Let's Consider Exponential (and Other) Rates of Change
- Famous example is "Moore's law"
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:
- 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"
