How Pi Can Help Sew A Costume

The following is an excerpt from Math In Drag by Kyne Santos.

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A lot of the clothes that I wear in drag are my own creations. I taught myself to sew because I got bored with the styles I was seeing in the ladies’ section at H&M, and I couldn’t afford anything from the ladies’ section at Versace. I’m by no means a professional, but I make do with some stretch fabric and a safety pin. I have to admit I had no idea how much math was required in sewing until I began learning myself. Sewing requires precise measurements, matching together congruent patterns, and a knowledge of how to geometrically translate a fit from a three-dimensional body onto a two-dimensional sheet of fabric. Pi (represented by the symbol π) is a necessity every time I want to sew a dress with a circular skirt attached to it.

To cut out a circle skirt, you have to make a donut shape using fabric—a big circle of fabric with a smaller circle cut out from the center. To fit snugly on my body, the small circle has to be the same size as my waist, and given that my waist measures 85 cm all around, the challenge is to cut out a circle that is exactly 85 cm all the way around (this is called the circle’s circumference). Drawing a circle freehand is not easy. You could trace around a circular object like a plate or a cup, but those circles have a fixed circumference that you can’t alter to fit your measurement needs. The best way by far is to choose a point to be the center and then fold the fabric up around it, using that point as the corner, as if you were cutting out a paper snowflake. If you make a curved cut at the corner, you can unfold and see that a circle is magically cut out. The key is to know the size of the radius to use—the radius is the length from the edge to the center. The farther from the center you cut, the larger the circle. A tiny 1 cm radius will produce a waist-hole only big enough for a doll, and a big 100 cm radius would make a hole big enough to fit around a bed. Happily, I don’t need to figure out exactly where my waist size falls between a Barbie and a TempurPedic mattress because when it comes to circles of all sizes, π is the answer to our problems.

Many minds conjure up the mystifying number π when we think of next-level mathematics. This is a number so popular that it has its own annual holiday—putting it up there with Abraham Lincoln, Jesus, and the Queen of England! Every March 14, we pay homage to this incomparable diva with memes, dad jokes, delectable desserts, and viral videos from yours truly! I’d go so far as to call π true mathematical royalty. The word “pi” comes from the Greek word periferia, which means the periphery, or circumference, of a circle. The number π is the ratio of a circle’s circumference to its diameter. I think our fascination with π may be thanks to the ubiquity of the circle in nature. There are circles in our eyes, in the sky, in ripples on the surface of a lake after being touched by a raindrop, and in a flower’s bloom as its petals unfurl. No matter which circle you choose, if you measure its circumference and divide that number by its diameter (the distance across it, or twice the radius), you will consistently get something close to 3.14, allowing for a small margin of error depending on how precise your measuring tape is.

Since π is approximately 3.14, we know that a circle with a circumference of 85 cm must have a diameter of about 27 cm (85/3.14). The radius, which is the distance from the edge to the center will thus be roughly 13.5 cm, since that’s half the diameter (remember the diameter is the distance from edge to edge). If I cut 13.5 cm away from the center, you get the perfect hole to fit my waist.

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But where does 3.14 come from? Archimedes of Syracuse, who lived around 200 BCE, devised a very clever way of calculating π that didn’t rely on any clunky measuring tape. He simply used the rules of geometry on a hypothetical circle, which allowed him to use exact, theoretical values. Archimedes imagined a theoretical circle with a diameter of 2 and said the circumference must be 2 × π whatever π may be. Then he drew a hexagon, a six-sided object, inside it and measured its perimeter, which was 6. Since the hexagon and the circle were close in size, Archimedes reasoned that 2 × π (the perimeter of the circle) was close to 6 (the perimeter of the hexagon). Solving this equation would mean that Π is approximately equal to 3.

Next, he added more corners to the hexagon by turning each straight edge into two new edges, resulting in a 12-sided polygon. By using the information about the hexagon along with more geometric rules (including the theorem of the controversial philosopher Pythagoras, a2 + b2 = c2), you can work out that the perimeter of this 12-sided figure is approximately 6.2, which gives π a new approximation of 3.1. As Archimedes moves from a 6-sided figure to a 12-sided one, and then later a 24-sided, a 48-sided, and finally a 96-sided polygon, the polygons get closer and closer to the true shape of the circle, and the approximations of π get closer to the true value. Archimedes stopped by saying that π was somewhere between 3.1408 and 3.1429. I guess his hands got tired; after all this was an age before calculators!

Archimedes turned what was once a matter of measurement into a matter of arithmetic, and a new age of mathematics was born. Now you might be saying, “But, Kyne, no matter how many sides a polygon has, it will never be a circle!” Which is exactly right, and that’s why we will never get closure on this issue because we’ll never find the exact value of π. It is an elusive, unknowable enchantress, with infinitely many digits. Mathematical paparazzi can chase her forever, but they will never capture her in her entirety.

If we are only concerned with using π to cut out dresses or build circular buildings, then we don’t actually need to know infinitely many digits. We can get by perfectly fine with just the first few. In fact, even the scientists at NASA only use up to 15 digits and, wilder still, it only takes about 40 digits to calculate the circumference of the visible universe with a margin of error thinner than a hydrogen atom!

And yet, like desperate lovers chasing a dream of unearthly beauty, mathematicians through the ages have taken up the challenge, picking up right where Archimedes left off, investigating π on a journey of pure obsession. About 1,500 years after Archimedes, the Indian astronomer Madhava of Sangamagrama made a huge leap in calculating π, discovering that it could be calculated using this formula:

π = 4 − (4/3) + (4/5) − (4/7) + (4/9) − . . .

The pattern starts with 4 (which is 4 divided by 1) and then alternates subtracting and adding 4 divided by the next odd number. Using only the first five terms, we can approximate π

we get:

π = 4 − (4/3) + (4/5) − (4/7) + (4/9) − (4/11) + (4/13) − (4/15) + (4/17) − (4/19) = 3.0418

The more terms we use, the closer we get to the true value of π. Another 400 years after Madhava of Sangamagrama, the quest to understand and calculate π was turned upside down by the development of the first computer. The computing power of the digital age has moved us from  knowing a few dozen digits of π to knowing millions. Although the brilliant mathematician Johann Lambert proved in the 1700s that π was an irrational number with infinitely many digits that never terminate nor repeat, people persist in searching for more. I suppose it’s for the same reason people try to break any sort of record. We do it for fun, for sport, or simply to test ourselves (or our computers).

Irrational numbers like π are special. With infinitely many nonrepeating decimal digits, they refuse to be written. To write down an irrational number, the Hindu–Arabic numerals 0 to 9 are not enough, nor can we use a fraction as we do with 1/3 or 1/7. We have to use mathematical equations to point at irrational numbers.

Another example of an irrational number is the square root of 2: 1.41421356. . . . Like π the square root of 2 cannot be writ- ten down in plain digits. We settle for calling it the square root of 2 ( 2). Unknowable like π with that irrational mystique, 2 could definitely be nominated for celebrity status. And yet, it turns out that there are infinitely more irrational numbers than rational ones! If you threw a dart at a random spot on the infinitely long number line, you are almost certainly guaranteed to land at an irrational number and not a rational one. Irrational numbers like π are not the unicorns of the number world. If anything, it is the rational numbers, numbers like 1, 2, and 3, that are the rare gems. Once again, we see that the definition of a number—what we think we know for sure—stretches far beyond our preconceived notions. Numbers are more than just the discrete, whole quantities that we can count—numbers form a vast, continuous spectrum that extends infinitely in both directions: the real number line. But in the same way that I am asking you to envision gender as an infinite multitude of possibilities, I am going to ask you to let your imagination guide you into a place beyond the infinite number line. Because as a matter of fact, most numbers don’t fit on the number line at all!

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From Math in Drag by Kyne Santos © Johns Hopkins University Press.