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07/03/2008
Projections, Perceptions and Illusions
Continuing my Phoenix watch, I imagine most of you have read
or heard about the wet chemistry that Phoenix carried out on
some Martian dirt. The results made a splash in the media when
the pH analyses revealed that the soil sample was somewhat
alkaline and, if here on Earth under warmer conditions, would be
suitable for growing asparagus! While Phoenix continues its
digging, heating and wet chemistry of samples on Mars, there
was the publication in the journal Nature of another finding of
spectacular proportions that came from data obtained by two of
the spacecraft orbiting Mars.
Jeffrey Andrews-Hanna and Maria Zuber at MIT and Bruce
Banerdt of NASA’s Jet Propulsion Laboratory reported data on
elevations and gravity indicating a huge impact crater on Mars.
The crater is over 5 thousand miles across and was caused by an
object roughly 1200 miles in diameter! That sucker was bigger
than the recently demoted planet Pluto. If confirmed, this is by
far the largest impact crater in our solar system and was formed
over 3.9 billion years ago. What a fireworks show that must
have been! For the sake of our planet, I hope there are no more
objects anywhere near that size headed our way.
Speaking of our planet, at this week’s Old Guard meeting I heard
a talk by one of our members and a DVD presentation on the
subject of maps. Making a map of our globe is not an easy thing
to do, the problem being trying to represent the features of a
sphere on a flat surface. No matter what you do, there’s bound to
be some distortion of reality. When I was in school, the standard
map of the world posted on the wall of the classroom was a
Mercator projection map. This type of map imagines in essence
what would be revealed if a globe was illuminated by a light
inside and the surface of the globe unfolded as a cylinder would
be unfolded.
There are problems with this approach. If I had paid attention, I
should have noticed that the Mercator map showed that
Greenland was roughly as large as Africa, maybe bigger.
Actually, Africa is some 14 times larger than Greenland! This
slight discrepancy wasn’t really of great concern to users of the
Mercator map, which was really meant for navigational use.
Draw a straight line between two points and, with proper
navigational expertise, you arrive at your destination. In 1974, a
historian and cartographer, Arno Peters, decided this type of map
should be replaced and he came up with the Peters map. The
Peters map employs a type of projection in which the relative
areas of countries and continents are accurately represented, as
are distances between points.
On a Peters map, Africa and South America look thin and
elongated. There are many other types of maps and projections,
some quite controversial, depending on your own point of view.
For example, a young student in Australia was told he would
flunk geography if he persisted in portraying the world “upside
down”. His map placed Australia on top and Europe and North
America, etc. “down under”! I forget his name but some years
later, he tried again, this time appealing to Australian pride.
After all, who’s to say north is “up” and south is “down”? The
map sold several hundred thousand copies!
Our Old Guard speaker had been in Japan and seen on the wall of
an office a map of the globe with Tokyo at its center. Why not?
On the DVD we saw a map with Toronto, Canada at its center,
with all the distances to other points on the globe to scale. The
Aussies might not have been happy with that map since Australia
looked more like Long Island than the continent with which we
are familiar.
Perspectives and illusions created by different projections can be
quite interesting. Take “The MPG Illusion”, the title of an article
by Richard Larrick and Jack Soll in the June 20 issue of Science.
MPG here stands for miles per gallon. With gasoline well over
$4 per gallon and rising, the number of miles per gallon we get
with our vehicles has become a figure of great importance.
Larrick and Soll point out that in using MPG as our measure of a
vehicle’s efficiency and contribution to our carbon footprint, we
are guilty of “linear” thinking on a “curvilinear” subject. They
say we should be using GPM, gallons per mile.
The authors grant that the ideal situation would be that everyone
drives a car, be it hybrid, diesel or conventional, that gets say 40
or more miles per gallon. However, let’s be realistic – we still
have all those SUVs on the road and in the showrooms. Here’s
where the “linear” thinking comes in. Suppose your neighbor
has one of these horrendous super SUVs that get only 10 MPG
and the government gives him a tax incentive to switch to a super
SUV hybrid that gets all of 12 MPG. On the other hand, you’re
driving a much smaller car that gets 25 MPG and the government
offers you the same tax incentive to switch to a hybrid that gets
40 MPG. You figure that’s not fair. You should get more of a
tax break because you’re getting 15 more MPG compared to only
2 more MPG in your neighbor’s case.
But wait. Let’s say you both drive 10,000 miles a year. Let’s do
the math. Your neighbor’s old SUV used 1,000 gallons of gas a
year (10,000/10). With his new hybrid SUV he’ll use 833
gallons (10,000/12). He’s saved 167 gallons of gas switching to
the hybrid SUV. On the other hand, your old car consumed 400
gallons a year (10,000/25) while your hybrid ate up 250 gallons
(10,000/40). You’ve only saved 150 gallons, 17 gallons less than
your neighbor! If your neighbor had switched to an SUV
capable of 15 MPG, he would have saved 333 gallons, more than
twice what you saved! The authors of the Science article, who
are at the Fuqua School of Business at Duke University, stress
that the use of gallons per mile, GPM, gives a true picture of gas
consumption. The use of MPG leads to underestimation of
savings by relatively small increments in vehicles with low
MPGs and overestimation of savings for larger increments in
higher MPG vehicles.
The researchers carried out studies on subjects evaluating their
willingness to purchase or switch vehicles based on the sort of
example given above. When given the MPG factors only, the
subjects behaved as expected, favoring more heavily the choices
that actually would have saved less gas. When given GPM
figures, the choices were more accurately based on actual
savings in gas consumption.
Oh what about that “curvilinear” thing? If you plot the gallons
of gas used to drive that 10,000 miles against MPG, it’s not a
straight line but curves decidedly upward as you go to lower and
lower MPGs. (Your neighbor’s 10-MPG SUV gulps down 1,000
gallons, while your 25-MPG car only takes only 400 gallons. If
you’re graphically inclined, you can calculate a couple more
points and you’ll see the curve. Meanwhile, it’s time for me to
cook dinner.)
Allen F. Bortrum
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