HS #18 2017.1.3
Lessons from Fixed
Points
Imagine
a hillside and a big round boulder.
Place the boulder on the side of the hill and it rolls down. Position it
at the very top of the hill or at the bottom of the valley and it remains in
place. These are called “fixed points.” However they are quite different from
each other. Nudge the boulder in the valley, and it rolls back into place.
Nudge the boulder at the top and it cascades down the slope. Hence the valley
is called a stable fixed point, while the zenith is called unstable. Make sense?
Appreciating
the difference between these two kinds of fixed points help in understanding
both nature and society. Let’s look at
nature first.
The
size of Lake Michigan is a stable fixed point. Why? If a heavy rainfall or
snowmelt causes Lake Michigan to grow in size, then the surface area also increases.
This causes water to evaporate at a greater rate, so the lake eventually reverts
back to its normal size. That is, the
lake tends to stay stable just like the boulder in the valley.
The
length of dog’s nails is stable. The longer the nails, the more there is to get
worn down when the dog walks on pavement. Eventually the nails wear down as
fast as they grow. Knowing this, I had a
dog for 12 years without ever cutting his nails – the length of Elvis’s nails
found a stable fixed point.
Other
natural phenomena are unstable. Try balancing a table knife on its point. As
soon as the knife tips just a bit, the force pulling it down increases, leading
the knife to tip even faster.
Or try
using one magnet to pull another one across a table without touching each other.
It’s almost impossible because as the magnets get closer the pull between them
increases, so they snap together. The only stable fixed point is for them to be
connected.
Some
fixed points are more complicated – and interesting. A boulder in a little divot on the top of a
hill will roll back into place if moved just a smidgeon, but if pushed too far,
will tumble down into the valley.
An
example of this? Consider your skin. If you get just a scratch or small sore,
your body will heal back to its original health. Too large of a wound, and it
will fester – leading to amputation or death.
The
earth’s climate is another example. If altered just a bit, experience and
models show that it returns to its original state. But if pushed too far, there is good reason
to think that, like the sore or boulder, it will cascade to a new position.
Why? Presently the polar ice caps reflect much of
the sun’s energy back into space thus helping to keep the earth from warming up
too much. As the icecaps melt, more solar energy is absorbed and the earth
heats up all the more.
Also,
there are huge areas of permafrost in the northern latitudes with carbon dioxide
frozen in the ground. As the earth continues to warm, these areas thaw and the
gas is released, leading to even more of the greenhouse effect. This is why
scientists believe that we have a short window of opportunity to keep the
climate from cascading to an entirely new fixed point. Worth noting.
Society
also has these fixed points. Yogi Berra once said of a restaurant, “Nobody goes
there anymore – it’s too busy.” In less humorous words, the busyness of the
restaurant was keeping others away. Thus
the number of patrons remained stable.
On the
other hand, consider Christmas toy crazes: Cabbage Patch dolls, Beanie Babies,
Tickle Me Elmo. These things created
their own demand. Folks wanted them because others had them. Thus, unlike the
restaurant, the more there are, the more others buy. Creating such an unstable fixed
point is every manufacturer’s dream.
A
social media story that goes “viral” is another example. As each viewer passes it on to friends, the
growth becomes exponential. That’s the definition of an explosion.
Gun ownership may have two stable fixed
points. The United Kingdom has a stable fixed point of essentially no guns. In the U.S., there is enough killing from guns
that others now want them for protection. So we are heading towards the
opposite fixed point. Two fixed points,
but different consequences: Our fixed
point results in per capita gun-deaths 50 times higher than England.
Worth
noting.
