Background:

"Low-Carb Diets Get Thermodynamic Defense", on Nature.com.

"Is a calorie a calorie? Biologically speaking, no" in Letters to Am. J. Clin. Nutr. 2004;80:1445-54.

Feinman RD and Fine EJ, Thermodynamics of Weight Loss Diets, Nutr. Metab. (Lond.) 2004 Dec 8;1(1):15.

Fine EJ and Feinman RD, "A calorie is a calorie" violates the second law of thermodynamics, Nutr. J. 2004 Jul 28;3:9.

Recently I gave my physics students the following scenario, basically just for fun:

Imagine what would happen if food scientists were to invent a kind of intense "supersweetener" with 3500 calories in a single ounce. If you were to ingest an ounce of this sweetener, then how much weight would you gain?

A few of the students knew offhand that 3500 calories is the equivalent of a pound of fat. So everyone figured, well, if you ingest 3500 calories, then you should gain a pound.

But obviously, if you ingest only one ounce of material, then your weight could not increase by any more than one ounce.

The students hated this answer! For one thing, many of them had forgotten (or never known in the first place) that mass is always conserved in chemical reactions. And they had also never really viewed human metabolism in the abstract as just one great big chemical reaction; even the biochemistry students were down in the details of ATP and such.

So I said, well, just imagine that you're standing on a scale when somebody hands you the ounce of supersweetener. As soon as they put it in your hand, the scale will tick up an ounce. And nothing more will happen when you swallow it. The scale doesn't know or care whether you're holding the supersweetener in your hand, or in your mouth cavity, or in your stomach cavity. And as you digest the supersweetener, the molecules will separate and go here and there, but until they emerge from your body and find their way into the environment, the scale reading won't change one bit.

There's a Fundamental Theorem of Calculus, a Fundamental Theorem of Algebra, and a Fundamental Theorem of Poker. I nominate the following as the Fundamental Theorem of Weight Loss:

Weight loss per day = mass in - mass out.

(You'll have to forgive the conflation of weight with mass; I'm assuming that all weight loss programs will take place in a static and uniform gravitational field, so that it will not cause a problem.)

According to the Fundamental Theorem of Weight Loss, if you want to lose weight, then your challenge is simply one of routinely defecating, urinating, sweating, vomiting, and exhaling more mass than you ingest on any given day. (You could also amputate something, deliver a baby, clip your toenails, get a haircut, or hawk a really big loogie.)

By the way, people suffering from eating disorders have long understood the Fundamental Theorem. With a ruthless logic, the anorexic minimizes the "mass in", while the bulemic maximizes the "mass out" using the ultimate weight-loss "foods", namely laxatives and purgatives, which trigger mass losses in excess of their own mass.

But if "mass in minus mass out" is all there is to weight loss, then why all the talk about counting calories? Don't calories make you fat?

Calories do in fact work well as an indicator of the kinds of foods that tend to make you fat. So in view of the Fundamental Theorem, calories must basically be a rating of how much the human digestive system "grabs onto" different foods. Eat a piece of celery, and most of its mass will find its way out of your body through defecation (cellulose passes through) or urination and respiration (much of the mass of celery is water). But eat a Snickers bar, and your body's going to say hey, let's hang onto that good stuff. Eat an extra two-ounce Snickers bar every day, at something like a fifty percent mass retention rate, and at the end of a year you'll be 20 pounds heavier.

So here's an idea: Instead of printing calorie counts on food labels, why not show the weight you will actually retain by eating the item in question? For example, the label on a candy bar with a net weight of 2 ounces could say something like, "Retained Weight 1 ounce." In other words, of the 2 ounces of input mass, your body's going to hang onto 1 ounce for the long term.

Labeling foods this way might make it easier psychologically for people to resist foods that are going to make them fat. The motivation factor would be clearer because you're no longer trying to avoid the abstract threat of a calorie; instead you're scoring yourself by the very same metric that shows up on the bathroom scale. If you knew exactly how much of that candy bar was still going to be with you in the morning, you might pass it up. (Thanks to the Fundamental Theorem, I can now visualize the act of eating a candy bar as amounting to a process of melting the chocolate down and smearing it all over my midsection. Want to eat a whole pizza? Why not just save time and staple it to your shirt front. Will that Twinkie go straight to your thighs? Well, not all of it - just half an ounce or so.)

Something else the students challenged me on is the question of exercise. Isn't the goal of exercising to burn calories? If "mass in minus mass out" is really all there is to weight loss, then how does exercising help you to lose weight?

Somehow, exercising must turn out to be an exercise in the expulsion of mass, the key mechanism presumably being breathing out CO2. CO2 molecules don't weigh much, but they weigh a lot more than the O2 molecules you breathe in to fuel the metabolic process--about 30% more.

There are charts that tell you how many calories you will burn by exercising in various ways for given lengths of time. But maybe the charts should cut to the chase and tell you how much mass you can expect to lose by exercising in various ways for given lengths of time. Personally, if my goal is to change what the scale says, then I'd prefer for everything in the conversation to be couched in the scale's units.

We might for example have a universal table like the following, which I sketched out using the rough conversion 8 Calories "=" 1 g of fat (sources here and here):

Butter: 90 grams retained out of every 100 grams consumed (sigh)

Bagels: 25 grams retained out of every 70 gram bagel consumed

Beef tenderloin: 1.2 ounces retained out of every 4 ounces consumed

Carrots: 0.2 ounces retained for every 4 ounces consumed

Jogging: 17 minutes to lose 1 ounce (for a 190-lb person)

Raking leaves: 33 minutes to lose 1 ounce (for a 190-lb person)

Rowing: 14 minutes to lose 1 ounce (for a 190-lb person)

***

Postscript: I first thought of the Fundamental Theorem of Weight Loss back in 2004, but this past semester was the first time I tried using the example in class. Well, my students were pretty skeptical of the whole notion. With my pride thus challenged, I went to the web later that night and found two experts, Dr. Richard Feinman of SUNY Brooklyn Health Sciences Center and Dr. Eugene Fine of the Albert Einstein College of Medicine, who have been publishing technical papers on metabolism and diet for quite some time. The papers linked to at the top are very much concerned with the question of whether "a calorie is a calorie is a calorie."

The two men were kind enough to respond to my emails, and I'm hoping that they will come to Bennington sometime to discuss their work. They verified the truth of the "mass in minus mass out" thesis - as an application of basic physical law, it could hardly have been wrong - and they also had many more interesting things to say. Two brief excerpts from their emails:

"Calories in the context of diet are a nutritional invention with many unfortunate and misleading consequences, but the concept has become so entrenched that it is impossible to discuss weight change without making reference to this usage." (Fine)

"This established, the remarkable thing is that, under most conditions, where careful measurements are made, a calorie IS a calorie, that is, the calories in food predicts weight gain or loss between diets. The above, however, means that this is not a thermodynamic effect but rather the specific characteristic of living systems." (Feinman)

So calories seem to work well as a proxy for mass retention.

## Sunday, May 27, 2007

## Sunday, May 6, 2007

### Understanding Exponential Growth

Ever since I saw this awful page about exponential growth (namely zebu.uoregon.edu/1999/es202/l3.html, which I complained about earlier), I've been considering the question of what a good piece of curriculum would look like for teaching exponential growth.

With apologies for diving right in, let me share a few working hypotheses:

1. Fluency with the mathematics of the exponential function does not automatically or inevitably lead to a rich intuitive grasp of exponential growth.

I arrived at this hypothesis by reflecting on my own educational trajectory. Though I mastered this body of mathematics as a teenager, I would say that my instinctive feel for exponential growth has become strong only in the past few years. (In fact, I wonder if my technical facility with mathematics actually shielded me from ever having to develop a rich mental idea of exponential growth.)

2. Fluency with the mathematics of the exponential function is not even necessary for having a rich intuitive grasp of exponential growth.

This is stating the case strongly; but for now I'm interested in pushing this perspective as far as I can.

3. A great piece of curriculum for exponential growth would be a valuable, eye-opening, and even transformative experience for a wide variety of audiences, including college students of all kinds, college faculty members, and adults outside of academia.

***

I don't have this magic piece of curriculum yet, but what I assembled recently for my Rediscovering Math class was perhaps a small start. A portion of what we covered is reproduced below. A leisurely ramble it may be; but I would also say that here & there it contains some real mathematical insights about exponential growth. (With respect to Hypothesis #1 above, I should say that I arrived at some of these insights for the first time as I prepared to teach this class!)

***

We begin, as one might expect, with vampires.

Vampires

A biologist on the Bennington faculty pointed me to this amusing paper by two physicists that aims to debunk various items of folklore about ghosts, vampires, and zombies. The vampire section was one of those surprising examples of exponential growth. The authors point out that according to standard (pre-Anne Rice) vampire lore, vampire-human ecology is simply a non-starter. The authors argue that with vampires feeding on people who turn into vampires who feed on people who turn into vampires who feed...and so on...then it would only take about three years for the entire world's food supply to run out!

When I saw this argument, my first reaction was embarassment that the absurdity of vampire population dynamics has always been right in front of my face without my ever having noticed it. My second reaction was to defensively poke holes in the argument. For example, the authors conclude by reductio ad absurdum that there's no such thing as vampires (or else we'd all be vampires by now); but we might alternatively conclude from the reductio that we're all just about to be vampires, or that vampires must have natural enemies, such as werewolves. (As in the "Blade" movies. Perhaps Hollywood understands exponential growth better than most.)

I encourage you to read the article to get a sense of how the numbers work out. But for the sake of time, let's move on.

Paradoxes of paper folding

There's an old saying that you can't fold a piece of paper more than seven times. In class, we tried it with 8.5x11 sheets of paper, and everybody managed the same number of folds—just six. Then we went out in the hallway and tried another folding experiment, this time folding a very long sheet of paper towels, over a hundred feet long. (We only used lengthwise folds in this case.) As it turned out, the difference between a single sheet of paper and a hundred-foot-long strip of paper was only a single fold! Seven, instead of six.

It was fascinating to enact the folding process for the long strip. After the first two or three folds, everything seemed to be going fine. Then, when we went from fold #5 to fold #6, the game was suddenly up. (More about the suddenness of exponential growth below.)

But why is paper folding an example of exponential growth at all? There's clearly some sort of doubling going on—or, what is the same, some sort of halving. And somehow this must be related to the difficulty of persisting in the folding process beyond a very few steps. But to draw the connection more clearly, I presented the students with a rough mathematical model of paper folding. The derivation is shown pictorially below; it leads to the equation

There's a nice echo of this in Philip and Phylis Morrison's excellent book The Ring of Truth, based on their 1987 PBS series. The Morrisons explain that expert Chinese chefs are able to make incredibly thin noodles called "Dragon's Beard" by repeatedly stretching the noodles by hand and cutting them in half at each step. In two minutes, the chef interviewed in the Morrisons' book has achieved twelve doublings, yielding about four miles of noodles, each noodle about twice the thickness of a human hair. Legendary chefs of the past were said to attain thirteen doublings.

Grains of rice

We all know the paradox of the Chinese emperor and the grains of rice. The way I tell the story, a wise man does a favor for the emperor, and the emperor asks what he might do in return. The wise man asks for 1 grain of rice to be placed on the first square of a chess board on the first day, 2 grains to be placed on the second square on the second day, 4 grains to placed on the third square on the third day, and so on, doubling the number of grains each day. The emperor agrees, and after a couple of weeks, all of the rice in the empire belongs to the wise man!

The night before class, I wondered whether I might attain a greater understanding of this paradox by acting out the wise man's challenge. So I sat on my kitchen floor and arranged 1+2+4+8+16+32+64+128+256 = 511 objects into a geometric pattern.

(The largest objects are dried beans, then it's rice grains, and then anise seeds.)

Just as in the experience of making "A Few Iron Posts of Obseration", I found it instructive to "think with my hands" for a while. I sat peacefully on my kitchen floor, pushing the pieces around, planning the next stage, and repairing damage from the occasional errant finger or too-vigorous exhaling of breath. With my hands busy, my mind was free to roam. I reflected on the way my ever-shrinking materials —beans, rice, seeds—resembled the ever-shrinking computer chips that carry out our society's calculations. If I needed to take that next step to an outer tier of 512 objects, how would I fit them into the structure? What objects could I use? Salt grains? How then would I manipulate such tiny objects and put them in the proper places? How would I better control my breathing and other destructive effects?

Likewise, how will we continue to shrink our processors to reach the next tier? What will we make them out of, and how will we assemble their circuits? How will we protect them from environmental interference? Can our ingenuity keep up with Moore's Law forever?

The Megamountain

The latest model I've come up with for explaining exponential growth is something I call "the Megamountain." Here's how it goes.

We're going to imagine climbing a mountain. First, hink about what it would be like if the mountain had the same steepness all the way up. What would this mountain look like? Try to draw it.

Next, think about what it would be like if the steepness of the hill kept increasing steadily as you went up. What would this mountain look like?

These first two mountains look something like the cartoons shown below.

Now for the mountain you don't want to climb: The Megamountain. On the Megamountain, the steepness of the mountain at any point is proportional to the altitude at that point.

That's the rule of the Megamountain: The steepness is proportional to the altitude.

If you think about this rule carefully, then you begin to realize that the Megamountain is a runaway situation. Because if you're high up, then [by the rule] it's steep; but, if it's steep, then because of that your next step gains a lot of altitude; but [by the rule] that means it's now going to be even steeper; but that means your next step will gain altitude even faster than before; but that means it'll now be even steeper; and...AAAHHH! It makes my head hurt to think about it!

When I think about what it would be like to climb the Megamountain, I actually get a panicky feeling that I can't possibly keep on going this way. I don't even want to take that next step, because every step is feeding a vicious cycle.

(By the way, you'll notice that I'm not going to try to draw the Megamountain. That's because it can't be drawn; not really. Sure, you can plot a graph of y = e^x, but in the end, you're going to find yourself plotting only the region around x-values of order unity. And in this region, the curve looks roughly similar to a parabola, so you haven't shown what is special, and terrifying, about runaway exponential growth.)

A collection of runaway situations

There are a lot of runaway situations like the Megamountain, including:

* Unchecked population growth: The number of babies born is proportional to the number of people already here. More babies make more people make more babies make more people....

* Gestation: Suppose you had to build a hundred billion houses in 9 months. I think you would quickly hit upon the idea of building houses that build houses. This is how we get from a single fertilized egg cell to a big fat baby in only 9 months. The cell is a house that builds houses. Like grains of rice on a chess board, the number of cells added is twice the number of cells that were there before. More cells make more cells make more cells make more cells....

* Chain reactions. A uranium nucleus splits into two, and the two products strike two more uranium nuclei, causing them both to split in two; their four products strike four more uranium nuclei, and so on. This process is called fission; it was actually named for the biological process of cell division (which was called fission first). In the same way that a big, chubby baby begins with a single cell, here a single subatomic "pop" is magnified, in a millisecond, into an explosion that can level a whole city.

Mathematically, a pregnancy is a runaway chain reaction in the uterus...an explosion of a kind, but one that takes 9 months to unfold.

* Compound interest. The amount of money credited to your account is proportional to the amount of money already there. More money makes more money makes more money makes more money....

But you know, I have had savings accounts, and I have never exactly had the feeling that my money was undergoing an explosive chain reaction! The reason is that the interest rate is so small. It's true that if you wait long enough, your money will double, then double again, and eventually bankrupt the Chinese empire. But the question is, how long will it take to double?

The rule of 72: Divide 72 by the interest rate, and that's how many years it will take to double.

Example: You have a CD earning 4 percent interest. Divide 72/4 = 18, so your money will take 18 years to double. After 18 more years, it will double again.

Warning: The cost of goods and services is also growing exponentially at a 4 percent rate (at least), so by doubling your money in 18 years you are really just keeping up. Your $100 today will double to $200, but that $200 will only buy what $100 buys today. Hence, if your money is not earning at least the same as inflation, the growing cost of goods and services will outstrip the value of your money, and you will actually be losing money in real terms. This is called "inflation risk," and it's the reason you have to put at least some of your money into higher-risk/higher-return investments.

The suddenness of exponential growth

I like to show people these two crude movies that I made a long time ago. Both movies are cartoon visions of what it might be like to ride in a spaceship that splashes down on the north pole. In the first movie, the spaceship moves at a constant speed. In the second movie, the spaceship moves at an exponentially increasing speed.

(The views are through a porthole on the spaceship. Sorry about the aspect ratio - something got screwed up when I put the videos on YouTube.)

* Note, a clearer version of the first movie is available here.

* Note, clearer version of the second movie is available here.

Whenever I show people these videos, they can hardly believe the second video. It says a thousand words about the way exponential growth can sneak up on you—and how unstoppable it is, once it gains momentum.

Where does the suddenness of exponential growth come from mathematically? One way to think about it is to recognize that when we "run the clock in reverse," exponential growth is a continual process of cutting in half. This means that any process of exponential growth must spend a very long time at very small values. And in any graphical or visual sense, the point is that one very small number is going to look visually just like another, even if the two numbers in question differ by many orders of magnitude. (On a graph with values ranging from 0 to 1, a value of 0.0003 is going to be indistinguishable from a value of 0.0000000008—even though you'd much rather your chance of winning the Lotto were 0.0003 instead of 0.0000000008!)

Additionally, when you throw in the fact that the rate of change of any changing quantity involves taking simple differences, you see why the rate of change can remain small even when the underlying numbers are actually growing by orders of magnitude. (The difference between two small numbers is necessarily small, even when the two numbers differ by orders of magnitude.)

All of this is why you can watch something "growing exponentially in time" and wonder why it's just sitting there. It's just sitting there, sitting there, sitting there, and BANG! All of a sudden it explodes. The explosion happens when your numbers take the crucial steps from "smaller" to "small" to order-unity. Prior to order-unity, it looks like nothing is happening; after order-unity, it's too late to do anything.

***

Ultimately, "understanding exponential growth" might have little to do with being able to solve certain classes of transcendental equations. It might instead depend on having a gnawing feeling in your gut that exponential growth is an unstoppable force: not only an unclimbable mountain, but an insatiable feeding machine that will devour anything in its path; and a monster that will lie in wait, lie in wait, lie in wait, and then leap forward in the blink of an eye.

***

But to end on a more positive note, we should also remember that exponential growth can also be a resource. (Anybody need four miles of noodles? Just fold 12 times!) In other words, the best weapon we have against the exponential function might be the exponential function itself. This idea arises for example in the theory of quantum computing, which, when it gets here, will be a process in which the computational resources at our disposal grow exponentially with the number of particles in the processor. Previously intractable problems will become solvable in an instant!

Exponential growth as a resource also comes up in tipping point phenomena - you tell two friends, and they tell two friends, and so on and so on and so on. The numbers stay small, but they're working their way up the orders of magnitude, until we reach the fateful stage of order-unity. We usually think of this model in connection with epidemics and fads. But what if the thing we're spreading is instead a message of positive social change: such as one about changing our habits of energy consumption? That would be fighting fire with fire.

With apologies for diving right in, let me share a few working hypotheses:

1. Fluency with the mathematics of the exponential function does not automatically or inevitably lead to a rich intuitive grasp of exponential growth.

I arrived at this hypothesis by reflecting on my own educational trajectory. Though I mastered this body of mathematics as a teenager, I would say that my instinctive feel for exponential growth has become strong only in the past few years. (In fact, I wonder if my technical facility with mathematics actually shielded me from ever having to develop a rich mental idea of exponential growth.)

2. Fluency with the mathematics of the exponential function is not even necessary for having a rich intuitive grasp of exponential growth.

This is stating the case strongly; but for now I'm interested in pushing this perspective as far as I can.

3. A great piece of curriculum for exponential growth would be a valuable, eye-opening, and even transformative experience for a wide variety of audiences, including college students of all kinds, college faculty members, and adults outside of academia.

***

I don't have this magic piece of curriculum yet, but what I assembled recently for my Rediscovering Math class was perhaps a small start. A portion of what we covered is reproduced below. A leisurely ramble it may be; but I would also say that here & there it contains some real mathematical insights about exponential growth. (With respect to Hypothesis #1 above, I should say that I arrived at some of these insights for the first time as I prepared to teach this class!)

***

We begin, as one might expect, with vampires.

Vampires

A biologist on the Bennington faculty pointed me to this amusing paper by two physicists that aims to debunk various items of folklore about ghosts, vampires, and zombies. The vampire section was one of those surprising examples of exponential growth. The authors point out that according to standard (pre-Anne Rice) vampire lore, vampire-human ecology is simply a non-starter. The authors argue that with vampires feeding on people who turn into vampires who feed on people who turn into vampires who feed...and so on...then it would only take about three years for the entire world's food supply to run out!

When I saw this argument, my first reaction was embarassment that the absurdity of vampire population dynamics has always been right in front of my face without my ever having noticed it. My second reaction was to defensively poke holes in the argument. For example, the authors conclude by reductio ad absurdum that there's no such thing as vampires (or else we'd all be vampires by now); but we might alternatively conclude from the reductio that we're all just about to be vampires, or that vampires must have natural enemies, such as werewolves. (As in the "Blade" movies. Perhaps Hollywood understands exponential growth better than most.)

I encourage you to read the article to get a sense of how the numbers work out. But for the sake of time, let's move on.

Paradoxes of paper folding

There's an old saying that you can't fold a piece of paper more than seven times. In class, we tried it with 8.5x11 sheets of paper, and everybody managed the same number of folds—just six. Then we went out in the hallway and tried another folding experiment, this time folding a very long sheet of paper towels, over a hundred feet long. (We only used lengthwise folds in this case.) As it turned out, the difference between a single sheet of paper and a hundred-foot-long strip of paper was only a single fold! Seven, instead of six.

It was fascinating to enact the folding process for the long strip. After the first two or three folds, everything seemed to be going fine. Then, when we went from fold #5 to fold #6, the game was suddenly up. (More about the suddenness of exponential growth below.)

But why is paper folding an example of exponential growth at all? There's clearly some sort of doubling going on—or, what is the same, some sort of halving. And somehow this must be related to the difficulty of persisting in the folding process beyond a very few steps. But to draw the connection more clearly, I presented the students with a rough mathematical model of paper folding. The derivation is shown pictorially below; it leads to the equation

*L*/*t*= 2^(2*N*), where*L*is the length of the strip of paper,*t*is the thickness of the paper, and*N*is the maximum number of folds obtainable. Though the model is crude, it does reveal the exponential nature of the process, and it shows*N*to be a function of the length-to-thickness ratio, as we would expect. Using this formula, we were able to estimate the number of times one would be able to fold a strip of paper that initially encircles the earth along the equator. (Guess how many!)There's a nice echo of this in Philip and Phylis Morrison's excellent book The Ring of Truth, based on their 1987 PBS series. The Morrisons explain that expert Chinese chefs are able to make incredibly thin noodles called "Dragon's Beard" by repeatedly stretching the noodles by hand and cutting them in half at each step. In two minutes, the chef interviewed in the Morrisons' book has achieved twelve doublings, yielding about four miles of noodles, each noodle about twice the thickness of a human hair. Legendary chefs of the past were said to attain thirteen doublings.

Grains of rice

We all know the paradox of the Chinese emperor and the grains of rice. The way I tell the story, a wise man does a favor for the emperor, and the emperor asks what he might do in return. The wise man asks for 1 grain of rice to be placed on the first square of a chess board on the first day, 2 grains to be placed on the second square on the second day, 4 grains to placed on the third square on the third day, and so on, doubling the number of grains each day. The emperor agrees, and after a couple of weeks, all of the rice in the empire belongs to the wise man!

The night before class, I wondered whether I might attain a greater understanding of this paradox by acting out the wise man's challenge. So I sat on my kitchen floor and arranged 1+2+4+8+16+32+64+128+256 = 511 objects into a geometric pattern.

(The largest objects are dried beans, then it's rice grains, and then anise seeds.)

Just as in the experience of making "A Few Iron Posts of Obseration", I found it instructive to "think with my hands" for a while. I sat peacefully on my kitchen floor, pushing the pieces around, planning the next stage, and repairing damage from the occasional errant finger or too-vigorous exhaling of breath. With my hands busy, my mind was free to roam. I reflected on the way my ever-shrinking materials —beans, rice, seeds—resembled the ever-shrinking computer chips that carry out our society's calculations. If I needed to take that next step to an outer tier of 512 objects, how would I fit them into the structure? What objects could I use? Salt grains? How then would I manipulate such tiny objects and put them in the proper places? How would I better control my breathing and other destructive effects?

Likewise, how will we continue to shrink our processors to reach the next tier? What will we make them out of, and how will we assemble their circuits? How will we protect them from environmental interference? Can our ingenuity keep up with Moore's Law forever?

The Megamountain

The latest model I've come up with for explaining exponential growth is something I call "the Megamountain." Here's how it goes.

We're going to imagine climbing a mountain. First, hink about what it would be like if the mountain had the same steepness all the way up. What would this mountain look like? Try to draw it.

Next, think about what it would be like if the steepness of the hill kept increasing steadily as you went up. What would this mountain look like?

These first two mountains look something like the cartoons shown below.

Now for the mountain you don't want to climb: The Megamountain. On the Megamountain, the steepness of the mountain at any point is proportional to the altitude at that point.

That's the rule of the Megamountain: The steepness is proportional to the altitude.

If you think about this rule carefully, then you begin to realize that the Megamountain is a runaway situation. Because if you're high up, then [by the rule] it's steep; but, if it's steep, then because of that your next step gains a lot of altitude; but [by the rule] that means it's now going to be even steeper; but that means your next step will gain altitude even faster than before; but that means it'll now be even steeper; and...AAAHHH! It makes my head hurt to think about it!

When I think about what it would be like to climb the Megamountain, I actually get a panicky feeling that I can't possibly keep on going this way. I don't even want to take that next step, because every step is feeding a vicious cycle.

(By the way, you'll notice that I'm not going to try to draw the Megamountain. That's because it can't be drawn; not really. Sure, you can plot a graph of y = e^x, but in the end, you're going to find yourself plotting only the region around x-values of order unity. And in this region, the curve looks roughly similar to a parabola, so you haven't shown what is special, and terrifying, about runaway exponential growth.)

A collection of runaway situations

There are a lot of runaway situations like the Megamountain, including:

* Unchecked population growth: The number of babies born is proportional to the number of people already here. More babies make more people make more babies make more people....

* Gestation: Suppose you had to build a hundred billion houses in 9 months. I think you would quickly hit upon the idea of building houses that build houses. This is how we get from a single fertilized egg cell to a big fat baby in only 9 months. The cell is a house that builds houses. Like grains of rice on a chess board, the number of cells added is twice the number of cells that were there before. More cells make more cells make more cells make more cells....

* Chain reactions. A uranium nucleus splits into two, and the two products strike two more uranium nuclei, causing them both to split in two; their four products strike four more uranium nuclei, and so on. This process is called fission; it was actually named for the biological process of cell division (which was called fission first). In the same way that a big, chubby baby begins with a single cell, here a single subatomic "pop" is magnified, in a millisecond, into an explosion that can level a whole city.

Mathematically, a pregnancy is a runaway chain reaction in the uterus...an explosion of a kind, but one that takes 9 months to unfold.

* Compound interest. The amount of money credited to your account is proportional to the amount of money already there. More money makes more money makes more money makes more money....

But you know, I have had savings accounts, and I have never exactly had the feeling that my money was undergoing an explosive chain reaction! The reason is that the interest rate is so small. It's true that if you wait long enough, your money will double, then double again, and eventually bankrupt the Chinese empire. But the question is, how long will it take to double?

The rule of 72: Divide 72 by the interest rate, and that's how many years it will take to double.

Example: You have a CD earning 4 percent interest. Divide 72/4 = 18, so your money will take 18 years to double. After 18 more years, it will double again.

Warning: The cost of goods and services is also growing exponentially at a 4 percent rate (at least), so by doubling your money in 18 years you are really just keeping up. Your $100 today will double to $200, but that $200 will only buy what $100 buys today. Hence, if your money is not earning at least the same as inflation, the growing cost of goods and services will outstrip the value of your money, and you will actually be losing money in real terms. This is called "inflation risk," and it's the reason you have to put at least some of your money into higher-risk/higher-return investments.

The suddenness of exponential growth

I like to show people these two crude movies that I made a long time ago. Both movies are cartoon visions of what it might be like to ride in a spaceship that splashes down on the north pole. In the first movie, the spaceship moves at a constant speed. In the second movie, the spaceship moves at an exponentially increasing speed.

(The views are through a porthole on the spaceship. Sorry about the aspect ratio - something got screwed up when I put the videos on YouTube.)

* Note, a clearer version of the first movie is available here.

* Note, clearer version of the second movie is available here.

Whenever I show people these videos, they can hardly believe the second video. It says a thousand words about the way exponential growth can sneak up on you—and how unstoppable it is, once it gains momentum.

Where does the suddenness of exponential growth come from mathematically? One way to think about it is to recognize that when we "run the clock in reverse," exponential growth is a continual process of cutting in half. This means that any process of exponential growth must spend a very long time at very small values. And in any graphical or visual sense, the point is that one very small number is going to look visually just like another, even if the two numbers in question differ by many orders of magnitude. (On a graph with values ranging from 0 to 1, a value of 0.0003 is going to be indistinguishable from a value of 0.0000000008—even though you'd much rather your chance of winning the Lotto were 0.0003 instead of 0.0000000008!)

Additionally, when you throw in the fact that the rate of change of any changing quantity involves taking simple differences, you see why the rate of change can remain small even when the underlying numbers are actually growing by orders of magnitude. (The difference between two small numbers is necessarily small, even when the two numbers differ by orders of magnitude.)

All of this is why you can watch something "growing exponentially in time" and wonder why it's just sitting there. It's just sitting there, sitting there, sitting there, and BANG! All of a sudden it explodes. The explosion happens when your numbers take the crucial steps from "smaller" to "small" to order-unity. Prior to order-unity, it looks like nothing is happening; after order-unity, it's too late to do anything.

***

Ultimately, "understanding exponential growth" might have little to do with being able to solve certain classes of transcendental equations. It might instead depend on having a gnawing feeling in your gut that exponential growth is an unstoppable force: not only an unclimbable mountain, but an insatiable feeding machine that will devour anything in its path; and a monster that will lie in wait, lie in wait, lie in wait, and then leap forward in the blink of an eye.

***

But to end on a more positive note, we should also remember that exponential growth can also be a resource. (Anybody need four miles of noodles? Just fold 12 times!) In other words, the best weapon we have against the exponential function might be the exponential function itself. This idea arises for example in the theory of quantum computing, which, when it gets here, will be a process in which the computational resources at our disposal grow exponentially with the number of particles in the processor. Previously intractable problems will become solvable in an instant!

Exponential growth as a resource also comes up in tipping point phenomena - you tell two friends, and they tell two friends, and so on and so on and so on. The numbers stay small, but they're working their way up the orders of magnitude, until we reach the fateful stage of order-unity. We usually think of this model in connection with epidemics and fads. But what if the thing we're spreading is instead a message of positive social change: such as one about changing our habits of energy consumption? That would be fighting fire with fire.

Subscribe to:
Posts (Atom)