How to Cut an Onion Optimally

I first became interested in the problem of cutting onions in a way to make the slices as uniform as possible at a gathering with friends. One of my friends and Washington mathematician colleagues, Gabe Feinberg, pointed me to a YouTube video from Chef Kenji López-Alt. In the video, López-Alt says he has a friend who is a mathematician, who claims that one should cut radially towards a point 60% of the radius below the center of the onion, and says that this is close to the reciprocal of the golden ratio, 0.61803398875...

I was intrigued by this and even began cutting onions at home with this technique, just because it made me happy. Each time I cut an onion for dinner, my mind would wander. I would think about why this is true and what techniques I could use to approach the problem. While this was meditative for me, these musings did not lead anywhere substantial over the span of two months. One day, however, my thoughts actually led me towards a way to approach this problem. Within two days, I had found the optimal way to cut an onion.

The goal of the optimization is to make the volume of each piece of onion as close to uniform as possible without having to move any part of the onion after it is cut in half. The idea is to cut the onion in half, then slice it perpendicularly to create the semicircular layers of the same width. For the final cut, instead of slicing vertically (as I used to do) or radially towards the center, one should slice radially towards a point below the center of the onion. The depth one should aim for depends on the number of layers of the onion. For a one-layer onion, one should cut towards the center of the onion. For two layers, one should cut slightly lower. 

I found it mathematically interesting to think about what happens as the number of layers gets larger and larger (and hence each layer gets thinner and thinner). What I found, with about seven pages of multivariable calculus to back it up, is that one should cut to a point 55.730669298566447885109305914592718083200030207273...% of the radius of the onion below the center of the onion for an onion with infinitely many layers (and slightly closer to the center of the onion for onions with finitely many layers). I call this number the onion constant, and I denote this number with the Hebrew character samekh, because the character looks most like an onion.

Finally, should readers cut onions this way? The answer is that it doesn’t matter at all for the taste of the food. However, if readers want to try cutting onions this way, I hope they will share in my mathematical joy while doing so.