The Mathematics Of Dieting

William Banting may have created the first very popular diet. In 1863 he published the Letter on Corpulence, Addressed to the Public, which outlined the details of a particular low-carbohydrate, low-calorie diet that had led to his own dramatic weight loss. The diet industry was born—see here for more on “banting”.

Today I want to have some fun with a major hard problem: how can we lose weight?

In our society many of us need to lost a pound or two or more. I for one recently worked hard over the last year to cut my weight down from to . It is now up to and I need to get back on the diet again. Oh well.

There are as many diet ideas out there as there are real numbers. Those of you who still believe that the reals are countable should try to make a list of the myriad numbers of diets available. Some use exotic foods, some use social pressure, some use phone apps, others use pills that increase this or reduce that. But there are an uncountable number of them.

Meatballs Are Round

I recently started to eat a certain type of modest-calorie microwave dinners. My favorite one is meatballs and pasta—I love meatballs. The meal is pretty good as far as microwave ones go, and I enjoy eating it. I just noticed the other day how they make it low-calorie. They use geometry.

A meatball, to a first approximation, is a sphere in three space. The volume of a sphere is of course well known, thanks to Archimedes of Syracuse, to be where is the radius of the sphere. Thus

Theorem: The volume of a meatball, or if you like, the amount of food in a meatball, is cubic in the diameter of the meatball.

The point of this is that the meatballs in the microwave dish are relatively small. They taste good, and there are several per meal. But their size is quite small. This saves a huge amount of calories. If their diameter is of a “normal” meatball, then they each have the food content.

The visual impression of size, however, comes from the diameter. If the diameter is unit, then the psychological impression is eating block of volume. But the actual volume is , so even with a “normal” meatball we actually ingest only about half the calories our eyes think we do.

Too bad we do not live in a world that has four spatial dimensions. Then we would have even better diet meals. The volume of a 4-sphere is . For unit diameter this is . Thus for unit visual satisfaction we would ingest less than a third of the stuff, eating 41% less overall than in our 3-D world.

People in higher dimensions are even healthier. The limit of the volume of a unit-diameter -ball to that of its enclosing -rcube approaches zero rapidly with . At some point the reduction would be too extreme, and people in -space would probably start to eat a lot of meatballs. This vanishing of the sphere is one of the most counter-intuitive facts about geometry. It matters in the statistics of high-dimensional systems, because it implies that most points are extreme outliers close to the boundary conditions in one or more respects.

My Own Diet Idea

I have invented a diet. It is not based on geometry, but is more a computer science type result. It uses the names of foods to control what you can eat. It is a syntax-based method. I call it the “R-less” diet. The diet is based on a simple rule:

You can eat anything that you want provided it does not use the letter r. You can’t get around this by using a fancy French name for a particular dish, say—you must also name the constituent foodstuffs in English.

For example:

• You cannot eat a hamburger.
• You can eat a piece of pumpkin pie.
• You cannot eat rice or french fries.
• You can eat mashed potatoes.
• And so on

Pizza is an important case of the “foodstuffs” clause. The tomato sauce and cheese and certain toppings such as sausage are OK, but the bread it is on is not. If you can find breadless pizza, go for it. You don’t have to take it down to the level of chemical compounds or elements, though the essence is to cut down on carbohydrates.

The advantage of this type of diet is that it is easy to remember what you can or cannot eat. Of course you do need to spell, but that should not be too hard. The key is the easy with which you can decide what is allowed on the diet and what is not. No complex list, no book of foods that are okay, no need to consult a website or a phone app. If its got an r then its forbidden, if it has no r then eat away.

Note there are variants of this diet based on other letters or even groups of letters. But I will leave that for others to ponder. The R-less diet—patent not pending—is one that I think you might enjoy trying. For me I am going to have some cookies for a snack, but alas no ice cream today.

Open Problems

What is your favorite diet? Is the R-less diet the next big thing in weight control?