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:

• Lightbulbs 
• The "AI singularity"

• Solar power
• PV in particular

• Reference: Figure 1 from JD Farmer and F Lafond, How predictable is technological progress? Research Policy, 45(2016):647-665.

•  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?

• (See http://www.eia.gov/todayinenergy/detail.cfm?id=18671)

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"