Exponential Effects: Humanity's Biggest Blind Spot

Intro

“The greatest shortcoming of the human race is our inability to understand the exponential function.”

Albert A. Bartlett

Sometimes the simplest ideas are the most powerful. Multiplying numbers appears to be a trivial process, but hiding behind that deceptive simplicity are forces that have the power to change the world. Human beings are notoriously bad at grasping the significance of exponential growth. For this reason, identifying examples of exponential growth and understanding their impact may be the most important lesson that is rarely learned.

NEXT: Warmup

Warmup

Bacteria Doubling Problem

A Petri dish contains some amount of bacteria. Every second, the amount of bacteria doubles, and the dish is completely full after one minute. When was the Petri dish half full?

NEXT: Wheat and Chessboard Legend

Wheat and Chessboard Legend

Perhaps the oldest story about exponential growth comes from ancient India. According to legend, a king offered the inventor of chess any reward for the invention of such a marvelous game. The inventor requested that a grain of wheat be placed on the first square of a chessboard (which has 64 squares), two grains on the second square, four on the third square, and so on (doubling the number of grains on each subsequent square). How many grains of wheat would be on the chessboard at the end?

NEXT: Folding Paper Experiment

Folding Paper Experiment

Take a piece of paper and fold it in half. Then fold it in half again. If you could fold it in half 45 times, how thick do you think it would be?

What is the most number of times that you can fold a piece of paper? What if you use very thin paper, like toilet paper? Share your answer with others and find out about the limits of what is possible in the Latest Lesson community.

NEXT: Domino Effect

Domino Effect

Most people know that when a domino falls it can knock over a nearby domino of the same size. It turns out that a domino can knock over another domino that is about 1.5 times as large in all dimensions as itself. As you probably realize by now, even if you start with a small domino that is 5 mm tall and line up just 29 dominoes, with each subsequent domino 1.5 times as big as the previous one, the last domino will be quite tall. If you knock over the first domino, the rest of them will fall, and each subsequent domino will release more energy than the previous one. How tall would the 29th domino be, and can we generate new energy by simply toppling heavy dominoes?

Answer:

NEXT: Carbon Dating

Carbon Dating

When some quantity decreases exponentially instead of increasing exponentially (exponential growth) it is called exponential decay, and it is the core concept underlying carbon dating, which allows scientists and historians to determine the age of objects containing organic material such as fossils and archaeological artifacts. Follow the interactive guide to learn how it works.

Go to the interactive guide

NEXT: Epidemics

Epidemics

COVID Simulation

An infectious disease like COVID-19 can spread exponentially, but exponential growth cannot continue forever. Learn about the limits of exponential growth and how diseases can be modeled and their effects mitigated by following the interactive guide and playing with the interactive COVID-19 simulations.

Go to the interactive guide

Logistic Curve

If indefinite exponential growth is not possible because eventually the resources that fuel that growth start running out, what curve describes that growth and eventual slowdown? Find out by watching this short animation and taking a deeper dive into the math behind the rise and decline of exponential growth.

NEXT: Implications

Implications

Philosophy

Suppose that the bacteria in the warmup problem could think, know how fast they were reproducing (doubling every second), know how much of the Petri dish was still unpopulated by them at any moment in time, and know that they would start dying when the dish was completely full (after one minute). When would they realize that they are facing a disaster?

Economy

Bacteria can't think and don't have to worry about impending doom, but humans have that ability. Do people understand when they are climbing an exponential curve and what it might mean? In this legendary lecture, Albert A. Bartlett answers that question and explains the perils of ignoring the simple arithmetic that leads to a population explosion and the collapse of energy resources that fuel the modern economy. This lecture may not have flashy graphics, but the powerful — and mostly ignored — message that it conveys more than makes up for that.

NEXT: Music

Music

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is determined by multiplying the previous one by a constant, non-zero number called the common ratio. An example is the sequence 1, 3, 9, 27, 81,...., where the common ratio is 3. When the common ratio is greater than 1, the result is exponential growth. Is there a connection between geometric progressions and music? Watch the video about the physics of playing guitar, to discover the answer.

Bach

As you saw earlier, Western music uses equal temperament, which is a tuning system that divides an octave into equal steps (the frequencies of the tones form a geometric progression). This was a relatively modern development, but the impetus for it may have been Johann Sebastian Bach's Well-Temepred Clavier, although there is still debate about how exactly Bach tuned his own instruments. Regardless of the historical nuances, the Well-Tempered Clavier is one of the crowning achievements of human civilization, which is why it was included in a recording aboard Voyager 1, a space probe that is the most distant man-made object from Earth. Listen to this recording (and watch the animations) for an otherworldly experience.

NEXT: Open-Ended Question

Open-Ended Question

Think of an example of exponential growth that has not been mentioned here but that has profound implications for humans, yet is not necessarily widely known or understood. Share your example with the Latest Lesson community, and see what others come up with.