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.
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.
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