Origin of the Problem

Mathematical Set Up

We want to find the depth below the onion to cut towards in order to minimize the variance of the volume of each onion slice.

Mathematical Set Up

The variance, , of a set of numbers whose average value is is

That is, the variance is the average of the square deviations from the mean (this will be important later).

Simplifying the Problem

For simplicity, consider a two-dimensional onion.

Simplifying the Problem

Insight: The depth to which you have to aim your knife for radial cuts depends on the number of layers.

Simplifying the Problem

So, we might as well consider the limiting case as the number of layers approaches infinity.

Simplifying the Problem

So, we might as well consider the limiting case as the number of layers approaches infinity.

Simplifying the Problem

Similarly, the number of cuts being made has an effect on the answer. So, for simplicity, we can think of making infinitely many cuts as well.

Live Mathematics

Inspiration: The Jacobian!

Rectangular Polar:

Inspiration: The Jacobian!

Problem: With infinitely many layers and cuts, the area of each piece of onion is zero. So, it is hard to measure variance.

Inspiration: The Jacobian!

Solution: Recognize that the Jacobian gives a measure of how big the infinitely small pieces are relative to each other. So, we can use the average value of the function as a stand-in for the average area.

Average of a Function

Fact from Integral Calculus: the average value, , of a function over a region is

Here, over a quarter onion of radius 1, the average "relative area", , is given by (any guesses?)

Average of a Function

Variance of a Function

To generalize the variance we saw earlier, we recall the variance is the average of the square deviations from the mean! So, the variance of our relative area is

Rest and Reflect

• All of this is great, but it doesn't answer the question!

• What allowed all this to work was a coordinate system whose axes cut the onion.

• Can we find a coordinate system that cuts the onion in the way described by Chef Kenji Lopez-Alt?

New Coordinate System

We make a coordinate system for cutting towards a point a distance below the center of the onion. In this coordinate system, we measure the angle from the point , while we measure the radius from the origin .

New Coordinate System

This coordinate system only works for the upper half plane, as there are now technically two points in the plane for a given point .

Game Plan

In order to mimic our computation for polar coordinates, we need to

1. define the region and
2. compute the Jacobian.

Define the Region

Fix . What is the range of ?

Define the Region

Define the Region

So, ranges from to 1.

Define the Region

Also, ranges from to

The Jacobian

In order to compute the Jacobian

We need to know what is the relationship between and in this coordinate system?

The Jacobian

Define to be the distance from the point to a given point (both in the rectangular coordinate system).

Using the law of cosines, we can calculate

The Jacobian

With

The Jacobian

From this, for a given depth , we can calculate the Jacobian as

Yikes!

Executing the Plan

Despite how difficult the Jacobian looks, we can proceed as we did before. Putting the pieces our plan together, we need to first compute

Amazingly, this is not as bad as it looks (even though Mathematica cannot do it).

Executing the Plan

Let's look at the numerator

Executing the Plan

Let's look at the numerator

If we go back to the rectangular coordinate system, this would simply be the integral of over the region we define to be the onion. That is, it's the area of the quarter onion so this integral is just .

Executing the Plan

Now let's look at the denominator

Executing the Plan

Now let's look at the denominator

This is fairly easy to do as well. We can see the denominator is equal to

Executing the Plan

In all, the average ``relative area'' of each piece is

Executing the Plan

Reducing the Problem

So, finding in a closed form reduces to evaluating

Reducing the Problem

• This integral cannot be evaluated by Mathematica. We can be clever, though.

• We can go back to the Cartesian coordinate system using and . The Jacobian for this transformation is .

Reducing the Problem

But, here we are integrating over a quarter disc. Wouldn't polar coordinates be easier?

Let and . Then, assuming ,

So, we now have a closed form for (don't make me write it out though).

Rest and Reflect

• Whew! We have done a lot of integration work with three different coordinate systems. What do we have at the end of it?

• We have an expression for the variance of the ``relative area'' as a function of the depth which we are cutting towards.

• Our goal is to minimize variance. What should we do?

• Take the derivative and set it equal to zero! (then test).

Putting it Together

where

The (2D) Onion Constant

define ס to be the unique root of over the interval . We call ס the onion constant. To fifty decimal places its value is 0.55730669298566447885109305914592718083200030207273...

Three Dimensions

• We can also do a similar analysis in three dimensions. We model the onion as nested spheres. We first cut the onion in half, as before.

• Usually, we cut an onion into slices perpendicular to the root. If we add a axis, the planes perpendicular to this access cut the onion in this way. So, we can use an analog of cylindrical coordinates (with our distorted "polar" coordinates) to answer the three-dimensional case.

Three Dimensions

• If we do the same analysis as before, we find the three-dimensional onion constant is approximately 0.484457.

• This is awfully close to 1/2 (an easy number for humans to estimate. What if we add in the fact that usually we cut off the ends of the onion?

Comprehensive Onion Cutting Guide

1. Cut off the ends of the onion, leaving approximately 82.8513% of the length of the onion in the vertical direction.

Comprehensive Onion Cutting Guide

2. Cut the onion in half perpendicular to the previous two cuts.

Comprehensive Onion Cutting Guide

2. Cut the onion in half perpendicular to the previous two cuts.

Comprehensive Onion Cutting Guide

3. Slice the onion parallel to the direction of your initial cuts and perpendicular to the cutting board up the half-onion.

Comprehensive Onion Cutting Guide

4. For the final cuts, cut perpendicularly to the previous cuts, but for each cut aim towards a spot 50% of the radius of the onion below the center of the onion.

Comprehensive Onion Cutting Guide

5. Profit

Concluding Thoughts

• Why do this?

• What now?

• What questions do you have?