Trajectories of the Future

(Updated 9/12/12)

Related reference:

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

Let’s apply the following to:

        a few ethics cases…

We will see how…

        Trajectories of change

In the short term, change appears linear

(what do the axes represent?)


      Last year you had 1 or 2 compact fluorescent bulbs

      This year you will “probably have 1-2 more”

In the longer term, change often looks exponential

Lightbulb example:

       you start with 1-2,

       but after a couple of years

       you’ve got a whole bunch

Change accelerates, in this case

      If you look at an exponential curve

      with a microscope,

      what does it look like?


       Complicated word

       Tricky math

       Simple concept

             Goes up faster and faster

             Has a doubling time

(Could also slow, have halving time)

Exponential curves explained

       Suppose something doubles every 3 years

       Popular example:

            Computer chip complexity doubles every 2 years

            New value after t years is starting value T times 2^(t/3)


                           Why does this double every 3 years?

         It can work for

               Any factor of increase

               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 shaped curve?

Think about quill pens, ball point pens, incandescent bulbs, compact fluorescents, etc., etc.

What do you think of these curves?

(Source:, 9/5/10)

How might this apply to technology-related ethics problems?


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