Benoit Mandelbrot was the mathematician who invented a way to "measure roughness," so to speak, coining the phrase "fractal equations" to describe what he was doing. His afterlife may contain an endless blackboard and chalk that never runs out, quite possibly heaven for a math professor.
Mandelbrot started thinking about the mathematical consequences of the idea of roughness when, as a young researcher, he considered the question, "How long is the coast of Britian?" Obviously, that could be measured in miles and an answer given. But, Mandelbrot thought, the smaller the scale of measurement, the more uncertainty went into the final result. The coast was not a straight line but a jagged one, zigging back and forth and adding distance to the measurement. Not only inlets and bays had to be considered, but when you dropped the scale small enough, large rocks could affect the final answer. Make it smaller, and even gravel had to be taken into account in order to be precise. In fact, the more Mandelbrot considered the problem, the closer he came to believing that there was no exact length of the coast, because the precise measurement would have to take the variations caused by subatomic particles into account, and that wasn't possible.
Dr. Mandelbrot leaves behind the idea of fractal equations and a particular equation called, in his honor, the Mandelbrot set. Mandelbrot sets create the different bizarre repeating shapes that we see in posters, album covers and T-shirts. Those drawings are actually the graphs of these strange equations. Some equations, when the answers are plotted on graph paper, make curves and are named after the shape of those curves. Thus, equations that plot out as "parabolas" are "parabolic equations," those that plot as hyperbolas are "hyperbolic equations," those that plot as ellipses are "elliptical equations," and so on. All of these are so-called smooth curves, though, and Mandelbrot brought a whole different set of circumstances to the table when he began dealing with equations that were not smooth, but were rather "rough," like things found in the real world.
Some equations do not have exact solutions and their graphs do not make simple curves. Instead, the graph is a pattern that repeats itself, usually smaller and smaller each time, but never actually finishes even though the pattern may become too small to measure even with microscopes. The equations which produce these pictures are often weird and have answers that seem to make no sense. For example, one such equation shows that there are an infinite number of numbers between 1 and 2, or 2 and 3, or any other pair of numbers. Half the numerical "distance" between 1 and 2 is 1.5. Half the "distance" between 1.5 and 2 is 1.75. Halfway between 1.75 and 2 is 1.875, and so on ad infinitum. The repetition of the equation is called "iteration," and fractals are equations that near -- but never reach -- zero as they continue to iterate. Since you can never find a pair of numbers between which you cannot slip another number, you literally have "infinity" between any two numbers you can name.
Reading a little about him at the obit and here, one learns Mandelbrot was a bit of an eccentric. That's kind of what you'd expect, though, from a guy who, if asked, "What's between 1 and 2?" might answer, "Everything."
(H/T University Diaries, which used the headline I wanted to use)
Sunday, October 17, 2010
Eternal June
Although she was probably far too much of a nice lady to have ever done this, I always wanted Barbara Billingsley to show up when someone complained about the ridiculousness of her cleaning her house, cooking, etc., in heels and pearls. I thought she could grab that person's collar, yank him or her in close and say, "I cleaned in heels and pearls because I am that...darn...good. Punk."
Ms. Billingsley is now a part of the place where there are no reruns, passing away at 94. My favorite role she played was "Jive Lady," in the 1980 comedy Airplane!. While it took the comedic genius of the Zucker-Abrahams crew to suggest that June Cleaver "spoke jive," Ms. Billingsley herself made the laugh one of the movie's best just by being her and speaking the words, "Jus' hang loose, blood," and by using the phrase "jive-ass dude."
Ms. Billingsley is now a part of the place where there are no reruns, passing away at 94. My favorite role she played was "Jive Lady," in the 1980 comedy Airplane!. While it took the comedic genius of the Zucker-Abrahams crew to suggest that June Cleaver "spoke jive," Ms. Billingsley herself made the laugh one of the movie's best just by being her and speaking the words, "Jus' hang loose, blood," and by using the phrase "jive-ass dude."
Thursday, October 14, 2010
A Case of Partial Extreme
Tuesday, October 12, 2010
There's Some Motion in the Ocean
When you're tooling out around Saturn, some 1.46 billion kilometers (900 million miles or so) from the sun, things are pretty cold. So if one of Saturn's moons has water on it, you'd expect that water to be ice.
Then one day in 2005, the Cassini probe flew by Enceladus -- the sixth-largest of Saturn's satellites -- and found something very interesting: It seemed to have sprung a leak. The four streaks spouting from the south pole of the moon in the photo here are actually gigantic plumes of water, shooting into space. Scientists found this out in 2008 after Cassini gathered and analyzed the ice crystals made by the plumes after they had refrozen. In fact, they're salty, which means water is liquid on Enceladus long enough to dissolve minerals into it, like in our oceans.
The problem is that Enceladus is too small and too far from the sun to maintain liquid water. A moon or planet has to be within a certain range of its sun or be large enough to generate enough of its own heat to keep chemicals and compounds from freezing. We're close enough to the sun, for example, that many substances are liquids at our average temperatures. Saturn and Jupiter are so large that, even though they are much farther away from the sun than we are, they generate enough internal heat to maintain the gaseous or liquid nature of the different substances that make them up.
Soo, how does this dinky little moon manage to have water, and in enough abundance that scientists think it might actually be possible for it to support some kind of life? Originally, scientists figured that the stress of tides generated during Enceladus' orbit of Saturn would generate heat. Enceladus is about three-fifths as far from Saturn as our moon is from us. Saturn is much, much larger than the earth, so its gravity creates tidal stresses much more powerful. The way those stresses pull the moon this way and that could provide enough friction within its own surface to heat the water to a liquid point. But the computer models didn't quite work, so they started looking for other factors, and they found a wobble.
The wobble not only generates more heat by making the gravitational stress on Enceladus stronger and increasing friction, it also shows scientists why the icy surface of the moon cracks from time to time and allows the giant streams of water to erupt into space. Coincidentally, in Greek mythology Enceladus was one of the Gigantes defeated by the Olympians, and was imprisoned by being buried under what is now Mt. Etna in Sicily. The volcano's fiery eruptions are supposed to be his breath and the tremors which shake the island from time to time were thought to be caused by Enceladus rolling to ease the pain from his wounds.
Without exploring Enceladus, there's no way to see if there really is life in its oceans, or if the conditions are right but nobody's shown up yet. It may be that closer exploration of the moon shows the water isn't trapped in caves or a part of an ocean but something else entirely.
Either way, whatever exploration takes place is likely to produce some spectacular scenery. The Cassini probe showed some of the plumes extended as much as 500 kilometers (about 300 miles) from the surface of the moon, meaning they are as "tall" as the moon itself is "wide." They don't show up that big in the photo because of the distance Cassini is from the moon when it was taken. Maybe in a hundred years or so, Niagara Falls will take a backseat as a honeymoon destination to the Fountains of Enceladus. And even if they don't, "Fountains of Enceladus" sounds like an awesome name for a science-fiction novel. I would have said, in my best Dave Barry fashion, a good name for a rock band, but Fountains of Wayne might not have thought that was so funny.
Then one day in 2005, the Cassini probe flew by Enceladus -- the sixth-largest of Saturn's satellites -- and found something very interesting: It seemed to have sprung a leak. The four streaks spouting from the south pole of the moon in the photo here are actually gigantic plumes of water, shooting into space. Scientists found this out in 2008 after Cassini gathered and analyzed the ice crystals made by the plumes after they had refrozen. In fact, they're salty, which means water is liquid on Enceladus long enough to dissolve minerals into it, like in our oceans.
The problem is that Enceladus is too small and too far from the sun to maintain liquid water. A moon or planet has to be within a certain range of its sun or be large enough to generate enough of its own heat to keep chemicals and compounds from freezing. We're close enough to the sun, for example, that many substances are liquids at our average temperatures. Saturn and Jupiter are so large that, even though they are much farther away from the sun than we are, they generate enough internal heat to maintain the gaseous or liquid nature of the different substances that make them up.
Soo, how does this dinky little moon manage to have water, and in enough abundance that scientists think it might actually be possible for it to support some kind of life? Originally, scientists figured that the stress of tides generated during Enceladus' orbit of Saturn would generate heat. Enceladus is about three-fifths as far from Saturn as our moon is from us. Saturn is much, much larger than the earth, so its gravity creates tidal stresses much more powerful. The way those stresses pull the moon this way and that could provide enough friction within its own surface to heat the water to a liquid point. But the computer models didn't quite work, so they started looking for other factors, and they found a wobble.
The wobble not only generates more heat by making the gravitational stress on Enceladus stronger and increasing friction, it also shows scientists why the icy surface of the moon cracks from time to time and allows the giant streams of water to erupt into space. Coincidentally, in Greek mythology Enceladus was one of the Gigantes defeated by the Olympians, and was imprisoned by being buried under what is now Mt. Etna in Sicily. The volcano's fiery eruptions are supposed to be his breath and the tremors which shake the island from time to time were thought to be caused by Enceladus rolling to ease the pain from his wounds.
Without exploring Enceladus, there's no way to see if there really is life in its oceans, or if the conditions are right but nobody's shown up yet. It may be that closer exploration of the moon shows the water isn't trapped in caves or a part of an ocean but something else entirely.
Either way, whatever exploration takes place is likely to produce some spectacular scenery. The Cassini probe showed some of the plumes extended as much as 500 kilometers (about 300 miles) from the surface of the moon, meaning they are as "tall" as the moon itself is "wide." They don't show up that big in the photo because of the distance Cassini is from the moon when it was taken. Maybe in a hundred years or so, Niagara Falls will take a backseat as a honeymoon destination to the Fountains of Enceladus. And even if they don't, "Fountains of Enceladus" sounds like an awesome name for a science-fiction novel. I would have said, in my best Dave Barry fashion, a good name for a rock band, but Fountains of Wayne might not have thought that was so funny.
Sunday, October 10, 2010
Hoping Darwin Was Wrong...
So Friday I parked at a Border's Books and Music and apparently didn't lock my truck. When I came back outside after making my purchase, I was short one gym bag and one portable jump-starter. The jump-starter is pushing five years old and I don't even know if it holds a charge anymore, and I have seen them in pawn shops for about ten bucks.
The gym bag held my workout clothes, deodorant, a nearly empty bottle of shampoo. It also held two items sure to be worthless to these thieves: A combination lock and a book. The combination lock is worthless because it was locked, can't be reset and the chances that someone will just happen on the combo are pretty minimal. The book is worthless because I'm going to presume someone who steals gym clothes capable of piloting the space shuttle before I presume them capable of understanding the concept of a book. The odds of deciphering the combination lock are better.
They left a $200 tool box that sits behind the seat of my truck, about which I am pleased and which further confirms to me that they possess suboptimal mental faculties. There is no way of knowing who they are, of course. But if prayer works, some clinic in the Norman area is going to be treating a couple of cases of painful genital warts within the next week, and those warts are going to recur. I opted against hoping for rectal warts because of the possibility that they could impede the minimal thinking of which these thieves might be capable.
The gym bag held my workout clothes, deodorant, a nearly empty bottle of shampoo. It also held two items sure to be worthless to these thieves: A combination lock and a book. The combination lock is worthless because it was locked, can't be reset and the chances that someone will just happen on the combo are pretty minimal. The book is worthless because I'm going to presume someone who steals gym clothes capable of piloting the space shuttle before I presume them capable of understanding the concept of a book. The odds of deciphering the combination lock are better.
They left a $200 tool box that sits behind the seat of my truck, about which I am pleased and which further confirms to me that they possess suboptimal mental faculties. There is no way of knowing who they are, of course. But if prayer works, some clinic in the Norman area is going to be treating a couple of cases of painful genital warts within the next week, and those warts are going to recur. I opted against hoping for rectal warts because of the possibility that they could impede the minimal thinking of which these thieves might be capable.
This Just In...
"Purdue Sells Souls to Devil"
"Wailing, Gnashing of Teeth Predicted
As Boilermakers to Literalize Nickname"
Unfortunately, as his favorite team begins their usual October antics, the Lord of Darkness's services appear to attract many other followers all too willing to exchange eternal paradise for the fleeting glory of a Saturday afternoon win.
Oh, the humanity!
"Wailing, Gnashing of Teeth Predicted
As Boilermakers to Literalize Nickname"
Unfortunately, as his favorite team begins their usual October antics, the Lord of Darkness's services appear to attract many other followers all too willing to exchange eternal paradise for the fleeting glory of a Saturday afternoon win.
Oh, the humanity!
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