The fourth dimension! The eighth dimension! The millionth dimension! Dimension is one of those math words that mathematicians don’t really bat an eye about but can seem baffling to the uninitiated.
It doesn't have to be. Mathematicians are kind of boring about how they think of dimension. To label a point in two dimensions, we use two coordinates—length and width, or x and y. To label a point in three dimensions, we use three coordinates—length, width, and height, or x, y, and z. So eight-dimensional space is labeled using eight coordinates. Mathematicians don’t really care that we don’t have words to describe the coordinates. We generally just label them (x1, x2, x3, … x8), which is a real snooze-fest, both bland and unenlightening. Being able to manipulate eight coordinates doesn’t tell me much about what eight-dimensional space is. In fact, geometrically, there isn’t much I can do to understand eight-dimensional space. A few people, including the fascinating Alice Boole Stott, could “see” four-dimensional space, and there are established ways to visualize it, but by eight dimensions, geometric understanding pretty much out goes the window.
It seems surprising, then, that high-dimensional spaces have any interest outside of purely theoretical mathematics. But they do, and I think we can understand where those applications might come from if we expand our horizons beyond geometry when we think about dimension.
Any context in which we make several distinct measurements being made can give us a high-dimensional space. I like to sew, so one context for me is in the measurements I have to take when I make a dress. (A similar analysis can be done on shirts, pants, or other garments, but I like sewing dresses the best.) The back of a pattern envelope concentrates on three measurements: bust, waist, and hip, so we can start there.
The bust, waist, and hip measurements make a person’s body into a point in three-dimensional space. Already, we can identify a mathematical reason it’s hard to find dresses that fit: a dress size is just one number, but our measurements give us a point in three-dimensional space. We are trying to use the one-dimensional spectrum of dress sizes to approximate the three-dimensional data set of people’s measurements. Sure, people with larger waists often have larger hips and busts, but there’s a lot of variation in human body measurements.
It gets worse! Those three measurements—bust, waist, hip—are crude at best. Someone with an A cup and someone with a D cup can have the same bust, waist, and hip measurements, but it’s very likely that the same dress will not look equally good on both of them. To make dresses that fit better, many people add to bust, waist, and hip the high bust measurement. This is measured around the top of the bust above the breasts, basically right under the armpit. And with that, body measurements enter the fourth dimension (and dress sizing gets even more difficult).
We don't have to stop there.There are quite a few other measurements, either of the body or of the garment, that are important if we want to make clothing that fits perfectly: torso length, skirt length, arm length, bicep circumference, and neck circumference all make a difference for some patterns. Taken together, we’re looking at nine-dimensional space. If we're more particular about fit or dress design, we might end up in an even higher-dimensional space.
I'd argue that sewing is one place where high-dimensional spaces naturally appear. These measurements are natural ones to take, and they are to some extent independent of one another. Why is that important? It would be kind of silly to think of the width and length of a square as separate dimensions, or of length, width, and area of a rectangle as separate measurements. But people who are six feet tall don't all have the same waist size or arm length, so our dress dimensions are not accidentally redundant.
Not every combination of nine positive numbers represents a dress that would fit someone: no one is 50 feet tall or has a 3-inch waist, so any points that have 50 feet for the skirt length or 3 inches for the waist length are not likely to become real dresses. But there is a region in nine-dimensional space that represents dresses people would wear. I'm going to call it pattern space. We can’t see pattern space visually, but it is a meaningful object sitting in nine-dimensional space, and understanding it is part of the job of fashion designers and pattern drafters.
If you, like many people, sometimes despair at how hard it is to find clothing that fits and flatters, it might be comforting to think that you’re trying to zero in on one point in nine-dimensional pattern space. It’s a marvel any dress has ever fit any body!
When you sew for yourself, you can force your garment to inhabit just the right point of pattern space, but it’s rarer to find one in the wild. Last Friday I was browsing through a vintage store, as I sometimes do at the end of a busy week. I ended up trying on five dresses. Two of them had perfect busts and hips, but my waist strained the fabric. The third one didn’t even fit over my head. The fourth was too big in the bust and too small in the hips. But the last one was perfect: I slid it over my head and knew I had found my special point in nine-dimensional space.