Monday, January 23, 2012

Trajectories of the Future (& Overpopulation on Mars)

Let's Consider Exponential (and Other) Rates of Change

  • Famous example is "Moore's law"
  • See the graph here


An Example of Extrapolation:
Overpopulation on Mars

Suppose we started a
self-sustaining colony
on the moon
or Mars (e.g. http://mars-one.com)

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!

Let's try it together
For components on a chip, it's around 41%/year
We could play with any %/year



Making and Discuss Predictions with Trajectories

       Method: Trajectories of change

. . . in the short term, change appears linear













Example 1:

  • Last year you used 1-2 e-books and the rest paper


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


Example 2:

  • This year maybe (???!) 1 or 2 of you have cordless chargers


  • Next year 1 or 2 more will?

Example 3:

  • Can you think of any other examples?











In the longer term,
change often looks
exponential




. . . 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

. . . Popular example: Moore's Law

. . . . . . (# transistors on a chip doubles every 1 1/2 to 2 years)

. . . Similar law proposed for digital camera resolution

. . . . . . (includes 7-second video)

. . . Suppose something doubles every 3 years

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

             Let's try that for, say, 3 and 6 years:

                   What should happen? What does?

. . . f(t)=t_o * 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
            What formula gives tripling time?
            What formula gives doubling time of 1.5 years? 10?

The hockey stick fallacy


. . .Exponential curves are
     a kind of "hockey stick curve"
. . . . . .Why?


. . .Why and how shall we pick the key point?

. . .Surprise! The curves are for the same function
. . . . . .x-axis: n
. . . . . .y-axis: 2^n

. . .All I did was

<-- s t r e t c h -->
         and
-->squeeze<--
         it!

. . . . . .My super-fancy graphical editor: MS Paint

. . .An important lesson:

. . . . . .the hockey stick "knee" is not a mathematical property

. . . . . .it is a purely graphical property! 

. . .Let's try another example...


. . .Where is the "knee" this time?






. . .Surprise! This is the exact same function:
                            y=2^x

. . .This time I just graphed it out to bigger x

. . . . . .spreadsheet automatically squeezed it downward more

. . . . . .(otherwise the graph would be way higher than the roof)

. . .One last example:


. . . . . .the two curves are the same size and shape
. . . . . . . . .(because overall graph size/shapes are the same)

. . . . . .they have different knee locations

. . . . . .they graph the exact same function

. . . . . . . . .one is graphed for x from 1-20

. . . . . . . . .other is graphed for x from 21-40

. . . . . . . . .curve shape & knee location vary with scale

. . . . . . . . . . . .scaling defined by width, height, & axis numbering

. . .Conclusion: 
     The hockey stick "knee" 
     is an optical illusion
     (for exponential curves)

. . . . . .It doesn't really exist

. . . . . .exponential curves accelerate
. . . . . .smoothly at a constant rate


The "singularity curve"

Example:


  • Solar power
    • PV in particular

    •  Grid parity enables:
      • Competitive production in sunniest locations, then 
      • competitive production in progressively less sunny areas, then
      • competitive overproduction in more and more areas, the
      • progressively more competitive energy transformation options 
        • batteries, 
        • mass transfer, 
        • liquid fuel production,, finally
      • withering of other energy sources.
  • The curve isn't really vertical, just "suddenly a lot steeper"


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, www.constructal.org/en/art/S-curve.pdf.


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

Think about transportation by horse,
pencils, compact fluorescent lightbulbs, etc.










.

.




What do you think of these curves?




There are other views of trajectories...

Gartner Hype Cycle











.


(Source:
http://en.wikipedia.org/wiki/File:Gartner_Hype_Cycle.svg)


How might this apply to some of our topics?

There is also the Technology Adoption Life Cycle

There are also other relevant curves for
"thinking quantitatively about technological progress"

No comments:

Post a Comment