by Matthew Cobb

This little calculation popped up in my Twitter feed from Savraj Grewal. TL;DR: a single E. coli cell, if left alone and with unlimited food/space, would in three days produce a sphere of bacteria the size of the solar system, expanding at faster than the speed of light…

https://twitter.com/SavrajGrewal/status/615381481335058432/photo/1

The calculation is by Pat O’Farrell and is taken from a recent article.

I remember doing a similar calculation, but with houseflies. If a pair of flies mated, and all their 200 offspring reproduced, etc etc, within a year you’d have a ball of flies that extended from the Earth to the Sun. Can’t find my working, but I’m sure it must be about right.

Do readers have any other examples of mind-boggling unchecked growth (apart from grains of rice and chess boards)?

Evangelicals

That’s what I was going to say. If evangelicals bread unfettered for 50 years the average IQ of the world would approach zero.

Is “bread unfettered” like matzoh?

Of course, according to evangelicals, humans are immune from this, and unchecked growth will never be a problem because resources exist in an infinite supply.

And technology is magic, and never has any downsides or unecpected consequences.

Using the maths we should be able to calculate the longest period that could have elapsed before Adam sinned and death entered into the world. Certainly, less than 2 days.

On a related note, I watched online a creationist talk where the person claimed the earth was very young because the current human population is X, the average reproductive rate is Y, and therefore by backwards extrapolation the time when there were two people on the planet was time Z. And the audience was all like: ‘oh gosh! This is amazing proof of a young earth!’

I once read a letter to the editor from a creationist. The human race couldn’t have been reproducing for millions of years, because if it did, the planet would be covered with people who had no room to lie down.

We’re at 7 billion people now and we’ve been doubling about once every 65 years over the past 2 centuries. We only need 32 doubles to get from 2 to 7 billion, therefore the Earth is no older than 2080 years (65 x 32) and Julius Caesar was the first man. Therefore, the Old Testament is pure fantasy. Checkmate, Creationists!

It is funny that things like the speed of light and radioactive decay are variable but things like population growth are constant…

Mine is not about unchecked growth but probably qualifies.

In Sam Kean’s book The Violinist’s Thumb he writes that if the DNA in an individual’s body was unwrapped it would stretch from Pluto to the Sun and back again.

More staggeringly, if all the DNA in the entire biosphere was unwrapped it would stretch across the known universe and back again a number of times.

What we would do to occupy our time while all this DNA stretching was going on he doesn’t say.

If you fold a thousandth-of-an-inch thick piece of paper over on itself fifty times, the resulting pile would be over three million miles thick. I think I read that in a Martin Gardener book years ago, but I did the math and it’s true.

But I bet you can’t do it even 10 times.

Dark Matters’s related cartoon take on why the fundamentalist Christian God had to cause The Fall/sin etc., “If Man Obeyed God”: https://www.youtube.com/watch?v=A_a6RjR_AHY

Of course, this calculation shows even Dark Matters’s cartoon greatly underestimates their problem!

Humans.

https://en.wikipedia.org/wiki/World_population

Two relevant charts. First, linear scale:

https://en.wikipedia.org/wiki/World_population#/media/File:Population_curve.svg

Looks frightening enough, as all exponential growth curves do…so let’s look at a logarithmic plot, shall we? Should give us close to a straight line, and the slope would tell us the annual percentage increase.

https://en.wikipedia.org/wiki/World_population#/media/File:World_population_growth_(lin-log_scale).png

Oh, shit.

When your

log scalegraph looks like an hockey stick, youknowyou’re well and truly fucked.Another scary example is just the “boring” roughly 3.5% historical annual growth rate you find pretty much everywhere…inflation, oil production, cost of living, population…nobody thinks anything of anything about a piddling 3.5% growth.

But, if you remember your Rule of 70…well, 70 / 3.5 = 20, or a doubling every generation. If your city’s population is growing at 3.5%, it’ll be twice as big when you go to college as when you were born, four times as big when you’re 40, eight times as big when you’re 60, sixteen times as big when you’re 80, and 32 times as big when you’re 100. You could be born in a sprawling metropolis of a “mere” million people, and die in a dystopian 32-million-people rat maze.

A final scary thought: global petroleum reserves are roughly equal to the total extracted since the start of the industrial revolution.

And if that fact doesn’t shake you to your very core…you flunked and need to go back to exponential math school.

(Imagine a bacterial colony in a flask that doubles once per minute. You start the experiment sometime in the morning with a single bacterium. At noon, the flask is full. At what time was the flask half full? How long will it take to fill a second flask? A third and fourth flask?)

Cheers,

b&

We discussed that mistake the other week (but then it was Jerry who made it), the population is not exponential. It was superexponential to the 60s (the hockey stick), thereafter it quickly went subexponential.

Importantly the reason for the subexponential growth was not resource constraints, so generic malthusian models has failed in tests. The reason was better education and less poverty, due to better economy, due to … better economical growth.

[Urbanization is another phenomena, but we need not discuss it specifically here. Urbanites use the same or less resources than non-urbanites. Except when they are producing in which case they are an order of magnitude more productive. Hence the phenomena helps, not hinder, decreasing population growth.]

Oy: “the population is not exponential” – the population growth is not exponential.

If you’re just looking at the past half-century, you’re falling into the same logic trap as the climate change deniers do when they pick a certain starting point eighteen years ago or whenever it was that corresponded with a local maxima…in other words, you’re focussing on noise.

Look again at that chart — the exponential growth chart that goes back to 12,000 BP.

For the first six thousand years of the chart, the human population was stable, with a total population ranging between the current populations of Phoenix and Toronto. It then started a very steady exponential growth, hitting about the population of Jakarta during the Egyptian dynastic period. At the time of the Caesars, global population about matched modern Bangladesh. That same growth rate continued through the Dark Ages, at which point we were up to about Bangladesh plus Paris, and another half a millennium or so to the Age of Exploration, when we were at about today’s population of Brazil.

And then it positively exploded.

For the past half a millennium, we’ve been creating more humans at a positively terrifying rate — again, to the point that I now live in a positively demographically unremarkable city with as many inhabitants as there were humans on the entire planet all the way up through the time the Pyramids were built.

That we’re currently in the midst of a minor regional generational blip is of no consequence. Even on the log graph, you can see far bigger dips and spikes on the order of centuries.

Realistically, I don’t see a sustainable human population over 100 million, and we’d probably be just fine — more than fine — with under 10 million.

Do we

reallyneed to live in a world with a thousand people for every one who lived at the height of the Roman Empire?b&

> Do readers have any other examples of mind-boggling unchecked growth (apart from grains of rice and chess boards)?

I’ve read this in a fantasy novel: A very, very old and powerful magician comes, after long absence (like, a thousand years) back to his hometown, where people are not happy about this, thinking they were doing just fine without him. So they present him a bill for the unpaid land tax on his tower, including interest and compound interest, assuming he can never pay … except that he can, which is where they’re backpedaling really quickly, as the sum amounts to burying the town under sixty metres of gold.

Many years ago I read a science fiction story with a similar theme, but unfortunately I remember neither the title nor the author. (Perhaps someone has a better memory than me.)

It starts with an eccentrically dressed man — call him Mr. Smith — walking into a medieval bank with a single gold coin which he deposits into an interest-bearing account. Every century or so, a man representing himself as the heir of Mr. Smith comes in to check on the balance in his account, which of course grows exponentially.

As the centuries go by the bankers naturally become concerned about how much of their wealth is dependent on this one client, and they resolve to learn as much as they can about this mysterious and reclusive family, hiring sketch artists, and later photographers, to make images of the successive Smith heirs.

By the early 21st century, the entire world economy depends, in one way or another, on the Smith estate. And the bankers have noticed several odd things. All of the Smiths look remarkably alike, as if they could be the same man. And that man’s style of dress no longer seems eccentric; it’s perfectly normal 21st-century garb. Finally, the original gold coin, which they still have, turns out to be a US gold eagle.

Could Mr. Smith be a time traveler? Through intermediaries, they discreetly approach a theoretical physicist to ask if such a thing is possible. The answer comes back: “Probably not, but let me think on it and get back to you.”

A few weeks later, the physicist walks into the bank, where he is immediately recognized as Mr. Smith. “I’ve changed my mind. Time travel is possible, but building the machine will cost trillions of dollars. Please start liquidating my accounts so I can begin construction.”

That does sound like a good story! My resort when failing to remember a story title is asking a literary studies person. No one but a human professional when it comes to fuzzy logic.

What if you started with

two cells? After that same time interval, how big would the sphere be? How fast would it be expanding?The difference will be the same as if you had started the experiment 20 minutes earlier.

b&

“We’re going to need a bigger orbital shaker.” -Roy Scheider

BTW, my copy of “Life’s Greatest Secret:…” is hopefully arriving today, can’t wait!

An old Power Point talk that Scott Gilbert posted online many years ago had this information on a slide:

Aphis fabae(aphid): 524,000,000,000 per year.Musca domestica(fly): 2 X 10^20 in five months.Elephas(elephant): 19,000,000 in 750 years.Staphylococcus aureus> Enough to cover the earth 7 feet deep in 48 hours.The slide came with this picture: Grim Reaper (hope it works).

Can anyone guess what movie and what scene it was from? Hint: it was, of course, a British comedy.

Awesome numbers! Even surprised by the elephants. That scene was from Monty Python’s The Meaning of Life, when Death comes to the farmhouse. Avoid the salmon mousse! (Or salmon acting like moose! Or moose acting like salmon!)

You can never be too careful around a mÃ²Ã³se. Give it even an hint of a chance, and it’ll bite your Å¡Ä¯ÅtÃªr. Pretending to be a ÅÃ¥ÅmÃ¸Ã± is just one of the many rÃ¼ÅÄÅ¡ it uses to sneak up on its victims. Your best bet? Learn how not to be seen.

b&

Correct, Anonymous, or whoever you are!

Slightly different take on mind-boggling numbers, but one that I still struggle to get my head round, even though the maths is clear-cut.

Assume the earth a perfect smooth sphere with circumference of 40,000km, with a loop of rope snug around the equator.

Break the loop, and add just 2 extra meters of rope (i.e. extend the 40,000,000 meter length by 2 meters) then wrap the loop round the equator again with an even gap.

That gap will be approx. 32cm the whole way round!

If you do the same with a loop of string round a tennis ball, you get the same result i.e. start with a loop the length of the circumference of the ball (approx. 25cm) add 200cm, make a circle of the string with ball concentrically in the middle, the gap between ball and string will be an even 32cm.

The maths shows the increase in the radius of a circle is directly proportionate to the increase in the circumference irrespective of the size of the starting circumference:

Assume starting circumference is C, starting radius is R

C = 2 x Pi x R

Extend C by 200cm, D is the increase in radius

C + 200 = 2 x Pi x (R + D)

C + 200 = (2 x Pi x R) + (2 x Pi x D)

C = (2 x Pi x R) + (2 x Pi x D) – 200

Substitute C = 2 x Pi x R from first equation

2 x Pi x R = (2 x Pi x R) + (2 x Pi x D) – 200

2 x Pi x D = 200

D = 100 / Pi

D = 32cm (approx.)

fater than the speed of light. I did not know that was possible???

Check out Sean Carroll’s books (or his website). It seams that space-time can expand faster than the speed of light, even though the particles within space-time cannot.

BTW, that’s my answer: space-time.

The difference is you can’t transmit information via expanding space. With this expanding glob of e coli, you conceivably could.

That is the point. (If not the bacterial mass being larger than Earth mass – the potential nutrient supply – is a tip off.)

It isn’t possible. All that shows is that it’s possible to calculate scenarios that cannot physically exist.

Because most of the classrooms in which I teach overlook a small patch of woods in which gray squirrels are common, and because students, even if no squirrels are visible out the window at the moment, are well familiar with them, I often use squirrels as an example of various biological phenomena. In discussing population growth, I often remark how without checks to their growth, we would soon be “hip deep in squirrels” (although I’ve never calculated exactly how soon that would be). This example puts the squirrels to shame.

That would endanger a lot of nuts.

If you wait but a few femtoseconds a water molecule will not remember the state it was in, i.e., position and momentum. Likewise, there is no known technology that can be conceived of that uses less than all of the available bits in the universe to comput the state of single molecules of water in a big jug of water, e.g., like humans. This is a naturalistic argument against the soul, as it depends on warm molecules in our head being reproduced and maintained forever as physical states of dark energy or whatever.

…unless you are an idiot who believes in homeopathy!

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My favorite example would be science. The growth rate of papers is exponential.

According to certain Christians, the Earth’s population was 8 less than 6,000 years ago.

How many kids must every woman have had since Noah’s time, in order to end up with 7 billion people today (assuming all children live to adulthood and reproduce)?

Addy Pross makes a similar calculation in

What Is Life: How Chemistry Becomes Biology. Mathematically, an auto-catalytic molecule that replicates itself every minute, will produce enough copies to exceed the mass of the universe in five hours.We are familiar with the creationist argument, “if it’s billions-to-one of happening once, what are the odds it will happen again.” RNA is auto-catalytic, which means it catalyzes itself. That makes the odds of a second RNA pretty close to one. And, the copies are subject to random errors: evolution at the molecular level!

Here’s some astounding unchecked growth, but in the other extreme.

Even as a mathematician, logarithmic growth still amazes me. It’s easy to see why a function like f(x) = log(x) must grow without bound precisely because it is the inverse function of the exponential function f^{-1}(x) = e^x, which is well defined for any real number x, no matter how large. With the help of examples like the ones above, we can get our heads around this type of exponential growth.

But the inverse growth? It’s harder to think of illustrative examples that help inform the intuition, where growth gets harder and harder the bigger things get, *but the quantity still grows without bound*. I think that’s because any example of logarithmic growth necessarily requires either an infinite population to start with (something unintuitive), or will eventually require thinking in fractions (also not very intuitive when those fractions get really small).

To me, the most illustrative example is still an abstract one: considering how the sum of the reciprocals of the positive integers adds up to infinity:

1 + 1/2 + 1/3 + 1/4 + 1/5 + … = infinity.

Why? Group the terms in a clever way and notice their sizes:

1/3 + 1/4 > 1/2,

1/5 + 1/6 + 1/7 + 1/8 > 1/2,

1/9 + … + 1/16 > 1/2,

etc.

These groups of summands are cooked up to always add up to more than 1/2, but it takes more summands each time to make the trick work (exponentially more): first two summands, then four, then eight, etc.

So

1 + 1/2 + 1/3 + 1/4 + 1/5 + …

> 1 + 1/2 + 1/2 + 1/2 + 1/2 + …

> 1 + 1 + 1 + 1 + …

= infinity.

It still amazes me.

Another way of looking at it: 1/2 + 1/4 + 1/8 + 1/16 … sums to 1, and in a very intuitively obvious visual way with a Golden Spiral. Therefore, any sum of fractions whose inter-relative sizes is bigger will have unbounded growth. Anything less will leave an hole that will never get filled.

b&

Hmmm, I don’t understand. What do you mean by “any sum of fractions whose inter-relative sizes is bigger”?

Perhaps Ben’s referring to the ratio test for convergent series. If the ratio of consecutive terms approaches a limit less than one, the series converges (has a finite sum).

In the sequence I gave (1/2 + 1/4 + 1/8 + …), each successive number is half the previous. In such cases, the final sum approaches twice the initial value (2 * 1/2 = 1, in this particular case).

If the next number in the series is more than half the previous one, the series will grow without limit.

If the next number is less than half the previous, the series will converge on some number less than twice the initial value.

…and, if some mathematician cares to correct me on that, please do so. I tend to do not bad at these sorts of things, but I’m also known for making mistrakes….

Cheers,

b&

As I indicated in my previous comment, the cutoff for convergence is r = 1, not r = 1/2. See the linked Wikipedia article.

Okay, I see what you were saying now. This is not quite correct though. Many series converge that have successive terms that are larger than half the previous term. For example, try summing up powers of 2/3:

2/3 + 4/9 + 8/27 + 16/81 + …

This series converges to the number 2. In general, any series of the form

\sum_{k=1}^{\infty} r^k

converges to a finite number if |r| < 1 (this is called a geometric series – "geometric" is synonymous with "exponential" in this context, more exponential growth/decay!). If you look at powers of 99/100, this series sums to 99. If you look at powers of 9999/10000, this series sums to 9999.

Your second statement is correct though: if you can bound each term of another sequence by the one you gave, then the sequence must sum to something less than what your series sums to. It's that the reverse logic doesn't hold.

I just had to spend some time fiddling around in Adobe Illustrator to convince myself…and, yes, of course.

Some rather interesting patterns when you stack squares or rectangles or other shapes in such a series. Well worth the experimentation.

Thanks!

b&

No problem! That’s a cool way to look at it – i.e. shape stacking. I’ll have to remember that technique. It does provide a really nice visual.

I think mathematics has unwarrantedly forsaken its geometrical heritage…take, for example, Pythagoras. We’re all taught that a^2 + b^2 = c^2, and that’s true and very important and very useful…but it’s only distantly related to what Pythagoras himself observed: that the sum of the areas of squares drawn on the adjacent sides of a right triangle have the same area as a square drawn on the hypotenuse. And, when you phrase it like that, it at least becomes obvious how you could go about proving (or falsifying!) it for yourself, and, once you’ve done so, it’s especially obvious why that relationship should be true. Really, it’s little more than stating some boring truths about diagonals and inscribed rectangles.

Embracing such elementary geometry not only in education but in professional circles would do lots of people lots of good, I strongly suspect.

Cheers,

b&

You can go the opposite way with the number of your possible ancestors. Two generations ago you have 2^2 direct relatives (your four grandparents). Three generations, 2^3 direct ancestors. 100 generations ago you have 2^100 direct ancestors.

The only caveat, I think, is that this represents an upper bound of direct ancestors.

100 generations back, there could be a few people that fill all the spots.

Actually one pair of ancestors could be all of your 100 generations back ancestors.

If all the girls in new York were laid end to end, erm…

I wouldn’t be surprised.

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When I was at school I did a similar calculation on the number of offspring that a female greenfly might have in a year… I cannot remember the details now though it should be an easy one to work out…

Everything alive will die someday, and that’s OK.

Anyone else having their comments eaten?

There’s the Busy Beaver Problem, but given we’ve only figured out values for the first five numbers of the set, I’ll go with something a little more quotidian.

Another old problem in computer science is the need to convert binary numbers to decimal Strings (for human consumption) and then convert those decimal strings back to binary (for computer consumption). The most common representation for decimal numbers are single precision floats (which are 4 bytes or 32 bits) and double precision floats (which are 8 bytes or 64 bits). In order to be sure your implementation that converts binary to decimal back to binary works, it can’t hurt to test all the values. For the single precision float, let’s say you have a processor and algorithm that can do one million conversions per second. Since there are 2^32 possible values, it’ll take your PC just under an hour and a quarter (2^32/10^6 seconds) to test everything.

Not so bad, all things considered.

Now, let’s merely double the storage size of the number to double precision (8 bytes). Your robustness test will now take 2^64/10^6 seconds to test every value. That’s in the ballpark of 584,000 years assuming you can keep your computer up and running and powered for that long. So, we have an input size that doubles correlating to a test time that increases by a factor of over four billion. It’s a pretty trivial example but underscores the problems engineers face when robustly testing a system (the brute force “test it all” solution is simply infeasible).

I remember reading the following excerpt in “The Challenge of Large Numbers” (Richard Crandall, Scientific American, Feb 1997, pp. 74-78). It still blows my mind.

“To show the existence of certain difficult-to-compute functions, mathematicians have invoked the Ackermann numbers (named after Wilhelm Ackermann of the Gymnasien in Luedenscheid, Germany), which compose a rapidly growing sequence that runs: 0, 1, 2^2, 3^3^^3^^^3…. The fourth Ackermann number, involving exponentiated 3’s, is approximately 10^3,638,334,640,024. The fifth one is so large that it could not be written on a universe-size sheet of paper, even using exponential notation! Compared with the fifth Ackermann number the mighty googolplex is but a spit in the proverbial bucket.)

Bees. Don’t mess with their natural habitat or they’ll come after your house!

http://www.cbc.ca/news/canada/kitchener-waterloo/hive-of-50-000-honeybees-removed-from-cambridge-ont-home-1.3132147

http://www.ctvnews.ca/canada/colony-of-50-000-bees-removed-from-ontario-home-1.2446530

http://www.stcatharinesstandard.ca/2012/07/29/ontario-house-a-honey-pit

As most of us may already know, the hexagonal shape of the honeycomb is optimal for making as many little ‘cribs’ for bee larvae. They sure can pack them in like sardines!