Energy and Human Ambitions on a Finite Planet, 2021a

1 Exponential Growth 4
The population doubles every ten minutes. If the original jar is filled
at 12:00, the population doubles to fill the second jar by 12:10. Another
doubling fills all four by 12:20. The celebration is short-lived.
Now we draw the inevitable parallels. A planet that has served us for
countless generations, <strong>and</strong> has seemed effectively infinite—imponderably
large—makes it difficult for us to conceive of hitting limits. Are we
half-full now? One-fourth? One-eighth? All three options are scary, to
different degrees. At a 2% rate of growth (in resource use), the doubling
time is 35 years, <strong>and</strong> we only have about a century, even if at 1/8 full
right now. 3
In relation to the bacteria parable, we’ve already done a fair bit of
exploring. We have no more jars. One planet rhymes with jars, but it is
hostile to human life, has no food, <strong>and</strong> is not within easy reach. We have
no meaningful outlet. 4 And even if we ignore the practical hardships,
how much time would a second planet buy us anyway for uninterrupted
growth? Another 35 years?
3: If we’re at 1/8 right now <strong>and</strong> double
every 35 years, we will be at 1/4 in 35 years,
1/2 in 70 years, <strong>and</strong> full in 105 years.
4: Chapter 4 addresses space realities.
1.1.1 Exponential Math
Box 1.1: Advice on Reading Math
This section is among the most mathematically sophisticated in
the book. Don’t let it intimidate you: just calmly take it in. Realize
that exponential growth obeys an unchanging set of rules, <strong>and</strong>
can be covered in just a few pages. Your brain can absorb it all
if you give it a chance. Read paragraphs multiple times <strong>and</strong> find
that each pass can add to your comprehension. Equations are just
shorth<strong>and</strong> sentences 5 capturing the essence of the concepts being
covered, so rather than reading them as algorithms to file <strong>and</strong> use
later when solving problems, work to comprehend the meaning
behind each one <strong>and</strong> its reason for being a part of the development.
In this way, what follows is not a disorganized jumble, recklessly
bouncing between math <strong>and</strong> words, but one continuous development
of thought expressed in two languages at once. The Preface offers
additional thoughts related to this theme, <strong>and</strong> Appendix A provides
a math refresher.
Experts habitually read complicated passages
multiple times before the material
sinks in. Maybe it’s this calm habit that
turns them into experts!
5: Unlike words/language, the symbols
chosen for equations are just labels <strong>and</strong>
carry no intrinsic meaning—so electing to
use x, n, t, b, M, etc. reflect arbitrary choices
<strong>and</strong> can be substituted at will, if done consistently.
The content is in the structure of
the equation/sentence.
The essential feature of exponential growth is that the scale goes as the
power of some base (just some number) raised to the time interval. In the
doubling sequence, we start at 1× the original scale, then go to 2×, then
4×, then 8×, etc. At each time interval, we multiply by 2 (the base). After
5 such intervals, for instance, we have 2 × 2 × 2 × 2 × 2,or2 5 32. More
generally, after n doubling times, we have increased by a factor of 2 n ,
where 2 is the base, <strong>and</strong> n is the number of doubling times. We might
formalize this as
M 2 n 2 t/t 2
, (1.1)
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.