Monday, April 7, 2008

Log-log Paper and a Straight Edge

The TV show Numb3rs started off magnificently, but has been predictably deteriorating. A cop show based on mathematics is a clever idea, and they managed to wring more story ideas out of it than I would have expected, but that’s been accomplished primarily by continually reducing the math content and increasing the amounts of character interplay, police procedural story lines, plus massive firepower.

They also “fine tuned” the ensemble, primarily by getting rid of Sabrina Lloyd’s character, Terry Lake, (photo at left)after Season One. Lloyd is fondly remembered by those of us who were hooked on Sports Night. Her character in Numb3rs was supposed to be a fairly cold and calculating “profiler,” which Lloyd did well, but cold women on TV don’t often go over, and I suspect that Terry Lake got written out of the show for that reason (the “official” explanation seems to be that it was Lloyd’s decision to leave the show, but I’m not buying it).

The replacement profiler on Numb3rs is Megan Reeves, played by Diane Farr (at right), and the first thing they did with her was to show her math illiteracy. She refers to something as “growing exponentially” and Charlie immediately smiles condescendingly and explains what “exponentially” really means: a rate of change that is proportional to size. So the math whiz puts down another dumb blonde. I know, I know, the show is actually better and more subtle than that, but I still have the suspicion that the exchange was supposed to make the new character less threatening, hence more appealing.

Exponential growth, or its twin, exponential decay is common as dirt, or at least the bacteria in dirt, which get to follow exponential growth curves pretty regularly. The typical “S” curve, in fact, is two exponential curves laid onto each other. The first one occurs when some growth phenomenon begins without significant limitation, like the first doublings of yeast in the brew vat. There are so few yeast cells relative to the nutrient that the growth at first doesn’t alter any of the factors that might affect the growth rate.

After time, of course, the yeast begins to run out of nutrients, or begins to be poisoned by their own waste products. At that point you switch to a case where the growth manages to get some fractional distance toward the inevitable limit with each cycle. As the old Firesign Theater bit goes, Antelope Freeway, one mile, Antelope Freeway, one half mile, Antelope Freeway, one quarter mile, Antelope Freeway, one-eighth mile, Antelope Freeway, one sixteenth mile… Xeno’s Paradox as logarithmic decay, in other words, “logarithmic” and “exponential” being two words for the same thing, a constant term added or subtracted to the logarithm of the quantity in question.

At Nasfic in Seattle a few years ago, I was on a panel with an energy consultant who had a good story to tell about the folly of assuming that exponential growth goes on forever. The title of this piece is a quote from that consultant: “…people who believe that you can predict the future with some log-log paper and a straight edge.” This, of course, identified him as an old foggy who remembers when log-log paper was used for that purpose. Now its all spreadsheets and curve fitting, but the principle remains.

My co-panelist told the story of a power company in the Pacific Northwest that did their predictions of electric power demand in the early 1970s and decided on that basis that they should build a nuclear plant. The Pacific Northwest has a lot of hydropower resources, but there came a time when all the easy dams had been built, so they decided that nuclear was the way to go. But nuclear power is pretty capital intensive; it costs a lot of money to build a nuclear plant, though not so much to keep one running. The PN power agency had a regulatory structure that allowed them to pass capital costs through to their customers, so that’s what they did. This raised their rates sufficiently that it effectively suppressed demand growth such that the anticipated growth in demand never occurred and the nuclear plant was an instant white elephant. They’d effectively saturated the market for low cost power and when it got more expensive, well, bye bye exponential growth. It didn’t do much good for the customers and stockholders, either, I’ll bet.

Before I’d ever heard of Moore’s Law, I read a book by Philip Sporn, who noted that electric power generation in the U.S. had doubled every decade in the 20th Century. That was in the early 1970s, just before the doubling stopped.

A goodly amount of both science fiction and futurism is basically just extrapolating growth curves indefinitely, but the really interesting stories are the ones where the exponential growth stops, the whys and wherefores. It takes more insight to do that, of course; it’s a lot easier to just use the log-log paper and the straight edge.


black dog barking said...

Once, racing a blizzard east on I-80, we found ourselves progressively 30 miles from home making 30 miles an hour, 20 miles from home making 20 mph, 10 making 10. Eventually we broke through although no one was sure of the exact moment — it was just white, cold, and dark.

James Killus said...

I am frightened to think how close you came to total destruction in a "divide zero by zero" incident...