**An Example of Extrapolation:**

**Overpopulation on Mars**

Suppose we started a

self-sustaining colony

on the moon

(see http://www.youtube.com/watch?v=EgOg0mzqGAM)

or Mars

(see 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!

**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?

Example 4:

- Last year you had 1 or 2 compact fluorescent bulbs

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

- This example used to work
- but no longer does!
- Why?

In the longer term,

change often looks

**exponential**

Lightbulb example:

. . . you start with 1-2, but then accelerate

. . . the rate of increase increases

. . . 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...

. . .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)

. . . . . .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

**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 fluorescents, 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"

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