BURN, BABY, BURN
Lulu Hunt Peters at least had a sound scientific basis for her weight-loss approach. At the time she wrote her bestselling diet book, it had been only twenty years since chemists Wilbur Atwater and Russell Chittenden came up with the notion of measuring food as units of heat that could be produced by burning it: calories. For instance, the calories contained in five pounds of spaghetti would yield enough energy to brew a pot of coffee, while those in a single slice of cherry cheesecake would operate a light bulb for an hour and a half. If one wished to drive eighty-eight miles to visit friends or family, one would need to burn the calories contained in 217 Big Macs. (Think about that when you’re planning your next road trip, and take a moment to appreciate the energy efficiency of burning fossil fuels.) If someone consumes 2,000 calories a day, that will yield just enough energy to power a 100-watt bulb for twenty-two hours—assuming 100 percent efficient conversion, which simply isn’t possible, as Carnot discovered back in the nineteenth century.
Our bodies evolved into incredibly efficient heat engines, optimized for survival, and we require far fewer calories to function than we realize. The standard method for determining how many calories we need to consume each day is called the Harris-Benedict equation, first developed in 1919. It relies on estimating a person’s basal metabolic rate, taking into account age, gender, height, and weight, and the resulting number is then multiplied by another number designating that person’s level of activity. This would range from 1.2 for those who never exercise, to 1.9 for, say, professional athletes who exercise strenuously as much as twice a day. A 120-pound woman should consume 1,300 to 1,800 calories a day, depending on age, height, and how active she is. The average 170-pound man should consume between 1,870 and 2,550 calories a day, with the same caveats.
The Harris-Benedict equation is not a perfect method, failing to account for the fact that those with excess muscle mass will burn slightly more calories than the equation suggests, while the opposite would be true for those with excess body fat. Still, the Harris-Benedict equation can be a useful tool for weight loss. All you need to do is reduce your daily caloric intake to a number below what the equation calls for—overestimating if you are muscular and underestimating if you have excess flab. Just remember that as you lose weight, you will need fewer calories to sustain your body at that lower weight (assuming all the other factors in the Harris-Benedict equation remain the same).
Even the most chronic yo-yo dieter can recite the mantra. If you don’t take in sufficient calories to give your body the energy it needs, it will begin converting fat cells into fuel—and you will lose weight. The converse is also true: If you consume more calories than your body needs, it will store that excess energy as fat. Stored fat is another fuel source for the body,46 just like the food you consume. There are 3,500 calories in a pound of body fat, so it is possible to reduce one’s daily caloric intake by 250 calories, burn off an extra 250 calories with daily exercise, and thus lose a pound per week.
So why is obesity so prevalent in our society? There are myriad rationales being bandied about, but from a thermodynamics standpoint, it is very simple: We are heavier than people in many other societies because we routinely consume more calories than we need for our bodies to function. This is difficult for many people to accept; they will claim they really don’t eat all that much and insist they must have a slow metabolism. Individual metabolic rates do indeed vary widely—and the Harris-Benedict equation takes this variation into account—as do body types, and no doubt genetics plays a role as well in determining one’s natural, healthy weight.
Those arguments don’t change the fundamental principle: People with lower metabolic rates need fewer calories. When they consume more calories than their bodies require—even if they eat less than “naturally” slim colleagues—they gain weight. It hardly seems fair. But who said physics was fair? Frankly, in times of famine, a low metabolism confers a distinct evolutionary advantage because it can do more with a small amount of fuel. It’s when food is plentiful that this superefficiency becomes a disadvantage.47
Psychologically, we easily can trick ourselves into thinking we eat less than we really do. Studies have shown that the vast majority of us routinely underestimate how many calories we consume. (It doesn’t take much to hit 2,000 calories, particularly if one is partial to fast food.) Brian Wansink is a professor at Cornell University who specializes in the study of consumer behavior and nutritional science, specifically how our environment influences our eating habits. In 2007, he and his colleague, Pierre Chandon, published the results of a study in the Journal of Consumer Research, demonstrating that people have become so conditioned to think that the Subway franchise’s food is healthier than McDonald’s that they underestimate how many calories they consume in a typical meal by as much as 21 percent. Famed Subway spokesman Jared may have lost a ton of weight by eating the chain’s sandwiches, but he chose the healthier options. A Subway twelve-inch Italian BMT sandwich has one third more calories than a McDonald’s Big Mac. Wansink and Chandon also found that people tended to choose high-calorie side orders with their Subway sandwiches.
For one of his earliest research studies, Wansink focused on automatic eating patterns. People would come to the lab and eat a meal while being videotaped, then answer questions about what and how much they ate. He found that people were often unaware of second or even third helpings they consumed and denied doing so—until they were shown the videotape. Other interesting findings: People will eat 16 to 23 percent more total calories if a product is stamped with a LOW-FAT label, and switching from a twelve-inch to a ten-inch dinner plate will cause people to eat 22 percent less. All this inspired Wansink to develop his own dietary secret: “The best diet is the one you don’t know you’re on.” In other words, small changes to the home environment and unconscious patterns can lead to big changes in your waistline.
The calories we consume are only part of the equation. At the same time, we routinely overestimate how many calories we burn when we exercise. The caloric numbers reported by the displays on exercise equipment feed into this misconception, because they are not always accurate, partly because they are often incorrectly calibrated and partly because when it comes to human metabolism, one size does not fit all. New York Times reporter Gina Kolata, author of Rethinking Thin, reported that while a given activity might burn an average of 100 calories per hour, for example, the range for different people could be as low as 70 or as high as 130.
Bad habits can also affect the total of calories burned. Are you one of those people who hang on to the bars while on the treadmill? You burn 40 to 50 percent fewer calories for that same activity. Do you do the same exercise routine for months at a time? As your body grows accustomed to that effort, it will need fewer calories to perform that routine. And most of the calculations used to determine the number of calories burned for various activities fail to subtract the number of calories the exerciser would be using even if they were simply sitting at home reading or watching TV.
“For moderate exercise, the type most people do, subtracting the resting metabolic rate can eliminate as much as 30 percent of the calories you think you used,” Kolata writes. Even those supposedly adept at math can fall victim to self-delusion in this area. Kolata tells the story of meeting a mathematician at a conference who figured he could indulge in a slice of pie because he’d just run a quarter of a mile. “At 100 calories a mile, he might have burned 25 calories. . . . A piece of pie could easily contain 400 calories.”
Personally, I adhere to the Thermodynamics Diet. The primary objective is to optimize two variables, diet and exercise, to ensure either that your weight remains constant (for maintenance) or that you steadily burn more calories than you consume so as to lose weight gradually. You don’t need calculus for that, just basic arithmetic. But if it really were that simple, everyone would be slim.
First, there are economic factors at play with regard to diet: The harsh reality is that healthier
foods actually cost more than junk food, so not everyone can afford a quality, well-balanced diet. Besides, some people really like pizza or French fries or a hot fudge sundae for dessert and would feel seriously deprived on a diet of lean protein, organic leafy greens, and whole grains. Surely quality of life must be factored into the equation as well. How do we find a balance?
Now calculus can be of service. In this case, we wish to maximize our “tastiness”: the pleasure we derive from our food intake, given a fixed number of calories we can consume per day and a fixed amount of money we can spend on groceries. To solve the conundrum, we can plot tastiness (designated by the variable y for “yummy”) as a function of diet, designated by f, for all the various foods we love that, taken together, comprise our diet. Given a diet restricted to 2,000 calories a day and a food budget of $40 per day, what small changes can we make among our current food items to maximize tastiness ( y) while staying within the boundaries imposed by those two constants?
For instance, we might love Snickers bars more than brown rice and carrot sticks, but if all we ate were Snickers bars, we would quickly exceed our caloric limit, and probably develop a vitamin deficiency in the bargain. Similarly, we might love the fresh organic mixed-greens salad with free-range chicken and a light vinaigrette available at our local health-food joint, but if that were all we ate, we would quickly exceed our food budget. So if we know what we’re eating each day now, what small change can we make in our diet to optimize how much we enjoy mealtimes?
This is a job for the derivative, with a twist. It is similar to the multivariable optimization problem we employed while house hunting, except in that case we had two variables constrained by cost; here our variables are constrained by cost and total number of calories. This makes it difficult to plot on a traditional Cartesian grid; there are simply too many dimensions to easily visualize. But we can think of it in terms of vectors, or directions of motion. There are any number of ways we can change what we eat, but some changes are not allowed because they exceed the stated limits to calories or cost; in other words, that particular vector is invalid. Other changes are allowed because they keep those two values fixed.
Normally we would take a derivative with respect to all possible values of f, but in this case we would take the derivative only with respect to those values for f that are allowed—namely, those that can be changed without exceeding our boundary conditions of total calories and cost.
What about the integral? It plays a role in the Thermodynamics Diet too, specifically with regard to how many calories we burn. It all comes down to the burn rate. We can take an integral of our rate of burning calories with respect to time and get the total number of calories burned—the calorie meter on an exercise machine at the gym is secretly doing this calculation. But as we’ve seen, that burn rate is affected by numerous variables: metabolism, level of exertion, muscle mass, and so forth, all of which complicate the equation. So most machines are incorrectly calibrated. The best those machines can manage is a ballpark figure.
THIS MORTAL CURVE
Not only is it possible to use math and calculus to optimize our diet and exercise regimen and maintain a healthy weight; we can also use it to determine the probability that we will die in any given year, thanks to the work of an obscure British actuary named Benjamin Gompertz. Gompertz hailed from a family of wealthy merchants who emigrated to England from Holland. Because he was Jewish, he was denied admission to university and thus was largely self-educated. He acquired his mathematical knowledge by reading Newton’s works, among others, thereby becoming proficient at calculus.
One day, when he was just eighteen, Gompertz stopped in at a secondhand bookstore, and struck up a friendship with the bookseller, John Griffiths, who was a mathematician. Initially Gompertz wished to be tutored, but Griffiths quickly realized the young man’s knowledge already outstripped his own. Instead, he introduced him to the Spitalfields Mathematical Society (later to become the London Mathematical Society), of which he was then president. Gompertz joined the Society, despite the fact that the minimum age was technically twenty-one, and found himself with more than enough math tutors at his disposal, enabling him to advance rapidly in his knowledge. (The society had a rule whereby, if a member asked another for help or information, the second member was required to provide that assistance or else be fined a penny.)
He married the daughter of another wealthy Jewish family with strong ties to the stock exchange, and that connection enabled him to join the exchange himself. He eventually became an actuary and head clerk for his brother-in-law’s nascent insurance company, where his mathematical skills proved very useful. Apparently he had a great capacity for “sustained complex computation” in compiling detailed actuarial tables. In particular, Gompertz found he could apply the principles of calculus to human mortality to determine the cost of life insurance. “It is possible that death may be the consequence of two generally co-existing causes,” he wrote around 1825. “The one, chance, without previous disposition to death or deterioration; the other, a deterioration, or an increased inability to withstand destruction.”
In other words, assuming one doesn’t meet with a fatal accident, such as being hit by a bus, it is possible to use calculus to model the probability of the likelihood that one would die in any given year—a probability that increases with age. Gompertz tested his hypothesis by comparing the proportion of people in different age groups in four cities in England and found that mortality increases exponentially as we age. Thus was born the Gompertz law of human mortality, which holds that whatever the odds that you will die in the next year—1 in 1,000, or 1 in 10,000—those odds will be twice as large eight years from now. In other words, the probability of dying increases exponentially with time.
The Gompertz mortality curve is another sigmoid function, wherein growth is slowest at the beginning and end of a given time period, much like epidemiological models; in fact, Gompertz based his model on the demographic model of Malthus. The slope of the tangent line at any given point (age) along that curve gives us the rate of actuarial aging in the form of a derivative. To get the probability of living to a certain age, all we have to do is integrate the mortality rate over time. The result is that sigmoid curve: the “Gompertz function.”
That’s right: The body has a built-in expiration date. For example, a twenty-seven-year-old American has a 1 in 3,000 probability of dying during the next year, but by the time the person is 35 that probability has increased to 1 in 1,500; by age 43, it has narrowed to about 1 in 750, and so on, so that, if one reaches 100, there is only a 50 percent chance one will live to see 101. The probability that you will die during any given year doubles every eight years. It still holds true today, despite all the advances in nutrition, medicine, and quality of life, and it holds across countries, centuries, even across species, once the different rates of aging are factored into the equation. Scientists don’t understand why this should be true, but it is—for the most part. There are certain age-independent factors at work as well, but in a low-mortality country, like Japan or the United States, this component is usually negligible. Gompertz himself died at the ripe old age of 86. He was working on a paper for publication in the journal of the newly formed London Mathematical Society when he suffered a paralytic seizure.
Working out and eating right to ensure better health is a noble endeavor, but sooner or later the Gompertz law of mortality kicks in. We’re all going to die one day. So it is quality of life that counts, and our overall degree of happiness; being healthy increases our quality of life. Perhaps that is the real benefit of the Green Microgym concept: It might not save the planet, one workout session at a time, but it saves the gym a bit of cash and makes people feel good about their efforts, in addition to keeping them fit. “There’s a certain satisfaction when you work out and feel like you’ve actually accomplished something, instead of just spinning your wheels,” Taggett has said.
Just in case good vibes aren’t enough, Boesel offers gym
members special bonus points: For every hour of electricity a member produces, she or he will earn coupon points redeemable at local businesses. Most important, as I found during my own brief session on Boesel’s retrofitted machine, the Green Microgym raises awareness of just how much energy we consume without thinking—and what it costs to generate that energy in the first place.
8
The Catenary Tales
As hangs the flexible chain, so but inverted stands the rigid arch.
—ROBERT HOOKE
It is a bitterly cold February afternoon in St. Louis, the kind of day that finds most people opting to curl up by a roaring fire with a good book and a nice cup of tea. But I am only in town for a few short days while attending a conference and thus join a handful of other hardy souls visiting the famed Gateway Arch. It is a landmark structure that opened in October 1965 to commemorate Thomas Jefferson’s Louisiana Purchase and now dominates the St. Louis skyline. Five of us cram into the little egg-shaped tram and ride to the peak of the arch, where we can gaze out over the frozen expanse of this Gateway to the West, named in honor of the early pioneers who migrated west through St. Louis on the first leg of the Oregon Trail.
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 17