The Mathematics of the Peeler

Warning! After you read this post, choosing vegetables at the supermarket will never be the same again!

A few days ago while I was preparing dinner, I was peeling carrots and noticed how much of the original thickness of the carrot I was peeling away.

The peeled carrots were significantly more slender than the carrots I started out with.

Suddenly I had a flash of intuition from the “sick of mathematics” side of my brain: the more spherical a vegetable is, the less volume is proportionally removed when it’s peeled.

Let me explain what I mean.

We can imagine that peeling is equivalent to removing a thin layer of thickness $dx$ from a vegetable with a surface $S$ and volume $V$.

The volume peeled off can be approximated by the value $dx\cdot S$ so that the proportion of removed volume relative to the total volume is given by:

$$\displaystyle \frac{\text{PeeledVolume}}{\text{TotalVolume}} = \frac{dx \cdot S}{V}$$

If you always use the same peeler, the $dx$ value is fixed. Let’s say that normally $dx$ may be equal to a couple of millimeters.

So, given the peeler, the removed volume is proportional to the $S/V$ ratio.

It’s well known that the geometric figure with the lowest $S/V$ ratio is the sphere.

This is connected to the circle’s analogous properties of being the planar geometric shape that maximizes the area given the perimeter.

Demonstrating this property is not as trivial as it may seem. The first to achieve results was the mathematician Jacob Steiner in 1838, and later mathematicians completed the demonstration.

The two main ideas behind the demonstration are the following:

1) If a planar figure is concave, then there is another figure with the same perimeter but with greater area

2) A planar figure that is not fully symmetrical can be deformed to create another flat shape with the same perimeter but with larger area

As a result of 1) and 2) the planar figure that maximizes the area given the perimeter must be convex and must have the greatest possible symmetry, and then it is the circumference (obviously this is just a sketch of the demonstration).

Formally the result is known as the isoperimetric inequality: each closed curve of length L and area A satisfies:

$$ 4\pi A \leq L^2$$

and the equality is true only for the circumference.

Then I would like to suggest the following result.

Peeler corollary to the isoperimetric inequality: given two vegetables with equal volume, the one whose shape is closer to a sphere is the one that minimizes the volume wasted peeling it.

Clearly this statement is somewhat vague because I haven’t defined what “closer to a sphere” means.

Things get complicated, however, if you have to buy a whole bag of potatoes of different sizes. In this case, in fact, to minimize the waste you should evaluate which bag has, given the same weight, the smallest total surface area (obtained from the sum of all the potatoes’ surfaces).

A bag with two large potatoes not at all spherical could have a smaller total area than a bag with many tiny perfectly spherical potatoes.

Reasoning in this way, in addition to reducing waste, will also minimize the time required for peeling all those potatoes (which is always proportional to the surface).

Take into account these considerations the next time you choose a sack of potatoes!