Wednesday, August 25, 2010

Trajectories of the Future

(To incorporate for next time:

An Example of Extrapolation:
Overpopulation on Mars

Suppose we started a
self-sustaining colony

100 colonists

What is your estimated
rate of increase per year?

What is the total population
capacity of Mars?

     Earth surface area = 510,072,000 km^2
     Earth land area     = 148,940,000 km^2
     Mars surface area = 144,798,500 km^2

                                     (0.284 of Earth)

How long do you think it will take
for Mars to overpopulate?

We can check this using a spreadsheet!

Just have the rows represent successive years
Each year has x% more people than the previous
See how many years go by until overpopulation!

Making and Discuss Predictions with Trajectories

           Method: Trajectories of change

. . . in the short term,
      change appears linear


Last year you had 1 or 2 compact fluorescent bulbs

This year you will "probably have 1-2 more"

In the longer term,
change may looks

Lightbulb example:

. . . you start with 1-2, but after a couple of years you've got a bunch

. . . change accelerates, in this case

. . . if you look at an exponential curve with a microscope, what does it look like?

. . . "Exponential": complicated word, tricky math, simple concept

. . . . . . goes up faster and faster

. . . . . . has a doubling time

Exponential curves explained

. . . Suppose something doubles every 3 years

. . . Popular example: computer CPU complexity doubles every 2 years

. . . new value after t years is original value v times 2^(t/3)

. . . f(t)=to * 2^(t/3)

. . . . . . where does the "doubles" appear?

. . . . . . where does the "every 3 years appear?

. . . . . . so it works for
            any factor of increase and
            any time constant

Longer term, things "level off": the S-curve

Also called "logistic curve"

Sort of "linear" early on

Then looks "exponential"

Then levels off

Justified by many, many diverse phenomena modelable as:
       Malthusian scenarios
       Constructal Theory scenarios
           A. Bejan and S. Lorente, The constructal law origin of the logistics S curve, Journal of Applied Physics, vol. 110 (2011), 024901,

Do you think an even longer-term view will look like a plateau curve?

Think about pencils, compact fluorescents, and college, etc., etc.

What do you think of these curves?
(Source:, 9/5/10)

There are other view of trajectories...

Gartner Hype Cycle


How might this apply to some of our topics?

There is also the Technology Adoption Life Cycle

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