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Malthusian Memes - An Exponentialist View
Viral Replicators - An Exponentialist View
Bacterial Replicators - An Exponentialist View
Cellular Replicators - An Exponentialist View
Rabbit Replicators - An Exponentialist View
Human Replicators - An Exponentialist View
Grey Goo - An Exponentialist View
Death By Replication
Exponential Assembly - An Exponentialist View

Population Doubling Mechanism
New Malthusian Scale
K. Eric Drexler - An Exponentialist View
Human Global Ecophagy

External Links:
Engines Of Creation - K. Eric Drexler

The Gray Goo Problem by Robert Freitas, on KurzweilAI.net

Exponential Assembly - Zyvex

Natural Selection and Differential Reproduction - from Replicators: Evolutionary Powerhouses

Grey Goo - An Exponentialist View

Introduction

Molecular nanotechnology, as described by Eric Drexler in Engines Of Creation, will become capable of building self-replicating entities known as assemblers. Such assemblers would naturally be capable of exponential growth. In  Chapter 4 "Engines Of Abundance" Drexler explores a simple, but scary, scenario of exponential growth with such assemblers (Drexler, 1990):

"Thus the first replicator assembles a copy in one thousand seconds, the two replicators then build two more in the next thousand seconds, the four build another four, and the eight build another eight. At the end of ten hours, there are not thirty-six new replicators, but over 68 billion. In less than a day, they would weigh a ton; in less than two days, they would outweigh the Earth; in another four hours, they would exceed the mass of the Sun and all the planets combined - if the bottle of chemicals hadn't run dry long before."

Grey goo (based on such assemblers) is the generic name given to a theoretical, and quite deadly, threat to all life. Drexler describes grey goo succinctly in Chapter 11 ("Engines Of Destruction") (Drexler, 1990):

"...early assembler-based replicators could beat the most advanced modern organisms. "Plants" with "leaves" no more efficient than today's solar cells could out-compete real plants, crowding the biosphere with an inedible foliage. Tough, omnivorous "bacteria" could out-compete real bacteria: they could spread like blowing pollen, replicate swiftly, and reduce the biosphere to dust in a matter of days. Dangerous replicators could easily be too tough, small, and rapidly spreading to stop - at least if we made no preparation. We have trouble enough controlling viruses and fruit flies."

Drexler's explanation of exponential growth starts by using the standard population doubling series, leaves numbers behind altogether with the mention of 68 billion, and then switches to using plain English. Using the population doubling timeframe of 1,000 seconds, let's use the New Malthusian Scale to help model a "grey goo" population scenario using just 11 easy to comprehend numbers:

Time 
(1,000s of seconds)
 0 1 2 3 4 5 6 7 8 9 10
Pops (individuals) 1 2 4 8 16 32 64 128 256 512 1024
Time
(1,000s of seconds)
10 11 12 13 14 15 16 17 18 19 20
Kilopops 1 2 4 8 16 32 64 128 256 512 1024
Time
(1,000s of seconds)
20 21 22 23 24 25 26 27 28 29 30
Megapops 1 2 4 8 16 32 64 128 256 512 1024
Time
(1,000s of seconds)
30 31 32 33 34 35 36 37 38 39 40
Gigapops 1 2 4 8 16 32 64 128 256 512 1024

Table 1. New Malthusian Scale used to measure the number of the early grey goo replicator population.

So, after 36,000 seconds (10 hours) there are 64 Gigapops (68,719,476,736 replicators). Each row on the New Malthusian Scale represents 10 population doublings. For our nanotechnology replicators doubling every 1,000 seconds, each row therefore equates to 10,000 seconds (2.78 hours) growing time. Before the first day was up, we're told our replicator grey goo would weigh a metric ton. Before the end of the next day, our grey goo would outweigh the Earth, and in another 4 hours it would outweigh the solar system.

Using the Alphabet Option For The New Malthusian Scale (constant rate), it is a simple matter to calculate precisely how many population doubles take place by the end of Day One or Day Two:

Day One: 24 hours is 86,400 seconds (24 x 60 x 60). The end of the H-pops row is reached after 80,000 seconds (22 hours, 14 minutes and 24 seconds). Each additional doubling adds 1,000 seconds. Therefore, the end of Day One is reached during the 87th population doubling from 64 to 128 I-pops.

Day Two: 48 hours is 172,800 seconds (48 x 60 x 60). The end of the Q-pops row is reached after 170,000 seconds. Each additional doubling adds 1,000 seconds. Therefore, the end of Day Two is reached during the 173rd population doubling from 4 to 8 R-pops.

That's the easy way to model the population growth of a grey goo growing at a constant rate. Mapping our model to Drexler's explanation, we have to be a little more inventive. 

New Malthusian Scale Using Metric Tons

Drexler's explanation switches from measuring the number of the population to measuring the weight of the population (which is perfectly valid). Starting towards the end of Day One with a grey goo weighing a ton (and sticking with the same population doubling time of 1,000 seconds) the following table shows that Earth's mass is reached during the 76th population doubling after roughly 21 hours (between 75,000 and 76,000 seconds):

A-ton 1 2 4 8 16 32   64 128 256 512 1024
B-ton 1 2 4 8 16 32   64 128 256 512 1024
C-ton 1 2 4 8 16 32   64 128 256 512 1024
D-ton 1 2 4 8 16 32   64 128 256 512 1024
E-ton 1 2 4 8 16 32   64 128 256 512 1024
F-ton 1 2 4 8 16 32   64 128 256 512 1024
G-ton 1 2 4 8 16 32   64 128 256 512 1024
H-ton 1 2 4 8 16 32 50.65 64 128 256 512 1024

Table 2. New Malthusian Scale now measuring the weight of the grey goo, from towards the end of Day One

Note:  1 A-ton = 1024 tons, 1 B-ton = 1024 A-tons, 1 C-ton = 1024 B-tons etc.

New Malthusian Scale Using Earth Masses

Object Mass (Scientific notation) Mass (New Malthusian Scale)
Earth 5.98 x 1021  tons 50.65 H-tons

Table 3. Measuring the mass of the Earth. Note: A metric ton = 1,000 kg. 

To help visualise continued exponential growth at such an enormous scale, and to continue Drexler's example, let's now change the New Malthusian Scale to measure Earth masses. The population doubling time is still 1,000 seconds. The following table shows that the mass of the Sun is reached during the 19th population doubling roughly 5 hours (between 18,000 and 19,000 seconds) after Earth mass is reached:

Earth Mass 1 2 4 8 16 32 64 128 256   512 1024
KiloEarth 1 2 4 8 16 32 64 128 256 325 512 1024

Table 4. From near the end of Day Two, the New Malthusian Scale is used to measure grey goo in Earth masses.

Note:  1 KiloEarth = 1024 Earths, 1 MegaEarth = 1024 KiloEarths etc.

Object Mass (Scientific notation) Mass (In Decimal / English) Mass (New Malthusian Scale)
The Sun 1.99 x 1027  tons 332,776 Earth Masses 325 KiloEarths

Table 5. Measuring the mass of the Sun.

The mass of the rest of the planets (and the Asteroid Belt, Kuiper Belt and Oort Cloud) would be swallowed somewhere in the 19th population doubling from 256 to 512 KiloEarths, along with the Sun.  

Constant Growth Rates versus Variable Growth Rates

This entire extrapolation assumes a constant rate of exponential growth, with regular doubling every 1,000 seconds. It is worth noting that constant growth rates (whether positive or negative) are the exception, and not the norm, for any population of replicators. Replicators include cells, algae, fungi, viruses, bacteria, trees, grass, birds, fish, whales, humans, elephants, beetles, butterflies and grey goo (to name but a few). For a growing population, variable growth rates are the norm in Nature. This means that population doubling times typically get longer, and growth slows. I can see no reason to suppose that things would be different for a grey goo. Hence, it would be more realistic to model the growth of a grey goo population using the Alphabet Option For The New Malthusian Scale (variable rate). If this were done, it could be seen that the grey goo would not consume the Earth or the Solar System quite as quickly as predicted.

Nonetheless, Drexler's grey goo does give a strong example of the power of exponential growth. And don't forget, Drexler's example represents the type of threat possible from just one grey goo! Each grey goo is a potentially deadly replication bomb, and could spread the "seeds" of other grey goos very quickly as described by Drexler.

Replication ("Birth") Rate and Death Rate 

Population doubling times are calculated by measuring the Replication and Death Rates for a given period, and then calculating the percentage growth rate for that period. See my article on the Population Doubling Mechanism for an explanation. 

Any extrapolation of exponential growth which fails to at least mention these basic facts is naive, and should be treated as such. Exponential growth deserves a clearer and more realistic explanation, especially when applied to populations of replicators.  

Active Shield

In Chapter 12 ("Strategies And Survival") of Engines Of Creation, Drexler discusses concepts such as the Active Shield. Think of it like the Earth's immune system, actively seeking out and destroying potential grey goo threats. The Active Shield will increase the death rate for a grey goo population, thus slowing the population doubling time down and hopefully turning a positive growth rate into a negative growth rate (thus destroying the grey goo). 

It would effectively be a Darwinian survival of the fittest between good goo and bad goo. The good goo could be programmed to stop replicating when it has destroyed the bad goo. See Natural Selection and Differential Reproduction - from Replicators: Evolutionary Powerhouses, for a short description of the principles involved.

Limits To Growth

Drexler shows a clear understanding of Malthus in Chapter 10 ("The Limits To Growth") of Engines of Creation All replicator populations face limits to growth, and grey goo is no exception. However, before you start feeling too cosy again (thinking grey goo scenarios can be restricted to Earth), consider Robert Bradbury's Matrioshka Brain page for a Solar System sized brain which could be built using molecular nanotechnology, or my own ideas on Nano Cells, Nano Sails and Heliovores.

Conclusion

Most scientists would probably still prefer to use scientific notation when dealing with large numbers and exponential growth. Then again, most scientists offer the public naive models of exponential growth which assume a constant growth rate rather than variable growth rates. If nothing else, I hope I have provided a useful alternative for the non-scientist and a more useful means of modelling the variable rates of population growth (which in turn lead to variable population doubling times).

If you forget the dreaded grey goo scenario for a moment, and focus on the impact of nanotechnology on the future of the process of evolution for Earth species, you might see that grey goo is not really the threat. For more, read the Nanotechnology and Darwin's Wedge section on Darwin - An Exponentialist View.

For further online reading about grey goo, I recommend Robert Freitas' The Gray Goo Problem on KurzweilAI.net in which Freitas discusses varieties of grey goo such as "gray plankton", "grey dust" and "gray lichens". The definitive work on self-replicating technologies in general is without doubt Kinematic Self-Replicating Machines (Freitas, Merkle, 2004).

For an exploration of a space-faring MNT-based life form - the heliovore - read A Crude Guide To Energy Levels In The Solar System from BNBG. The heliovores, and the Nanosphere, are mentioned towards the end of the article.

For an similar nightmare ecophagy scenario to grey goo but involving a human population see my article Human Global Ecophagy.

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References

Drexler, K. Eric. Engines Of Creation - The Coming Era of Nanotechnology. Oxford University Press. 1990.

Freitas Jr, Robert A., Some Limits to Global Ecophagy by Biovorous Nanoreplicators, with Public Policy Recommendations. The Foresight Institute (website 14th September, 2009). 2000

Freitas Jr, Robert A & Merkle, Ral[ph C. Kinematic Self-Replicating Machines. Landes Bioscience. 2004.


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Copyright 2001 David A. Coutts
Last modified: 10 July, 2012