Lessons from Fixed Points

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.