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Primer On Exponentials

There has been a lot of talk in the news about exponential growth, especially as it relates to epidemics. We wanted to provide some information to the public. We will try to keep it simple. 

Firstly, there is a natural human tendency to perceive change in systems as gradual or even constant, even in disasters: "Water level in the river is rising by a foot a day. When it gets to six feet, I'm flooded out. I have less than a week to get everything I cherish to higher ground." Exponentials are different. If the flood water were to rise exponentially, it would be something like this: "Water level rise is doubling every day. Two days ago it was six inches. Yesterday it was a foot. I need to bug out now!"

The salient quality of a true exponential is that no matter how small it starts (so long as it's not zero) and no matter how slow it grows (so long as it's not zero) the exponential will reach any large number you choose very quickly. One of our favorite examples, no doubt apocryphal but highly illustrative, comes from chess, that game played on a board with 64 squares. It is said that inventor of the game showed it to a great Raja who controlled a vast empire. And the Raja, duly impressed asked the man: "what can I pay you for bringing me this wonderful game?" The man said: "I ask not much. Just one grain of rice for the first square on the chessboard, and two for the second, and four grains for the third and so on." With a smile, the Raja agreed, and they had a deal. And by the 45th square, the man owned all the Raja's empire. By the final square, the man would have owned a thousand worlds and all their empires had the Raja not beheaded him. And so exponentials can cut both ways. Here we show a chart of "The Raja's Curse"




This chart doesn't look very interesting until you get out past 45 squares, but the scale of the vertical axis makes that deceiving. If we show The Raja's Curse for just the first 20 squares, the shape of the chart is almost exactly the same. The only thing that's really qualitatively different is the scale of the vertical axis. We show that below:



Of course, to the Raja the numbers on the vertical axis make a huge difference. If it's just the first 20 squares, he's only out 10 pounds of rice, but if it's all 64 squares, he's out a thousand worlds of kingdoms. 


What Causes Exponential Growth in Nature?

Exponential growth in nature is almost always caused by feedback processes. Let's say I start with ten rabbits. They pair off into five pairs. Each pair makes two baby bunnies. The ten baby bunnies hang out together and make five more pairs that are added to the rabbit patch. Now we have ten pairs. They breed ten new pairs, making 20 pairs. They breed and add 20 new pairs making 40 total pairs, and so on. One way to think about this is that opportunities for interaction create new opportunities for interaction. In the case of a virus, a couple infected people infect a couple others, now all of them infect others, and so on.

The same thing works in reverse. Opportunities for interaction can destroy opportunities for interaction. This happens in chemical reactions. If you put some hydrogen and some oxygen in a jar and add some energy, hydrogen atoms tend to bond with oxygen atoms when they collide. That takes the hydrogen atoms out of play, making for fewer opportunities for interaction. But when an interaction does occur, a bond forms taking that hydrogen atom out of play. Etc. This is a "negative exponential", which we show below. 



Whereas the exponential takes you from any low value to any high value in a short amount of time, the negative exponential can take you from any high value to any low value in a short amount of time, more or less. Negative exponentials are actually a lot more common in the real world and we'll explain that in the next section. 

How do Exponentials Work in the Real World?

Exponentials are a bit unnatural. No exponential can go on for long. If one did, it would soon absorb all of the energy in the universe. Looking back to our chess example for illustration, even if the Raja brought hadn't killed the clever man who foisted the exponential on him, it would have been brought to a halt when all of the world's resources were expended to pay his debt. It's not the same for negative exponentials because they use fewer and fewer resources as time goes on. In our chemical reaction, there are pretty much always going to be a couple unbonded hydrogen atoms, and the fewer there are, the fewer the number of interactions between them so the slower they dissipate.  

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