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One quibble: The butterfly effect is not that the butterfly causes the tornado in texas. It is that the butterfly causes unpredictable changes everywhere.

That is, take two copies of the earth, identical in every detail except that one earth has one extra flapping butterfly in the amazon. Wait a sufficient time (probably a couple of months for this system), and the two earths will have completely different weather. They will be the same in overall climate, and in the big, general features (which we also see repeated each year), but all the day to day details will be different.

In particular, one earth may have a tornado in texas at a particular time and place; the other earth may have a tornado on a different day in a different town. The two Texases will probably suffer pretty much the same severity of tornado season that year.

Thanks for the clarification! Quibbling corrections and additional insighs are always welcome, especially from those with specific expertise.

At the risk of stating the obvious, artists' techniques change over time. (Not all Seurat's paintings are pointillist and the difference between early Beethoven and late Beethoven is as night and day.) So, the absence of the expected scaling behavior in these new-found paintings may be attributable to the fact that they were executed by Pollock at a different period from that of the paintings that Taylor had analyzed earlier.


With regard to the apparent non-randomness of the iPod's "shuffle play" feature, here is the link to the original article by Steven Levy that you seemed to be referring to:

http://msnbc.msn.com/id/6854309/site/newsweek/

Here is my letter to him explaining why finding 3 or more songs from the same album when using the iPod's "shuffle play" feature is not unreasonable:

------------

Sent: Saturday, January 29, 2005 11:31 PM
To: steven.levy@newsweek.com
Subject: 3 songs from the same album is more likely than not with your iPod Shuffle


From Steven Levy's "Does Your iPod Play Favorites?":
"[...] The question will be even more important to owners of the new tiny iPod Shuffles. These use a new feature called autofill to load the one-ounce players with a supposedly random selection of 120 or so songs from much larger collections. The first few times I tried this, I found some disturbing clusters in the songs chosen. More than once the "random" playlist included three tracks from the same album! Since there are more than 3,000 tunes in my library, this seemed to defy the odds. [...]"


Let's assume there are exactly 3000 songs in Levy's library and that they come from 300 albums, each with exactly 10 songs each. Further, let's assume that we choose exactly 120 songs at random from this library of 3000 songs. There are Binomial[3000, 120] ways of doing this. The number of ways of choosing 120 songs such that no two come from the same album is just the number of ways of choosing 120 albums out of the 300 (Binomial[300, 120]) multiplied by the number of ways of choosing one of the 10 songs from each of the 120 albums chosen (10^120). Therefore the probability that no two songs come from the same album is just (Binomial[300, 120] * 10^120)/Binomial[3000, 120] = 9.46 * 10^–12. This means that the probability that at least two songs of the 120 chosen come from the same album is 1 – 9.46 * 10^–12 = 0.99999999999054, practically a dead cert.

With slightly more involved calculations (Mathematica helps here), it can be shown that the probability that at least three songs of the 120 chosen come from the same album is 0.858, more than 85 % and far more likely than not.

Levy need not be concerned—the odds are not being defied.

------------

And here is Levy's reply (I trust he won't mind my reproducing it here):

Thanks. Of several mathematical breakdowns I got on this, yours is the clearest. Plus, your 85% probability seems more in line than the I one got saying that the certainty of three songs from the same album is so close to 100% that if autofills were atoms, only a few in the universe would not have three songs from the same album.

In any case, a decent percentage of my tunes are not part of a complete-album set, so my milage varies.

Steven

Have to agree with Levy that this is the clearest mathematical breakdown on the topic. Thanks for sharing it! I, for one, found it extremely edifying. If anyone can explain the math behind Amazon rankings or Google search results, feel free to share that as well. :)

As for Taylor, he is himself an artist (has a Master's degree in addition to his physics PhD) so he's well aware of such subjective criteria as changing artistic styles. I didn't mention in my post that he's analyzed Pollock's paintings over time, and found that the fractal dimension actually increases in later paintings -- but the fractal nature of the splatters is always there, regardless of the period(s) he looked at. So I'm quite sure he took such considerations into account.

I was planning on contacting him for a brief update as to his latest science/art stuff. If I find out anything interesting on this front, I'll write a follow-up post.

Just ran across an article in the New York Times about an art forger who has become so notorious, he is able to openly sell his fakes: http://www.nyyimes.com/2006/03/04/international/europe/04myatt.html

John Myatt is his name, and for years he painted fake masterpieces that sold for a pretty penny and helped him keep his kids fed and clothed. He now gives lectures on art forgery, and says he can't believe his fakes fooled anybody since they were so poorly done.

It's a great argument for more objective, scientific methods of evaluation/authentication of fine art... :)

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    • Heavy G
      The perfect pick-me-up when gravity gets you down.
      2 oz Tequila
      2 oz Triple sec
      2 oz Rose's sweetened lime juice
      7-Up or Sprite
      Mix tequila, triple sec and lime juice in a shaker and pour into a margarita glass. (Salted rim and ice are optional.) Top off with 7-Up/Sprite and let the weight of the world lift off your shoulders.
    • Listening to the Drums of Feynman
      The perfect nightcap after a long day struggling with QED equations.
      1 oz dark rum
      1/2 oz light rum
      1 oz Tia Maria
      2 oz light cream
      Crushed ice
      1/8 tsp ground nutmeg
      In a shaker half-filled with ice, combine the dark and light rum, Tia Maria, and cream. Shake well. Strain into an old fashioned glass almost filled with crushed ice. Dust with the nutmeg, and serve. Bongos optional.
    • Combustible Edison
      Electrify your friends with amazing pyrotechnics!
      2 oz brandy
      1 oz Campari
      1 oz fresh lemon juice
      Combine Campari and lemon juice in shaker filled with cracked ice. Shake and strain into chilled cocktail glass. Heat brandy in chafing dish, then ignite and pour into glass. Cocktail Go BOOM! Plus, Fire = Pretty!
    • Hiroshima Bomber
      Dr. Strangelove's drink of choice.
      3/4 Triple sec
      1/4 oz Bailey's Irish Cream
      2-3 drops Grenadine
      Fill shot glass 3/4 with Triple Sec. Layer Bailey's on top. Drop Grenadine in center of shot; it should billow up like a mushroom cloud. Remember to "duck and cover."
    • Mad Scientist
      Any mad scientist will tell you that flames make drinking more fun. What good is science if no one gets hurt?
      1 oz Midori melon liqueur
      1-1/2 oz sour mix
      1 splash soda water
      151 proof rum
      Mix melon liqueur, sour mix and soda water with ice in shaker. Shake and strain into martini glass. Top with rum and ignite. Try to take over the world.
    • Laser Beam
      Warning: may result in amplified stimulated emission.
      1 oz Southern Comfort
      1/2 oz Amaretto
      1/2 oz sloe gin
      1/2 oz vodka
      1/2 oz Triple sec
      7 oz orange juice
      Combine all liquor in a full glass of ice. Shake well. Garnish with orange and cherry. Serve to attractive target of choice.
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      3/4 oz Rum
      1/2 oz Strega
      1/4 oz Grand Marnier
      2 oz Pineapple juice
      Fill with Sweet and sour
      Pour rum, strega and Grand Marnier into a collins glass. Add pineapple and fill with sweet and sour. Sip until all the day's super-positioned states disappear.
    • The Black Hole
      So called because after one of these, you have already passed the event horizon of inebriation.
      1 oz. Kahlua
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      .5 oz. Cointreau or Triple Sec
      .5 oz. dark rum
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      Pour into an old-fashioned glass over (scant) ice. Stir gently. Watch time slow.