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The concept that succesive iterations of a system bring about growth of the form X(n+1)=X(n)^Y.

Moore's Law is an example of exponential growth, with X being processing speed, n being an 18 month period, and Y being 2.

Exponential Growth is in essence any geometric progression with a common ratio greater than 1!

If the common ratio was less than 1 then it would become exponential decay instead!

The exponential function is a special case, where the common ratio is equal to e.

Comment on mrichich's writeup:
As I recall, Moore's law states that the number of cirquits that can be fitted on a given area of silicon will double in eighteen months. That means the function would look something like
X(n)=X(n-1)*2
or, non-iteratively
X(n)=X(0)*2^n
A rate of growth of an organism, a part of an organism, or a population of organisms which, when graphed, produces an exponential or logarithmic curve. Such a rate occurs, for example, during the exponential growth phase, when a population of bacterial (or other) cells divide at a constant rate so that the total number of cells doubles with each division.

From the BioTech Dictionary at http://biotech.icmb.utexas.edu/. For further information see the BioTech homenode.

First,

a story:

Once upon a time,

there lived a very wise man in India.

He presented a beautiful chessboard to his king. The delighted king promised him anything he desired.

The wise man asked for a single grain of rice to be placed on the first square of the chessboard, two grains of rice on the next square, four grains on the next, and so on.

The king readily agreed. He ordered his slaves to bring a bushel of rice and place it on the chessboard squares just as the wise man had requested. Each new square received twice as much rice as the square before it. By the second row, the slaves were counting hundreds of grains of rice. The courtiers watched as the piles grew higher and higher.

Just as fine sand trickles through an hourglass, and the boy who watches the golden thread in wonder feels his eyes grow heavy, dull, leaden, as thick hours are distilled into seconds, and the seconds are heaped into minutes, and the minutes are garnered again into hours, and the boy is lulled by the plaything, and at last his eyes are sealed with sleep, so too did brown grains of rice slip through brown fingers, so too did the king grow tired and bow his head. His crown slipped from his brow and clanged on the marble floor. The king started at the sound and picked up the crown at his feet. "I have had enough," said the king, rising from his throne, "I command you to continue."

So the king retired. The slaves added grains of rice. The courtiers left one by one. Only the wise man remained to watch. He waited silently, patiently. The day grew late. Left to themselves, the slaves discovered ways to count more quickly. Each insight left them working with greater zeal. In this way they kept pace with their work. When the twentieth square had been filled, the bushel bag had less than a handful of rice at the bottom. One slave strode out of the room. He returned bearing another bushel of rice on his back. Without saying a word he dropped the entire bag onto the pile. Grains of rice leapt into the air and scattered across the marble floor. The others flinched. They stared at the bag. Slowly, they looked up at the wise man. The wise man smiled back at them. And that was square twenty-one. Soon, two more bushel bags lay on the chessboard, then four more, then eight: the third row was completed. Shortly after that the men brought an oxcart of bushels, then two oxcarts, then four. The men worked late into the night. They recruited more men to help them. At dawn, the wise man greeted the king and escorted him to the court. The doors of the throne room opened to a solid wall of rice stacked in bushels. "I see that Your Majesty has filled thirty-eight squares," said the wise man, "And when you have filled thirty-three million such chambers, I shall return to collect my prize."


Exponential growth

is an idea so simple, so powerful, yet so unintuitive that it seems likely that we will perish in part from our inability to comprehend it.

So what is it?

To put it simply, exponential growth is a pattern of increase over time based on multiplication rather than addition.

I say 'rather than addition' because human beings seem to be more comfortable with the idea of how things grow by adding up or accumulating. Every year I add +1 to my age; in five years I will add +5 to my age. In five hours, a ship that cruises at 30 mph will travel 150 miles. The mathematical name for this type of increase is linear growth, so-called because a chart of age over time or distance over time will graph as a straight, unbroken line (barring death, time travel, icebergs, or pirates).

Things that continually multiply themselves (by a factor greater than one) are more unwieldy. This sort of graph begins as a gentle, rising slope that curves upward and ascends rapidly in an attempt to slap infinity in the face.

Here's another way to imagine exponential growth. Take a thin piece of paper. Let's suppose it is .0032 inches thick. Fold it in half once: now it is two sheets thick. Fold that in half: now it's four sheets thick. It is difficult to fold it in half eight times, but it would then be 2^7=128 sheets thick. Ignoring empty space, your paper wad is now a little less than half an inch thick. Not that impressive? Keep folding it, or imagine you could. 20 more folds and now your tortured little sheet of paper is almost seven miles high, much taller than Mt. Everest. Nine folds after that and it towers past the moon. A sheet of paper folded a total of 105 times is longer than the diameter of the observable universe! Knowing that makes me want to jump up and dance around the room.*

Why does any of this matter?

Physical and biological laws moderate all instances of exponential growth we see in reality. Material constraints, competition, and death are the usual suspects here. Our species has learned a couple of neat tricks, but we tamper with these forces at our peril.

All populations grow exponentially. Colonies of bacteria double each time every cell reproduces by dividing. Viruses multiply themselves a thousandfold every time they successfully hijack a cell. Humans too are fruitful, and multiply.

If our inherited instincts of how things grow and change are calibrated more to predict linear growth patterns, we will perhaps respond too late or with half-measures against a breakaway cycle of exponential growth.

World leaders have set a precedent for passing on mounting crises to their successors and most citizens seem unmotivated to address inter-generational problems. The issues that we face are much more complicated than tallying rice, but the parable teaches us something about unexpected consequences.

So take my story and message of doom with a grain of salt. And then two more. And then double to taste.



* The chessboard problem was a little different than this because that story adds together all of the rice from the previous generations. To apply the same trick to the paper example, you could imagine someone building a pyramid of folded paper. At the bottom would be a single, unfolded sheet. Above that would be a brand new sheet of paper folded once. On top of that would be a new sheet of paper folded two times, etc. Actually, it would seem less like a pyramid and more like a wicked looking spike piercing through the sky, vanishing as the paper columns tapered thinner than a spiderweb, thinner than an atom, thinner than an electron, and then much thinner still.

Be sure to take a look at Oolong's brother's dire warning about exponentially growing hamsters. Ooh, it's animated! The threat is real, people.

http://waynesword.palomar.edu/lmexer9.htm
http://mathforum.org/~sanders/geometry/GP11Fable.html
http://www.vaughns-1-pagers.com/computer/powers-of-2.htm

Visualizing exponential growth is something we're currently all doing; we look at the covid-19 graphs either worldwide or, for the most part, our own countries (China and South Korea are exceptions) and we see that we're somewhere along the way up a very steep slope in terms of new cases. Even when we're not talking about the spread of a disease, seeing a graph that has exponential growth (of anything at all) can be scary.

I came across this really neat YouTube video that gives us another way to look at exponential growth. So, let's start with the typical curve you can see if you're looking at the data from Johns Hopkins University. It's plotting cumulative number of cases on the y-axis and time on the x-axis. With very few exceptions, every country is on the steep part of the exponential curve.

What this video suggests though (and this just makes sense anyway) is to use a logarithmic scale for the y-axis (the number of new cases). This makes sense, the inverse of an exponential is a logarithm and will flatten out the curve. We could do the same for the x-axis (time) but then the graph would become very misleading as each interval would represent, for example, a doubling of the previous value. So, the first interval would represent 1 day, the next 2, the next 4, the next 8, and by the time we got to the 10th interval you'd be looking at a period of a little short of 3 years. So, scratch that, don't use a logarithmic scale for time unless you really mean it.

Now, let me go back to something I hid back there; the y-axis is used for the number of new cases. Even this is, in many countries, growing exponentially so it still makes sense to use a logarithmic scale. Now, on the x-axis, they use the total number of cases (also on a logarithmic scale). Now, points on the graph represent the ratio of new cases to existing cases and, when both scales are logarithmic comes out a a fairly straight line. Well, fairly straight until you get things under control, then the curve plummets as the number of new cases drops in comparison to the total number of cases.

Why care? Well, first of all, it's reassuring that there's a way to visualize things that's helpful. Second, there are some important caveats (e.g., time is implied by progression along the curve). But, third, and most important, this is a way to visualize any kind of exponential growth which eventually reaches a limit. Even if it doesn't reach a limit, what you'd see would be the straight line curve continuing on. If growth stops, the line drops. Simple.

I'm considering using this kind of visualization for, for example, measures of defect growth in software systems.

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