Definition:

A pattern of data that shows greater increases with passing time, creating the curve of an exponential function. [REF]

Deconstruction:

Various forms of exponential growth are experienced in everyday life.  Mold evolving on a loaf of bread in your kitchen.  Accumulating compound interest of an investment account.  Viral videos spreading via social media across the internet.  Drastic improvements in personal computing power, per Moore’s Law predicting integrated circuit innovation. 

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The shape of an exponential growth curve is commonly known, even if the term is not.  It’s basically an arc that starts shallow and horizontal at the bottom left, gradually ramping upwards to a nearly vertical line at the top right of the graph.  

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The sharpness of the curvature varies based on the starting level, and the multiplication rate, but the inevitable result of exponential expansion is predicable.  You start with very few of something, and over the long term, end up with more of that item than you would ever need, or want, depending on how useful the entity is.  This inexorable property of exponential growth is something that can be either a blessing, or a curse. 

The concept also applies in reverse, a phenomenon known as exponential decay.  Several interesting examples in this vein are the half-life of radioactive isotopes in a nuclear reactor, the diffusion of caffeine in the human body, and the decrease in atmospheric pressure when climbing a mountain.

In nature, exponential growth and decay are common, but often eventually hindered by fundamental capacity constraints.  Invasive plants like English ivy spread uncontrollably, until they consume the entire forest around them.  Rabbits, known for their breeding proclivity, experience massive population surges, inevitably checked by the food and space limitations of their surroundings.  In early-stage tumors, cancer cells unfortunately divide at an exponential rate, attacking the body aggressively unless counteracted with treatment, or when they achieve their morbid goal.

One of the most interesting, and mathematically useful, exponential growth curves is the natural function.  Per the diagram below, this is defined as the Euler number “e”, a quantity of roughly 2.718, raised to “x”, the x-axis value.  

This function has the unique property that its calculated result matches both the slope at any point along the line, as well as the area under the curve up to the input value, making it very useful for the Calculus branch of mathematics.  

Like many other analytical terms that involve repeated multiplication, the massive rate of change over time is difficult for the human mind to comprehend.  This is because strict mathematical equations are theoretical, and their predictions often exceed the physical laws governing our world.  The best way to understand exponential growth is with a few simple, but unfathomable, examples.

Example #1:

A savvy child convinces her stingy father to change her allowance from $10 per week, to a penny a day, doubling each morning, and resetting back to the original single cent at the beginning of each new month.  At the end of the first week, the father smirks, as he tosses his daughter a handful of coins, totaling just 64 cents.  

However, by the second week, the girl goes to her dad and asks for payment, which calculates out to $81.92, more than double her usually monthly total, which the man begrudgingly pays.  Fortunately, the accommodative lass is nice enough to let her father of the hook at this point, content to have taught her elder a valuable lesson about math.  

Good thing.  Even in February, the shortest month, during a leap year, on the 28th day, the father would have owed his daughter $1.3 million for the 4th week’s allowance.  That’s a lot of ice cream and video games.  

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Example #2:  

Bored, you pick up a sheet of standard white printer paper, and start folding it in half aimlessly.  The first few pleats are easy, your fingernail creating a crisp crease along the edge.  After the 5th fold, the original page now looks like a standard matchbox, with the next manipulation resembling a bulky UBS stick.  

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On the 7th fold, as the rectangular prism’s aspect ratio switches from horizontal to vertical, you realize this is the last operation you can physically execute.  The wad sitting in front of you is now 128 layers thick, more ball-like, as opposed to faceted, with rounded paper corners, due to your folding ineptitude.

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Intrigued, you envision what would happen if you could keep going.  Physics aside, with unlimited bending dexterity, and infinite crease force, around the 15th iteration, the paper on your desk slices through the ceiling of your bedroom, the cross-section already as thin as a fine human hair.  Just 5 folds later, the increasingly narrow stack of paper now extends past the roof of the 25-story apartment high rise you live in.  

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Then, the experiment really gets interesting.  By the 27th crease, the needle is directly in airplane transcontinental flight paths, and from there its space bound.  On Fold #30, the paper tower leaves Earth’s atmosphere, Fold #32 needs to dodge the International Space Station, and Fold #42 rockets out past the moon.  

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What started mundanely bending a piece of paper over and creasing in, has created unique origami structure, which can reach to the Sun, in just over 50 simple acts.

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Exponential growth surrounds us, and is one of the most powerful factors in our universe.  It has led to cell phone technology more widespread and functional than anyone could have dreamed 50 years ago, but also contributed to the climate and environmental issues resulting from explosive human population increase over the past century.  

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Don’t underestimate exponential growth’s simple, relentless cadence, which imparts both positive, and negative, consequences throughout our world.  

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Details:

• 10 examples of exponential growth in everyday life. [REF]

• Background on the origin of Moore’s Law, and how this prediction has panned out in actual computer technology development. [REF]

• Details on the Euler number, and the natural exponential function. [REF]

• A Dr. Dark After Dark podcast which discusses several very unique examples of exponential growth starting at the 10:15 minute mark. [REF]

• Additional paper folding thought experiment information. [REF]