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In problem solving as in street-fighting:
Rules are for fools!

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(Laughter)

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(Applause)

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Let's see how far we can go
by bending rules

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as we estimate the fuel efficiency,


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the miles per gallon of a 747.

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The fuel is used to fight drag,


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the force of air resistance,

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what you would feel

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if you stuck your hand 
out of a moving car --

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don't try this at home --

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or try to run in a swimming pool.

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There are at least two ways 
that you can use

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to figure out the drag.

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You could spend 
10 years learning physics

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and you write down 
the Navier–Stokes equations:

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the differential equations
of fluid dynamics.

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And then you spend another
10 years learning mathematics

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to solve for the pressure.

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And whereupon you find

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that actually there's no exact solution


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for the flow around a 747,

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or, in fact, for most of the situations


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which you want to know.

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Rigor, the rigorous approach,

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the exact approach
has produced paralysis,

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rigor mortis.

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(Laughter)

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We need a different way.

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The street-fighting way,

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which starts with
a home experiment.

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Chair please.

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Props please.

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(Laughter)

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Small cone, big cone. Coffee filters.

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They're the same shape,

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but this one has
one-fourth the area.

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This one has four times
the area, twice the diameter,

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but otherwise the same shape.

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When I drop them,

41
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how fast do they fall 
relative to one another?

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Is the big one roughly twice as fast?

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Are they comparable in speed?

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Or is the small one
roughly twice as fast?

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Take ten seconds and think.

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What do you believe?
What does your gut tell you?

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And then we'll take a vote.

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Check with your neighbor.

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(Laughter)

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(Crowd murmuring)

51
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OK, let's take a vote.

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You don't have to agree 
with your neighbor.

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(Laughter)

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That's the beauty of democracy.

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So, cheer if you believe
that the big cone

56
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will fall roughly twice as fast 
as the small cone.

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(Faint cheering)

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OK, I hear a few.

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Cheer if you believe 
that they'll be roughly comparable.

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(Louder cheer)

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And cheer if you believe the small cone


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will be roughly twice as fast.

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(Loudest cheer)

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A lot of cheering for that one.

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OK, well, as Feynman said and believed,

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in science we have a
supreme court: experiment.

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So, let's do the experiment!

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One, two, three.

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(Cheering) (Applause)

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They're almost the same.

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Within experimental error.

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So what does that mean?


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What can we use 
that experiment to tell us?

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Well,

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the cones fell at the same speed.

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They fall in the same air.
It has the same density.

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The same properties.
The same viscosity.

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The only things different 
between the two cones

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is this one has four times the area,

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the cross sectional area of this one,

81
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and their drag force is different.


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How different?

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Well, the drag force
is equal to the weight.

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Because they were falling
at a steady speed with no acceleration.

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So the drag and the weight cancel.

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So we have a very sensitive measure

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of the drag force
without any force sensors.

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All we do is measure the weight.

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So this one has four times
as much paper as this one.

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So it's four times heavier,
four times the drag.

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Only change, four times the area.

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The conclusion:
drag is proportional to area.

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Not square root of area,
not the square of the area.

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but just the area.

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That's the result 
of our home experiment

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without the rigorous 
rigor mortis method.

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How can we use that?

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Well, that one constraint,

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along with the next street-fighting tool

100
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of dimensional analysis,
solves the drag force.

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We match their dimensions.

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We match the dimensions of force,
drag force on one side

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with what we have on the other,


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which is area, density, speed 
and viscosity.

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But we already know how to put in
the area, just one of them.

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That gives us length squared,
meters squared.

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Now we look and we say,
"Oh, there's kilograms over here,

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we have to get a kilogram over here."

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The only place to get it from is density.

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Speed and viscosity, the kinematic
viscosity, have no mass in them.

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So we put in one density.

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Now what we need still 
is meter squared / second squared,

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out of speed and viscosity.

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The only way to make it is speed squared.

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So there is our drag force.

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One experiment for a constraint.

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Dimensional analysis
for the rest of the constraints.

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Drag Force =
Area x Density x Speed squared.

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How can we use this?

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Well, the fuel consumption 
is proportional to the drag force.

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So, let's compare the fuel consumption
of a plane with a car.

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Rather than calculating the plane
from scratch, compare it to a car.

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Another street-fighting technique.

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So there're three factors in
the comparison, in the ratio:

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the area, the air density
and the speed squared.

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Do them one at a time.

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So, the area. Well, 
in the old days of plane travel,

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you could lie down on three seats

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and there were three sets of those seats.

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So three people wide.
Plane is about three people high.

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So it's nine square people.

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A car: Well, 
from nocturnal activities in cars

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you know you can sort of lie down
in cars a bit uncomfortably.

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(Laughter)

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And you can stand up.
So it's one square person.

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So it's roughly a ratio of ten,

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maybe nine or ten.

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So the plane is 10 times
less fuel efficient for that.

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What about air density?

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Well, the planes fly high, 
about Mt. Everest.

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So the density is about one third.

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So that helps the plane.

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But they fly about ten times faster,


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600 miles an hour versus 60.

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That means planes pay a factor 
of a hundred, 10 squared.

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The result is planes are 300 times


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less fuel efficient than cars.

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Oh, no. By flying here, 
did I damage the environment

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300 times compared to driving? 
(Gasp)

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What saves it?

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300 people on my plane!

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So the conclusion is planes and cars

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are roughly equally fuel efficient.

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(Laughter)

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All from that.

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(Applause)

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So let's say the plane is 
30 miles per gallon.

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Crossing the country
back and forth 6,000 miles,

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30 miles per gallon, 2 dollars a gallon.


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That's 400 dollars of gasoline.

161
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That's not that different than
the price of my plane ticket,

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which may explain why 
airline companies teeter on bankruptcy

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and why they charge us for peanuts.

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(Laughter)

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So connection between
the 747 and the cones.

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They increase our enjoyment of the world


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and expand our perception.

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And that, making connections here 


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was enabled by street-fighting reasoning,

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by getting away from rigor mortis.

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Making connections is so important


172
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because it builds ideas and isolated facts


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into a coherent story.

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Imagine each dot is an idea

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and the lines are the connections 
between them.

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As I increase the fraction of connections


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from 40% to 50%, to 60%,

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the big story, the red connection network,


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grows to fill the whole space.

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That's the long lasting learning.

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That's what we want 
to build in our thinking

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and in our teaching.

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The goal of teaching should be

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to implant a way of thinking
that enables a student

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to learn in one year
what the teacher learned in two years.

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Only in that way 
can we continue to advance

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from one generation to the next.

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In fifty years, all education

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will, I believe and dream,
be based on this principle.

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Richard Feynman, I think,
would have agreed.

191
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Thank you.

192
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(Applause)