitted that his first attempt was not at all successful, but after spending much longer than anticipated, he produced a dress on his sewing machine that pleased his wife, according to her note at the end of the article.

 

I also found that while knowing differential geometry helped me to think clearly about making the dress, it didn’t really help me in any great detail. What helped much more was the dresses I already own, whose construction I could study and copy. Unlike Zeeman, I was under no illusion that mathematical theory would help me more than the combined wisdom of centuries of dressmakers. It’s a mistake to conflate theory with practice. 

But neither mathematics nor copying was ever going to make a dress that fit me well: The only way to do that is to pin it. Ready-made dresses are designed to fit a supposed average body shape, or perhaps an ideal, which is why they never fit me properly. It’s possible that they don’t fit anyone properly, because every body has a different shape, and there is no mathematical formula to deal with that. 

 

What did help me, more than any mathematical theory, were other skills that I have acquired as a mathematician: detailed analysis, an ability to be methodical, patience with a slow process and the belief that I can understand a structure by careful study. Knowing mathematics is helpful, but knowing its limitations is also important.