Asimov - An Exponentialist View
Bartlett - An Exponentialist View
Darwin - An Exponentialist View
Dawkins - An Exponentialist View
Drexler - An Exponentialist View
Ehrlich - An Exponentialist View
Malthus - An Exponentialist View
Sagan - The Secrets of the Universe
Savage - An Exponentialist View
Turchin - An Exponentialist View
Wallace - An Exponentialist View
Witting - An Exponentialist View

External Links:

Carl Sagan's Influence: Favourite Quotes from Readers - Universe Today
Legal Metrology, Metrication and Bytes - David Coutts
Day of 6 Billion - UNFPA
Day of 7 Billion - UNFPA
Total Midyear World Population for 1950 to 2050 AD from the US Census Bureau

Wikipedia articles:

Binary Prefixes for bytes
Rule of 70



 Carl Sagan - The Secrets of the Universe

 "If you understand exponentials, the key to many of the secrets of the Universe is in your hand." (Sagan,1997, p.23)


I've always loved that quote from Carl Sagan, and my examination of all things exponential whilst writing my Exponentialist web site has certainly been illuminating for me at least. In this article, as with similar articles for those I term Famous Exponentialists, I will explore Sagan's understanding of "exponentials". Most of Sagan's thoughts on the matter are contained in just one book, Billions and Billions, in the first two chapters. Let's see if Sagan is right...let's look at exponentials and see if they truly do unlock "...many of the secrets of the universe". I'll also mention some secrets, or potential secrets, that Sagan did not include.

I've been a huge fan of Sagan's work, both science-fiction and non-fiction. Being pro-space myself, for me it all started with the TV series for Cosmos, followed by the excellent book Pale Blue Dot. In this article then I will also explore Sagan's pro-space advocacy.

Above all, I value his work on the nature and value of science.

Scientific Notation

Appropriately enough for a book called Billions and Billions, Sagan starts by providing a useful introduction to the concept of scientific notation (or exponential notation, as he terms it) for big numbers through expressing a number to the power of another number.

A simple example is 102 (10 to the power of 2 = 10 x 10 = 100). Exponential laws then allow you to easily multiply large numbers by adding the exponents together - the first example Sagan gives is 1,000 x 1,000,000,000 = 103 x 109 = 1012. Another example with bigger numbers are 1011 galaxies with 1011 stars each would mean there are 1022 stars in the universe.

A reverse rule applies for division, not mentioned by Sagan, whereby you subtract exponentials instead. So 1,000,000,000  / 1,000 = 109  / 103 = 106 (1,000,0000).

I have no problem with this system, though Sagan admits it does make those shy of mathematics a little "jittery". Sagan also provides an amazing table (p.9) that estimates how long it might take to count from zero (with one count per second non-stop night and day). So, for example a Quintillion = 1,000,000,000,000,000,000 or 1018 in scientific notation (which is how he now refers to it in his table). It would take you a staggering 32 billion years!

Sagan acknowledges that although the use of scientific notation is a relatively recent phenomenon there were precursors in Mayan, Indian and Chinese cultures.

The Persian Chessboard

In the second chapter, called The Persian Chessboard (Sagan acknowledges the story may have originated in India or even China) Sagan gives his version of a very common story how exponential powers of two were a very "unpleasant surprise" for a Persian King outwitted by his Grand Vizier. Having invented the marvellous game of chess for his King, the Grand Vizier was asked to name his reward. He asked for 1 grain of wheat for the first square on the chess board, two for the second, 4 for the third, 8 for the fourth and so on...until he receives payment for all 64 squares on the board. The King apparently argues that the reward is too modest, but agrees to pay.

Well, 264 is 18.6 quintillion grains of wheat (actually 18,446,744,073,709,551,616)...which Sagan calculates would weigh 75 billion metric tons or 150 years of today's modern wheat production. Unable to pay, the King is effectively bankrupted and - depending on what version of the story you prefer - may have been forced to hand his kingdom over to the Grand Vizier.


In his chess example, Sagan explains that 210 = 1024. Working in computing I am very familiar with this number being applied to bytes:

Bytes 1 2 4 8 16 32 64 128 256 512 1024
Kilobytes 1 2 4 8 16 32 64 128 256 512 1024
Megabytes 1 2 4 8 16 32 64 128 256 512 1024
Gigabytes 1 2 4 8 16 32 64 128 256 512 1024

Table A - Commonly used binary multiples of Bytes. 

Note:  1 Kilobyte = 1024 bytes, 1 Megabyte = 1024KB, 1 Gigabyte = 1024 MB and 1 Terabyte = 1024GB.

In fact, a new international standard has declared that these binary multiples of bytes are technically kibibyte, Mebibyte and Gibibyte for 1,024 bytes, 1,024 kilobytes and 1,024 Mebibytes and so on. Kilobyte should strictly be regarded as 1,000 bytes, Megabyte as 1,000,000 bytes etc to adhere more closely with proper usage of SI Unit prefixes having a decimal meaning. Refer to this Wikipedia link Binary Prefixes for bytes for more. For my own views on this matter, refer to my article Legal Metrology, Metrication and Bytes.

Returning to the Persian Chessboard

The main reason I mention this is because my understanding of the essentially binary nature of bytes has lead me to create what I call the New Malthusian Scale. I don't claim that this in any way replaces the need or value of scientific notation. However, I think it provides a useful alternative aid for non-mathematicians to visualise what Sagan has explained. Normally I apply it to populations (and there's another example later on) and these days choose to avoid prefixes such as Mega or Mebi and hence use the terminology A-Pop, B-Pop etc which I think works well. Hear I've applied it to grains of wheat based on Sagan's story of the Persian Chessboard:

Individual grains 1 2 4 8 16 32 64 128 256 512 1024
A-grains 1 2 4 8 16 32 64 128 256 512 1024
B-grains 1 2 4 8 16 32 64 128 256 512 1024
C-grains 1 2 4 8 16 32 64 128 256 512 1024
D-grains 1 2 4 8 16 32 64 128 256 512 1024
E-grains 1 2 4 8 16 32 64 128 256 512 1024
F-grains 1 2 4 8 16 32 64 128 256 512 1024

Table B. New Malthusian Scale  Projected doubling of grains of wheat for each of the 64 squares on a chess board

Note:  1A-grain = 1024 grains, 1B-grain = 1024 A-grains, 1 C-grain = 1024 B-grains etc.

This combination of binary (doublings) and decimal (each row is 10 doublings) makes things easy, as 6 rows is 60 doublings which takes us to the end of the E-grains row or the beginning of the F-grains row (1 F-grain = 1,024 E-grains). Four more doublings means the total number of grains is 16 F-grains. To test this, divide 18,446,744,073,709,551,616 by 1024  repeatedly (for a total of 6 times) with 16 being the remainder.

Of course, you could just try this example with a chess board (use anything, maybe grains of rice) but then you're mixing binary (doublings) and octal (each row on the chess board is only 8 doublings) which is less familiar and clumsier (we often think in decimal, and binary, but rarely in octal).

I feel that the New Malthusian Scale comes into its own when we consider populations (see later) as population growth occurs over time whereas there is no time scale for the Grand Viziers exponential victory over his King.


Sagan states (Sagan, 1997, p.15):

"Exponentials show up in all sorts of important areas, unfamiliar and familiar - for example compound interest."

Sagan goes on to give a pretty simple example of $10 invested at 5% for 200 years resulting in a windfall of $172,925.81.

As Sagan has already demonstrated doubling is a useful way of visualising exponential growth. Another easy way of very quickly estimating the return on your investment here is to use the Rule of 70. Simply divide the interest rate into 70 to get the doubling period - in this case 14 years. Hence your money will double roughly every 14 years at 5%.

200 years divided by 14 is just over 14 (14 x 14 = 196). Ignoring just 4 years of compound interest, in 196 years your money would have doubled 14 times from a starting value of $10 to a windfall of $16,3840:

10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840

Sagan also explain how inflation is a similar but reverse process, as it involves negative growth such that for a 5% inflation rate your dollar is reduced in value by 5% every year. In many ways, inflation is similar to the Radioactive Half-Life discussed later. Also, see my article Negative Growth - Constant Rate for a quick overview of such growth.

I explore Compound Interest in greater detail in my article Understanding Compound Interest and related articles including one on the exponential nature of the stock market as illustrated by the Google share price over time (see Exponential Brownian Motion). I also explore the Rule of 70 in greater detail (including negative growth) in my article The Scales of 70.

Our Common Ancestral Tree

Sagan draws a parallel between the Persian chessboard story and your ancestral tree (Sagan, 1997, p.21):

"Everybody has two parents, four grandparents, eight great-grandparents, sixteen great-great-grandparents etc."

Assuming 25 years per generation Sagan calculates that just 1,600 years ago (around 400 A.D. and the end of the Roman Empire in the West) you would have had 64 generations of ancestors...yes, that's right, 18,446,744,073,709,551,616 ancestors (the same number as the grains of wheat owed to the King in the Persian chessboard story). He notes that this is more than the number of people alive then or indeed more than the number of people who have ever lived.

He draws the inevitable conclusion that "we are all cousins". Hence, whenever someone claims a famous person in their family tree you can be assured that both that person and their famous ancestor are also in your family tree....somewhere. The only difference is whether or not you can actually trace your lineage...but it's there, nonetheless.

I have explored this hidden treasure of the world of exponentials in my own article Generations and Population Doublings with some amusing quotes from Bill Bryson as he struggles with the same concept.

Population Crisis

I have a deep and abiding respect for Sagan's ability to tackle all the big issues with reason, including perhaps the most thorny issue of all...overpopulation (Sagan, 1997, p.18):

"Exponentials are also the central idea behind the world population crisis. For most of the time humans have been on Earth the population was stable, with births and deaths almost perfectly in balance. This is called a "steady state". After the invention of agriculture ... the human population of this planet began increasing, entering an exponential phase which is very far from a steady state. Right now the doubling time of the world' population is about 40 years. Every forty years there will be twice as many of us. As the English clergyman Thomas Malthus pointed out in 1798, a population increasing exponentially - Malthus described it as a geometrical progression - will outstrip any conceivable increase in food supply. No Green Revolution, no hydroponics, no making the deserts bloom can beat an exponential population growth."

I do take slight issue with Sagan here, in his only mention of the first Exponentialist - Malthus. I simply don't like the concept of an exponential phase, even though I know what Sagan means (that the annual growth rate was pretty constant at this time). Refer The Myth of The Exponential Phase for more.

I argue that all populations of all species grow via variable rate compound interest (which is a force with exponential power) - see What Is Exponential? for more. This applies not just to a population but equally to all sources of food for all populations of all species, including the food supply for human populations. Hence, food supply is capable - but is not necessarily guaranteed - to keep up with (or even exceeding) exponential population growth. I argue that famine is the periodic result of  a negative imbalance between the exponential power of a population and it's exponential food supply (for many possible reasons this can be periodically less than needed by the population). Hence, Sagan is wrong to assert that nothing can beat an exponential population growth as another exponential population growth (of the food supply itself) can exceed what is required by a population and produce a glut .

Sagan provide a brief example of human population growth expressed via populations doublings (Sagan, 1997, p.20):

"At present there are around 6 billion humans. In 40 years, if the doubling time stays constant, there will be 12 billion; in 80 years, 24 billion; in 120 years, 48 billion. . . . But few believe the Earth can support so many people. Because of the power of this exponential increase, dealing with global poverty now will be much cheaper and much more humane, it seems, than whatever solutions will be available to us many decades hence. Our job is to bring about a worldwide demographic transition and flatten out that exponential curve— by eliminating grinding poverty, making safe and effective birth control methods widely available, and extending real political power (executive, legislative, judicial, military, and in institutions influencing public opinion) to women. If we fail, some other process, less under our control, will do it for us."

Sagan's ominous warning if we fail to slow population growth that "...some other process, less under our control, will do it for us" echoes Malthus' warning of that periodic leveller of human populations - Gigantic Inevitable Famine (Malthus, 1798). Even though food supply is also exponential, and is therefore capable of keeping up with population, there is no guarantee that it will do so...

Here I've translated and extended Sagan's brief example onto my New Malthusian Scale:

Doubling Period   40   40   40   40   40   40 40 40 40 40
of humans
1 2 3 4 6 8 12 16 24 32 48 64 128 256 512 1024
Doubling Period   40   40   40   40   40   40 40 40 40 40
A-pops 1 2   4 6 8   16   32   64 128 256 512 1024
Doubling Period   40   40   40   40   40   40 40 40 40 40
B-pops 1 2   4   8   16   32   64 128 256 512 1024

Table C. New Malthusian Scale  Projected doubling human global population

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops, 1 C-pop = 1024 B-pops etc.

This shows our 6 billion starting population and the three doublings each 40 years to 48 billion, all nested neatly within the same classic exponential series used for the Persian chessboard example. This is an important point, as at 6 billion we were part way through doubling from 4 billion to 8 billion. Also, to get to 6 billion we doubled from 3 billion in 1960 to 6 billion in 1999...just 39 years. So 6 billion is just a moment in time in the classic exponential series. The Day of 6 Billion was 12th October, 1999, as it happens, with the Day of 7 Billion officially recognised by the UNFPA on 31st October 2012...just  over 13 years later.

Of course, even though Sagan assumes a constant rate of growth (and hence a constant doubling period of 40 years), this does not happen in practice. Growth rates vary every year as epitomised in the table presented in Total Midyear World Population for 1950 to 2050 AD from the US Census Bureau. Essentially, variable rates of population growth lead to variable population doubling times in a manner approximated in my The Scales of 70 and precisely represented in my Scales of e. Again, think variable rate compound interest...just four words...and you will understand how all populations of all species grow and shrink over time.

Human Global Ecophagy

I've added in 6 A-Pops in Table C to illustrate a 1024 fold increase in population from Sagan's starting 6 billion. This is 6 x 1024 = 6144 billion or 6.144 trillion people. Sagan thinks that few would believe that the Earth can sustain a human population of over 48 billion. So you'd be insane to promote human population growth to the level of 6 trillion people or more on Earth...you'd think so (I did) but you'd be wrong. Here's a post from my Exponentialist blog condemning pro-space advocate Robert Zubrin for doing just that...and denying that limits to growth exist even on an Earth for which he provides an accurate and finite definition of 6 trillion trillion kilograms (Zubrin, 2012, pp.123-124)!

Or read my Human Global Ecophagy which takes it to the extreme and calculate the theoretical time it would take humanity to consume the entire Earth and everything on it. Alternately, read Grey Goo - An Exponentialist View for a look at a similar scenario involving futuristic molecular nanotechnology.

Zubrin tries to avoid fully exploring the issue of growth within limits by then leaping into space (Zubrin, 2012, p.124). I've got nothing against the idea of humanity spreading through the Solar System and beyond to the stars (and I'm sure Sagan would be fine with it too, having read his excellent Pale Blue Dot). I even admired Zubrin for his pro-space books A Case For Mars and Entering Space. But as Sagan says (Sagan, 1997, p.19):

"There is no extraterrestrial solution to this problem....Even if it were possible to ship everyone off to planets of distant stars on ships that travel faster than light, almost nothing would have changed - all the habitable planets in the Milky Way would be full up in a millennium or so. Unless we slow our rate of reproduction. Never underestimate an exponential."  (my bolding)

This is uncannily like Malthus' own comment from over two centuries ago (Malthus, 1798)

"The germs of existence contained in this spot of earth, with ample food, and ample room to expand in, would fill millions of worlds in the course of a few thousand years. Necessity, that imperious all pervading law of nature, restrains them within the prescribed bounds. The race of plants and the race of animals shrink under this great restrictive law. And the race of man cannot, by any efforts of reason, escape from it. "

In the book that accompanied the TV series Cosmos  Sagan demonstrates once again both his humanity and humility (Sagan, 1980, p.103):

"Our intelligence and our technology have given us the power to affect the climate. How will we use this power? Are we willing to tolerate ignorance and complacency in matters that affect the entire human family? Do we value short-term advantages above the welfare of the Earth? Or will we think on longer time scales, with concern for our children and our grandchildren, to understand and protect the complex life-support systems of our planet? The Earth is a tiny and fragile world. It needs to be cherished."

Yet Zubrin and his supporters would label Sagan (and me) a Merchant of Despair, Radical Environmentalist, Criminal Pseudo-Scientist, and/or Antihumanist. Nothing could be further from the truth. I agree wholeheartedly with Sagan that the Earth must be cherished, and the biosphere protected.

Sagan argues that humanity has become a danger to itself on Earth and repeats his concerns on global warming in Pale Blue Dot adding ozone depletion, nuclear weapons, asteroids, biological warfare, and that (Sagan, 1994, p.304):

"We humans have already precipitated extinctions of species on scale unprecedented since the end of the Cretaceous Period."

I share Sagan's concerns and find that they contrast sharply with those of Robert Zubrin. For example, Zubrin argues that climate change is having a positive affect on Earth's biosphere (Zubrin, 2012)! Sagan's passionate plea for humanity, and "on behalf of Earthlife", is that we adopt a balanced approach (Sagan, 1994, p.312):

"...safeguard the Earth from otherwise catastrophic impacts, hedging our bets on the many other threats, known and unknown, to the environment that sustains us. Without these arguments, a compelling case for sending humans to Mars and elsewhere might be lacking. But with them - and the buttressing arguments involving science, education, perspective, and hope - I think a strong case can be made. If our long-term survival is at stake, we have a basic responsibility to our species to venture to other worlds."

We need more people like Carl Sagan in this world! I miss that voice of reason and humanity.

Space Colonisation

In Cosmos  Sagan eloquently explains his vision for a space faring species, and openly acknowledges the need to limit population growth on the home planet in order to avoid global ecophagy and crash into the inevitable Malthusian limits to growth, despite the constraints this imposes (Sagan, 1980, p.310):

"No civilization can possibly survive to an interstellar spacefaring phase unless it limits its numbers. Any society with a marked population explosion will be forced to devote all its energies and technological skills to feeding and caring for the population on its home planet. This is a very powerful conclusion and is in no way based on the idiosyncrasies of a particular civilization. On any planet, no matter what its biology or social system, an exponential increase in population will swallow every resource. Conversely, any civilization that engages in serious interstellar exploration and colonization must have exercised zero population growth or something very close to it for many generations. But a civilization with a low population growth rate will take a long time to colonize many worlds, even if the strictures on rapid population growth are eased after reaching some lush Eden." (my bolding)

I believe we can both learn to live sustainably within limits to growth whilst at the same time take every opportunity to expand beyond the Earth. It's not one or the other - we can do both. It's a difficult balancing act, as alluded to by Sagan, but here's no fundamental reason why we can't succeed.

Other Famous Exponentialists that successfully fuse Malthusian population theory and space exploration include the following (with links to respective articles):


Bacteria are often trotted out as an example of exponential growth precisely because people like Sagan have noted their ability to double their population regularly and relatively quickly (Carl Sagan, 1997):

"The most common circumstance in which repeated doublings, and therefore exponential growth, occurs is in biological reproduction. Consider first the simple case of a bacterium that reproduces by dividing itself in two. After a while, each of the two daughter bacteria divides as well. As long as there's enough food and no poisons in the environment, the bacterial colony will grow exponentially. Under very favorable circumstances, there can be a doubling every 15 minutes or so. That means 4 doublings an hour and 96 doublings a day. Although a bacterium weighs only about a trillionth of a gram, its descendants, after a day of wild asexual abandon, will collectively weigh as much as a mountain; in a little over a day and a half as much as the Earth; in two days more than the Sun. . . . And before very long, everything in the Universe will be made of bacteria. This is not a very happy prospect, and fortunately it never happens. Why not? Because exponential growth of this sort always bumps into some natural obstacle." (my bolding)

Doubling period
  15 15 15 15 15 15 15 15 15 15
Pops 1 2 4 8 16 32 64 128 256 512 1024
A-pops 1 2 4 8 16 32 64 128 256 512 1024
B-pops 1 2 4 8 16 32 64 128 256 512 1024
C-pops 1 2 4 8 16 32 64 128 256 512 1024
D-pops 1 2 4 8 16 32 64 128 256 512 1024
E-pops 1 2 4 8 16 32 64 128 256 512 1024
F-pops 1 2 4 8 16 32 64 128 256 512 1024
G-pops 1 2 4 8 16 32 64 128 256 512 1024
H-pops 1 2 4 8 16 32 64 128 256 512 1024
I-pops 1 2 4 8 16 32 64 128 256 512 1024
J-pops 1 2 4 8 16 32 64 128 256 512 1024
K-pops 1 2 4 8 16 32 64 128 256 512 1024
L-pops 1 2 4 8 16 32 64 128 256 512 1024
M-pops 1 2 4 8 16 32 64 128 256 512 1024
N-pops 1 2 4 8 16 32 64 128 256 512 1024
O-pops 1 2 4 8 16 32 64 128 256 512 1024
P-pops 1 2 4 8 16 32 64 128 256 512 1024
Q-pops 1 2 4 8 16 32 64 128 256 512 1024
R-pops 1 2 4 8 16 32 64 128 256 512 1024
S-pops 1 2 4 8 16 32 64 128 256 512 1024

Table D. New Malthusian Scale Sagan's bacterial population doubling example. A row is 2.5 hours.
At the end of the 1st day there are 64 I-pops (equal to the mass of a mountain),
at the end of a day and a half there are 16 N-pops (almost the mass of the Earth)
and at the end of the 2nd day there are 4 S-pops (equal to the mass of our Sun).

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops, 1 C-pop = 1024 B-pops etc.

Whilst bacteria may be one of the most obvious examples of exponential growth there is no fundamental difference between the population doubling of the human species (or any other species) and bacteria. The same law of nature applies, and the same growth model based on variable rate compound interest. Not even bacteria can sustain a constant rate of exponential growth, as Sagan points out. In fact, nothing grows indefinitely at a constant rate. Brief periods of constant rate growth (the "exponential phase") are follow by periods of growth at other rates. That is the same as stating that the population grows via variable rates.

Some even claim that "life is bacteria", though I disagree. See my article Bacterial Replicators - An Exponentialist View for more.


Although Sagan doesn't explicitly consider population doubling and viral replication in the one context, when it comes to AIDS he does examine the annual doubling of AIDS cases (Sagan, 1997, p.17):

"Right now, in many countries the number of people with AIDS symptoms is growing exponentially. The doubling time is around a year. That is, every year there are twice as many AIDS cases as there were in the previous year."

Sagan projects this explosive exponential increase of AIDS victims, explaining that after one decade there would 1,000 fold increase (actually 210 = 1,024) and after the second decade there'd be 1,000,000 fold increase (actually 220 = 1,048,576). It's interesting how we humans still prefer to use the decimal system, even the nature of the increase is itself binary (base 2). A binary perspective though is useful when it comes to viruses, especially given that when they replicate they produce thousands more viruses and not just two "daughters" as bacteria do.

Sagan projects this explosive spread of AIDS saying that if it continued it would wipe out everyone on Earth very soon. However he also predicts that this won't happen, in no small part because a minority of people are naturally immune. Hence he expects the exponential curve to naturally flatten out, though he admits that this is little comfort to those that suffer because of AIDS (Sagan, 1997, p.18).  

The exponential nature of the spread of AIDS is nonetheless revealing as it suggests that viruses follow the same laws of Nature as all living creatures on Earth. This is controversial, however, as not everyone regards viruses as living at all. I do consider them to be living (even though they're good at playing dead), in part because they obey the same laws of population, as well as the same biological basis, as other living creatures. Thinking of them as living helps us in our understanding of them, and helps us respect the exponential power of viral replication. See Viral Replicators - An Exponentialist View for more.

Nuclear Fission

Sagan explains the origins of our understanding of nuclear fission, tracing it back to Hungarian physicist Leo Szilard working in London in 1933. Szilard's epiphany came whilst sitting in his car in at traffic lights when (Sagan, 1997, p.21):

"...it dawned on him that there might be some substance, some chemical element, which spat out two neutrons when it was hit by one. Each of those neutrons could eject more neutrons, and there suddenly appeared in Szilard's mind the vision of a nuclear chain reaction, with exponentiating numbers of neutrons produced and atoms falling to pieces left and right."

That substance was uranium. Szilard appreciated that nuclear fission could be used as a source of energy as well as a destructive force, and it was Szilard  who persuaded Einstein to write to Roosevelt which lead to the United States of America building the first atomic bomb. Szilard worked on the first uranium chain reaction in 1942. Szilard then spent the rest of his life warning against the dangers of nuclear weapons. For better and for worse, as Sagan puts it (Sagan, 1997, p.21):

"He had found, in yet another way, the awesome power of the exponential."

Radioactive Half-Life

The Sagan quote that heads this article comes from Sagan's examination of how exponentials apply to the concept of a radioactive half-life. This is where an unstable radioactive element gradually decays until it effectively becomes another element which itself has its own half-life and so on (Sagan, 1997, p.22):

"It represents an exponential decay, in the same way that the Persian chessboard represents an exponential increase."

This is the basis of carbon-dating, for example, and the dating of the past. The sorts of secrets revealed by this method are that the Shroud of Turn is a 14th century hoax, that humans made campfires millions of years ago, that life started at least 3.5 billion years ago according to the fossil record, that the age of the Earth is 4.6 billion years old, and even the age of the universe (Sagan, 1997, pp.22-23).

Sagan ends the Persian Chessboard chapter by urging the reader not to be afraid of quantification, to see hidden beauty and power and gain understanding (Sagan, 1997, p.23):

"Being afraid of quantification is tantamount to disenfranchising yourself, giving up on one of the most potent prospects for understanding and changing the world."

Hopefully my Exponentialist web site will help illuminate Malthus' Principle of Population and thus assist humanity in the quantification of matters pertaining to exponentials.


In tackling yet another controversial topic  - this time abortion - in his usual considered and reasonable manner Sagan describes cellular replication and how a single fertilised cell becomes a human being. I do not intend to discuss the pro-life and pro-choice debate here, but rather Sagan's brief examination of early cellular replication in humans which follows the familiar pattern of 1 becomes 2, 2 becomes 4 and so on...what Sagan calls "an exponentiation of base-2 arithmetic (Sagan, 1997, pp.176-177)."

The landmarks that Sagan notes are (Sagan, 1997, pp.176-177):

It takes roughly 45 to 46 cell population doublings for an adult human (at 18 years old) of about 50,000 billion cells to be formed. For a more detailed look at cell replication see Cellular Replicators - An Exponentialist View.

Another replicator related to all life on is the gene. Sagan has nothing to say on the exponential power of genetic replication. However, a good place to start with Richard Dawkins - see Dawkins - An Exponentialist View for more.

Death By Replication

Sagan sadly passed away from Myelodplasia  - a bone marrow disease that impairs blood cell reproduction due to abnormal reproduction of bone marrow stem cells - with pneumonia complications on 20th December 1996 aged 62. In the chapter entitled In the Valley of the Shadow Sagan calmly discusses his own death. At one point he mentions the bad news his physician has for him (Sagan, 1997, p.224):

"My bone marrow had revealed he presence of a new population of dangerous, rapidly reproducing cells."

Cells do reproduce rapidly - either when they're prokaryote cells (bacteria) or eukaryote cells (animals and plants). Sometimes cells reproduce abnormally, as in the case of cancer cells or defective bone marrow stem cells. Viruses, a cause of much human suffering, can replicate even faster. See Death By Replication for more.


Sagan has certainly proven his point that the study of exponentials is the key to understanding many of the secrets of the universe.

Exponential power is the force behind fitness - populations with highest growth rate - in the survival of the fittest, and it also has the power to compel those populations to face inevitable limits to growth. What is clear is that reproduction (and replication) is the key exponential power behind life and death on Earth. This exponential power operates at the level of the gene, the virus, the cell, and for all populations of all species all of the time. This is perhaps one of the greatest secrets of the universe revealed by the study of exponentials. Hence being about to quantify and understand the nature of exponential growth is of vital importance to understanding the past, the present and the future of all living creatures.

As Sagan cautions, never underestimate an exponential.


Malthus, Thomas Robert, An Essay on the Principle of Population. J. Johnson. 1798. (1st edition) Library of Economics and Liberty.  

Sagan, Carl. Cosmos. Random House.1980

Sagan, Carl. Pale Blue Dot: A Vision of the Human Future in Space. Random House.1994

Sagan, Carl. The Demon Haunted World: Science As A Candle In The Dark. Headline Publishing.1996

Sagan, Carl. Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium. Headline Publishing. 1997

Zubrin, Robert. Merchants of Despair Radical Environmentalists, Criminal Pseudo-Scientists, and the Fatal Cult of Antihumanism. New Atlantis. 2012.

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Copyright 2012 David A. Coutts
Last modified: 08 August, 2012