In this month’s column, the cookbook author Kenji López-Alt answers a question of his own: What’s the mathematically best way to cut an onion?
I didn’t go to culinary school. My first real instructor was a book, specifically Jacques Pepin’s “La Technique.” In what feels like a precursor to today’s internet videos, it demonstrates every essential French technique with thousands of hands-only photographs. The onion-cutting method I learned from it is the same one taught at top Western culinary schools, and it’s the method used by most of the cooks at every restaurant I’ve ever worked at.
You trim off the onion top (the side opposite the hairy root), then split the onion from end to end. Next, you peel the resulting halves — Mr. Pepin has told me that he also likes to remove the first layer, as it can be tougher than the rest — and lay them flat on the cutting board. There’s nothing particularly contentious so far. The next step is where disagreements creep in, but we’ll get back to them.
Working with one half at a time, Mr. Pepin has you next make cuts in three different orientations, essentially dividing the onion into cubes. Your first cuts are straight up and down with your knife tip pointed toward the root end of the onion. The goal is to cut through most of the onion but leave each “slice” connected near the root end.
Because of how an onion’s layers naturally subdivide it, if you look at the cut face, you’ll see that your vertical cuts have divided the layers into wonderfully even squares in the middle, but as you get toward the edges, the shapes come out increasingly stretched and elongated — far taller than they are wide.
This is where a second set of cuts, made with your knife held horizontally, comes in. Once again, the goal is to cut through most of the onion but keep the pieces connected at the root end. (Smaller onions may need only a single horizontal cut, while a larger one may require two or three.)
Finally, your third move is a series of parallel vertical cuts. If all goes well, when you start making these cuts, the onion should tumble onto your board in a fine, supposedly even dice.
Over the years, I’ve had questions about this process: Is this the most efficient way, and is it the most even way to dice an onion? Or is there a better, more effective way to get more even dice with fewer strokes?
Modeling an Onion
There’s an alternative slicing method I’ve seen recently, recommended by Food52 and Alton Brown, called the “radial” or “Lyonnaise” cut. Instead of starting with parallel vertical cuts, you instead angle your knife blade so that it is always aimed at the center point of the onion (that is, it should always enter the outer layer at a 90-degree angle). You wind up with a series of cuts fanned out like a sunburst. The idea is that because an onion is radially symmetrical (that is, it does not change shape if you rotate it around it’s axis), you should be adjusting your knife strokes to match that shape.
To figure out which method is better, years ago I enlisted the help of Rui Viana, an old friend with a degree in math and computer science from Massachusetts Institute of Technology, to build computer models of the cross section of an onion. We modeled the cross section of half an onion lying flat on the cutting board as a series of concentric half circles representing its layers. On top of this model, we could then lay a series of lines representing knife strokes, using the computer to adjust the spacing and angle of the strokes. This allowed us to simulate various cutting geometries and to calculate basic information, such as the number of pieces cut with each method, their average size and the standard deviation from the norm within that group (a number that represents how similarly sized the individual are pieces are in the group). Years later, Dylan Poulsen, an associate professor of mathematics at Washington College in Maryland, created an even more robust model to help make these calculations.
The Result: Neither Method Is Ideal
As it turns out, cutting radially is, in fact, marginally worse than the traditional method. With all your knife strokes converging at a single central point, the thin onion wedges of onion that you create with your first strokes taper drastically as they get toward the center, resulting in large dice cut from the outer layers and much larger dice from the center. But even the classic method doesn’t produce particularly even dice, with a standard deviation of about 48 percent.
So we know that the classic method doesn’t produce very even dice, and that the radial method is even worse — is there a better method?
For the next set of simulations, I wondered what would happen if, instead of making radial cuts with the knife pointed directly at the circle’s center, we aimed our knife at an imaginary point somewhere below the surface of the cutting board, producing cuts somewhere between perfectly vertical and completely radial.
This proved to be key. By plotting the standard deviation of the onion pieces against the point below the cutting board surface at which your knife is aimed, Dr. Poulsen produced a chart that revealed the ideal point to be exactly .557 onion radiuses below the surface of the cutting board. Or, if it’s easier: Angle your knife toward a point roughly six-tenths of an onion’s height below the surface of the cutting board. If you want to be even more lax about it, making sure your knife isn’t quite oriented vertically or radially for those initial cuts is enough to make a measurable difference in dice evenness.
The Takeaway
The most efficient, even way to dice an onion is to trim off the top, split it in half, peel it, then lay one half down on a cutting board.
If we call the radius of the onion one (that is, its height above the cutting board as it’s lying flat on its cut surface), you’ll want to start with a series of vertical cuts made with your knife aimed toward an imaginary point about .6 of an onion’s height below the surface of the cutting board.
Follow that up with a single horizontal slice and a series of crosswise vertical slices to divide it into dice.
Why does it matter?
Will a recipe made with onions that are 30 percent more evenly diced taste 30 percent better? I cooked a handful of dishes side by side with onions cut with various methods and found that unless I went out of my way to produce wildly uneven dice, most of the little differences ended up washing out in the mix.
At the end of the day, people have been cutting onions and cooking with them using a huge range of techniques, and none will make or break your dish. Still, I personally find great satisfaction in answering these kinds of questions, whether I apply their answer or choose to knowingly ignore them.
