Feeds:
Posts
Comments

As promised, the solutions…

1.   681472 [Um, Didn’t we answer this one earlier?]

2.   3927.27272… seconds This represents the amount of time it takes the minute hand of a clock to lap the hour hand.  For example, the hands coincide at midnight; they next coincide 3927.27272 seconds later, or at about 1:05:27 AM.

3.   23.14069… This is just e^π.

4.   2.1656 x 10^185 This is how many cubic planck lengths fit in the observable universe…basically, if our universe were a 3D computer, this is how many pixels you’d need.

5.   1.03 light year/year^2 This is the acceleration due to gravity g, in non-standard units.  It has the following interpretation: if you ignored relativity and accelerated at a rate of 1 g (reasonable for a starship), after a year you’d have reached the speed of light.

6.   133956 This is the number of possible combinations of two birthdays, since 133956 = 366^2.  If everyone on Earth had a significant other, there would be over 26,000 couples with the exact same two birthdays as you and your other.

7.   About 19.5 million people The number of people on Earth who share your birthday.

8.   0.739085… This is called the “Dottie number”…an irrational number that solves the equation cos x = x.

9.   1.72048 m^2 The area of a pentagon with sides of 1 m.

10.   0.004295 % This is what percent of Earth’s history homo sapiens has been around.

As promised, here are hints to puzzle #3…

1.   681472 [Um, I did this one as an example?  Do you really need a hint?]

2.   3927.27272… seconds [clocks]

3.   23.14069… [transcendentals]

4.   2.1656 x 10^185 [universe]

5.   1.03 light year/year^2 [Newton’s apple]

6.   133956 [birthdays]

7.   About 19.5 million people [birthdays]

8.   0.739085… [trigonometry]

9.   1.72048 m^2 [area]

10.   0.004295 % [humanity]

Detailed solutions to follow…

Many Worlds Puzzle #3

Today there are really 10 puzzles. Can you figure out the significance of each number below? I’ve answered the first to get you started.

1.   681472

This number has a prime factorization of 2^9 x 11^3, which indicates that it equals 88^3. There are 88 keys on a piano…so one obvious interpretation is that the number 681472 is the number of possible three-note permutations that could start any piece on a piano (not counting rests, and ignoring duration). I wonder how many of the permutations have actually ever been played over the years?

2.   3927.27272… seconds

3.   23.14069…

4.   2.1656 x 10^185

5.   1.03 light year/year^2

6.   133956

7.   About 19.5 million people

8.   0.739085…

9.   1.72048 m^2

10.   0.004295 %

Because many of these problems are challenging, I will post hints in a week or so.

Here I present three new mathematical discoveries for your edification.

1. According to Alexander Pope, “The proper study of mankind is man.” Symbolically,

S(mankind) = man,

where S(x) is the study of x. Now, Aldous Huxley tells us that “The proper study of mankind is books,” or

S(mankind) = books,

from which we can use the transitive relation to see that

man = books.

Of course, “Man is the measure of all things,” [Protagoras] so we immediately find that

man = books = μ(ξ).

Recall that μ(x) is the notation for the measure on a set, and we’ll use ξ to denote the universal set (ignoring Russell’s paradox as being too annoying). We already have a number of new apothegms, including

  • The proper study of mankind is the measure of all things
  • Books are the measure of all things
  • Women are books

where in the final example we have used Henry Adams’ quote “The proper study of mankind is woman”. Of course, the astute reader will note that Cicero’s quote “So many books, so little time” then takes on a whole new meaning, as noted by the Robert Cray band.

2. We now move on to the observation that theology is the study of theology, a fact which is self-evident. In our notation this becomes

S(theology) = theology.

We can then do multiple substitutions to learn that

theology = S(S(S(S(S(S(S(···))))))).

It is now evident that theology, at its core, is the study of an ellipsis; it’s turtles all the way down.

Barney-The-Dinosaur-Creator-Malibu-Shooting-by-son3. We end with the following logical proof. Consider Nietzsche’s observation that “that which does not kill you makes you stronger.” Let

K = something which kills you,

                   S = something which makes you stronger.

Then Nietzsche’s quote is simply

~K → S.

Applying the contrapositive, we see that

~S → K,

meaning that anything which does not make you stronger must kill you. Barney the dinosaur certainly doesn’t make anyone stronger; therefore Barney kills.

You’re welcome.

Magnus Carlsen is the current world chess champion. He’s the best in the world at something. Not that many people can make that claim, can they?

Magnus Carlsen at FIDE World Chess Championship

Then again, there are lots of things in the world that you could be best at. Whistling, lemur training, lemon-pie-making, juggling, lying, rock climbing, sleepwalking. Somewhere in the world, there is “the best in the world” at each of these pursuits. Maybe my chances of being best at something are not so bad, after all? Maybe I just have to find the right thing…

Consider the modern pentathlon. In this sport, athletes compete in five events—fencing, shooting, swimming, running, and horse jumping—to achieve the overall best combined score. The winner need not be the best at any one specific event, but must have proficiency in all five.

Let’s say I am in the 99th percentile in all five events: very good, but not world class. [Here I am assuming that I’m in the 99th percentile of all humans, not just people who fence.] Taken individually, I wouldn’t have a prayer of making the Olympics. For example, the 99th percentile in épée fencing would still mean that there are

(0.01)^1 * 7,000,000,000 = 70,000,000

people with a similar proficiency around the world. Doesn’t seem that impressive, does yet? But I’m in the 99th percentile in all five events, right? So in reality there are only

(0.01)^5 * 7,000,000,000 = 0.7

people like me. That is, there’s just me. I’m probably the best at this combination of events. I should medal in the modern pentathlon.

And this brings me to my broader point. If you can think of five events in which you are in the 99th percentile individually, then in all likelihood you would be world champion if these events were combined into a single composite event. For those scoring at home, here’s where the number five comes from:

(0.01)^N * 7,000,000,000 = 1 (a single champion)

N ln (0.01) = ln [1/(7 x 10^9)]

N = [–ln (7 x 10^9)] / [ln (0.01)] = 4.9 ≈ 5

Let’s take my own skill set and see how I would do. I am certainly in the 99th percentile when it comes to physics. (Remember, I am comparing myself to the general population, not just physicists. I would never claim to be in the 99th percentile of people with physics PhD’s.) I am probably in the 99th percentile when it comes to chess (considering that I am in the 85th percentile for tournament players based on an 1800 rating). But am I good, really good, at anything else?

I will claim without proof that I am also in the 99th percentile (among the general population) in the following additional skills:

  • Knowledge of classical music
  • Playing the recorder
  • Geometry

Remember, I am not claiming any particularly high proficiency in any of these things. I just claim a 99th percentile rank in the general population. And individually, any one of these skills would only put me in the company of some 70 million others.

But now: make a hybrid event, where competitors have to take a battery of tests on physics, geometry, and classical music, then perform on the recorder, and then play chess… I believe I may do well in such an event. I might even be world champion.

Of course, nothing is that simple. I have ignored the fact that some of these skills may be correlated. Anyone who can play the recorder will probably also know about classical music. And many physicists will also be good at geometry. This means that my competition will be stiffer than I suppose, since if the events aren’t mutually exclusive then I’ve calculated the probabilities incorrectly. But I can improve my chances by making the five events as disparate as possible. I might change “Geometry” to “Movie Trivia”, for example.  My chances of becoming world champion would thereby be increased.

If you think that “99th percentile” is too high a bar, we could lower it to 90th percentile. Most people are in the top 10% at several things. Redoing our calculation, we get N = 9.8 in this case. So if you can find ten things you’re fairly good at and combine them, you too can be a world champion.

Of course, you also have to convince the Olympic governing body that that particular concatenation of events is worthy of a medal. But hey, that’s your problem.

I have some geometry to do.

I liked this post so much (from Sean Carroll at CalTech) I couldn’t help sharing:

http://www.preposterousuniverse.com/blog/2014/06/30/why-the-many-worlds-formulation-of-quantum-mechanics-is-probably-correct/

multiverse-4

Screen Shot 2014-07-22 at 2.00.36 PM

Photo credit: brusspup

Recently I came across the following “optical illusion” on the normally good website I Fucking Love Science:

http://www.iflscience.com/brain/cycloid-optical-illusion-will-boggle-your-mind

I encourage you to read the article and think about the video (otherwise the rest of this post will be less than illuminating).

Supposedly, there is an “optical illusion” in the video because you think there’s a wheel when in “reality” the dots are moving linearly.

This is bullshit.

This is not an illusion. The dots are moving linearly, that’s true. But there is also a wheel. If this causes you cognitive dissonance, so be it. It is not a paradox, however. It is the case that some wheels, when spinning inside other wheels, have points on them which travel linearly.

I admit, this is a field of mathematics close to my heart. Much of my recent work has involved geometric phase, which has connections to Spirographs, epitrochoids and hypotrochoids as mentioned in the IFLS article. I have battered notebooks with over 500 pages of algebra devoted to such things. This is something I know something about.

One way to see that this isn’t an illusion at all is to watch it being drawn. Go to the following Spirograph applet:

http://www.personal.psu.edu/dpl14/java/parametricequations/spirograph/

Play around a little bit. Then create the following specific Spirograph (which is exactly the one in the IFLS “illusion”):
Radius1 = 60
Radius2 = -30
Position = 29
Velocity = 8
If you need to, CLEAR the picture, input the above parameters, and hit DRAW. Hit DRAW again to watch it all over again.

There’s no illusion. A circle is rotating inside another circle. Simultaneously, a particular part of that circle is traversing a straight line. This comes as a surprise to many people: there’s a frisson of incredulity from the idea that your motion can be simultaneously linear, but curved as well. But there’s an easy explanation. Your motion is different in different frames of reference.

Consider an ant on the wheel. With respect to the center of that wheel, he just orbits in a circular manner. But with respect to us, outside of the contraption, he moves back and forth along a single line. This is not an optical illusion. It’s the relativity of geometric shapes.  And I think that’s even cooler than some cognitive trickery.

With the World Cup in full swing, I thought I’d try to tackle that age-old question: why do so many Americans hate soccer?  Maybe if I can get to the bottom of that question, I can help some Americans find joy in the “beautiful game”…at least until World Cup 2014 is over.

First, a little context.  I am an American who loves sports of all kinds but, I will admit, I hated soccer when I was younger.  The reasons for this are many.  I like to think that I’m typical in my soccer-aversion—typical of many other Americans—and this is what gives me some credibility in writing this blog post.  But what’s interesting is that I eventually came to enjoy soccer, and it is partly the journey from hatred to enjoyment that I wish to share with you.

Why did I not like soccer?  I can think of at least 4 reasons:

1)      Turnovers.  To an American immersed in the culture of American football (henceforth just called football) and basketball, it seems as though (in soccer) teams commit turnovers every five seconds or so.  A little bit of background: a turnover (in any sport but soccer, really) occurs when one team gives up control of the ball.  Normally, in most sports, a turnover is a major thing; statisticians keep track of turnovers, and the team that “turns the ball over” more loses most of the time.  In basketball, a turnover often leads to a “fast break” (an exciting play usually leading to a score).  In football, turnovers are catastrophic; fumbles and interceptions are often the most exciting plays in a game.  They represent huge reversals of fortune.  A football team which commits six turnovers in a game will almost always lose.

So imagine an American kid like me, watching soccer on TV for the first time (something that didn’t happen until I was almost in college, by the way).  I see Spain playing Belgium in the World Cup.  Spain has the ball…but within five seconds Belgium has the ball…but then within five seconds Spain has the ball…ad infinitum.  An American football announcer could not possibly keep up: “Spain turns it over!  Belgium kicks it…and turns it over!  Now Spain has it but…oh no, they’ve turned it over!  Belgium has a chance here…nice pass to Ceulemans…but he turns it over!”  If you grew up watching football and basketball, this turnoverfest is maddening.  It appears random, like pinball.

What I failed to realize, back in 1986, is that soccer is a game of averages, of field position, of drift velocity.  It doesn’t really matter in soccer if the ball is “turned over” often.  As long as (on average) the ball tends towards one end of the field or the other, one team will have an advantage.

drift-velocity

Soccer is a game of drift velocity.

It’s like an electron in a copper wire, under the influence of an electric field: the motion of the electron is mostly random, but over time it tends to move in the opposite direction as E.  If Brazil has a better team than Cameroon, then—despite the large number of apparent “turnovers”—the ball will tend to drift towards the Cameroonian goal.  This drift velocity was apparent in the final stats from Monday: Brazil had the ball 54% of the time, and had 19 shots on goal (compared to 12).

I’ve learned to enjoy soccer, in part, by turning off my instinctual aversion to turnovers.  When I watch soccer now, I am watching the semi-random kicking of an electron, which will tend (over time) to drift in one direction or the other, due to the superior ability of one of the teams.  It’s a game of statistical mechanics; it’s irrelevant whether you keep the ball continuously for any particular length of time.

 

2)      Low scoring.  To an American used to basketball scores like 95-92, or football scores like 35-28, soccer seems boring, in part because scoring is so rare.  But the “low scoring” of a soccer game should be taken in context.

For one thing, football isn’t as high scoring as you might think.  The average number of points scored by American football teams in 2013 was 23.4.  Consider that a touchdown (analogous to a goal in soccer) is worth a de facto 7 points (since the extra point is almost always successful).  To compare football scoring to soccer scoring in any meaningful way, football scores should be normalized by dividing by 7.  A score of 35-28 is analogous to a soccer score of 5-4.  High scoring, sure, but not overly so.  And a defensive battle like the Panthers/49’ers game last November, which ended with a Carolina victory of 10-9, is much like a soccer score of 1-1.

As for basketball, well, goals come so often that (individually) they lose almost all meaning.  I like basketball, but a soccer goal is much more exciting for being so rare.  Of course, it’s possible to make scoring too rare: I imagine that a game of Ullamaliztli was pretty boring indeed.  You can only use your hips, and have to get a 9 pound ball into a tiny goal?

Tlachtli

The losers are executed.

Which brings us to a tangential point.  Basketball is a very pixillated sport, since the “quantum of scoring” (one point) is so meaningless.  In soccer, the quantum of scoring (one goal) is a much, much bigger deal.  This makes soccer goals more entertaining, on a 1-1 basis, than  basketball goals; but it also means that you’re measuring the worth of individual teams with a very blunt instrument.  A football victory, 10-9, becomes a draw in soccer (when normalized) because the goals are not finely-tuned enough to “detect” a difference in such evenly matched teams.  Whether this is a good thing or not is up to debate.

 

3)      Red cards.  To an American, penalties are a common and necessary part of having a physical game.  But in soccer, the penalties seem very out of proportion to the offenses committed.

Consider a tackle in soccer.  It’s OK to tackle the opponent if I get my foot on the ball.  But if I miss the ball, I’m going to get penalized.  And if the referee thinks that I was trying to trip the opponent on purpose (a very subjective thing), I’ll get a yellow card waved in my face.  Two yellow cards equals a red card, and I’m out…and my team is now down one player.

Seriously?  Down one player for the entire game?

The same thing happens in ice hockey.  It’s called a power play.  And when the other team scores, the penalized team gets the player back.  The power play ends, and everything is fair again.  Why can’t it be like that in soccer?

I’ve always felt that your entire team losing a player for the rest of the game should be the nuclear option of penalties, such as if one of your players bites another on the shoulder.  It shouldn’t be used against a player that commits two ticky-tack penalties.  This is especially true in an era when diving (called flopping in the USA) has become a cottage industry.  Why not dive, when you have a good chance of ejecting a player from the game entirely?

o-LUIS-SUAREZ-570

In football, you have to do something egregious to get tossed out of a game, like throwing a punch.  Even then, your team is not down a player; a substitution is allowed.  In NBA basketball, you can commit up to 5 personal fouls; you’re tossed out on the 6th (this is called “fouling out”).  Again, when you foul out, your team isn’t penalized unduly…they put in someone else to take your place.

How does an American learn to accept the harshness of the red card system?

With difficulty, I admit.  I still don’t like it.  But I sort of understand it.  After all, how else can you penalize a team in a game in which there’s no stopping of the clock?  If players were allowed five, or four, or even three yellow cards before being tossed out, I daresay there would be more tripping, more pushing, more dangerous plays…and more injuries.  Then again, there would be less diving…

 

4)      Offside.  This might be the hardest aspect of soccer to fathom, to a person raised on Michael Jordan fast breaks and Dan Flutie Hail Mary passes.  Why do you penalize a team for having a player in scoring position?  Get rid of the offside penalty (the idea goes) and scoring would go up, and the number of exciting plays would increase.

Oh, who am I kidding.  I still hate the offside rule.

“But wait!” the soccer aficionado says.  “You get rid of offside penalties, and people will just park in the goal, waiting for a ball.  What’s the excitement of that?”

Um, that happens already.  It’s called a corner kick.  And corner kicks are exciting.

Sure it would change the game.  There would be no more beautiful offside traps.  Instead, there would be fast breaks.  Which is more likely to end up on a highlight reel: a well-executed offside trap, or a well-executed fast break?  I’ll let you decide.

Which brings me to soccer’s flaws (yes, it has flaws, just like every game and sport does.)  Not only should the offside rule be tossed out (or at least relaxed), but shootouts to decide a game are ridiculous.  Why?  Consider that a shootout contest has little relation to the actual game of soccer.  It is, if you will, a different (but related) sport entirely.  Settling a game with a shootout is like settling a basketball game with a free-throw shooting contest.  Why anyone thinks that shootouts are a good idea is anyone’s guess.  Sure, they can be exciting…but settling a soccer game with a spin of the roulette wheel would be “exciting” too—that doesn’t mean we should actually do it.  Just have extra periods until someone scores a golden goal.  And if you’re concerned with players getting too tired, well…there are a lot of players sitting over on that bench.  Don’t you think some of them would like a chance to play?

Ultimately, I like soccer, despite its flaws.  I’ve gotten used to the offside rule; I recognize it as a rule that purposely rewards passing and open-field play, at the expense of shots-on-goal.  It’s a choice, to make soccer a particular kind of game, no better or no worse than the (different) game you’d get without the rule.  Similarly, I’ve learned to embrace the shootout: they are rare, after all, and only occur after an extra period has failed to designate a winner.  In such a case, the teams are so evenly matched that we might as well use a flip of the coin.  And we’ll call that coin flip a shootout.

Note: I’ve made no mention of baseball in this discussion.  The reason?  Come on.  Baseball is just boring.

*********************************************************************************************

If you enjoyed this post, you may also enjoy my book Why Is There Anything? which is available for the Kindle on Amazon.com.

sargasso

I am also currently collaborating on a multi-volume novel of speculative hard science fiction and futuristic deep-space horror called Sargasso Nova.  Publication of the first installment will be January 2015; further details will be released on Facebook, Twitter, or via email: SargassoNova (at) gmail.com.

[Note: this is the article that I wish Nate Silver would write.  He’d be so much better at the number crunching.  Of course, he’s also paid a lot more.]

I’m tired of hearing pundits spew their opinions about this, that, or the other.  Let’s face it, most people don’t know anything.

Of course, that includes me, but hey: this is my blog.

I’m using hyperbole; it’s true.  And yet in a country where 9% of people think that space aliens may have caused the disappearance of Malaysia Air flight 370; in a country where 55.56% of Supreme Court justices are completely ignorant about the First Amendment; in a country where sizable numbers of people believe in a 10,000 year-old Earth, and where numerous people doubt facts like global warming, evolution by natural selection, and the supremacy of Matt Damon…in such a country, how can you really take anyone seriously?

OK; I take Neil deGrasse Tyson seriously.  But he’s earned it.

People (of every political ideology) spew forth talking points without any facts to back them up.  Hell, they spew forth talking points without any justification at all.  I have liberal friends who are against GMO foods…even though there is no reason to think they could ever be harmful, and in fact have already saved millions of lives.  (Trivia question: how did Norman Borlaug save a billion people from starvation and subsequently win the Nobel Peace prize?  Answer: genetically modified wheat.)

On the other side, all kinds of nifty-sounding talking points spew from the conservative font, again without even a shred of justification other than “well, that sounds right”.  For example, people claim that less government is good.  Shrinking government is a goal of Tea Baggers.  “Democrats want more government, Republicans want less government…everyone knows that.”  It’s become a cliche, and people don’t even question it any more.

But this is the information age.  We don’t have to rely on our gut feelings, or even our supposed “knowledge”, to evaluate claims like “less government is good”.  We have data.  Why don’t people look at the data and then make up their minds?

I’ll tell you why: because looking at data is hard work.  Let’s face it; most people just can’t do it.  Most people want to be told what to believe.  But I just got tenure, so I have some time.  Let’s try to get to the bottom of this.

It didn’t take me long to find this website, which has nifty (exportable) data on all kinds of economic indicators.  Hey, look, the USA is ranked 12th in economic freedom out of more than 165 countries.  Yay!  We’re doing OK.

What about the size of government?  This is harder to quantify, since it means different things to different people (for example, many Republicans want to “reduce” the size of the US government without touching our defense budget, which is a little like trying to lower world sea levels by draining the Mediterranean Sea).  However, the indicator “Gov’t Expenditure % of GDP” seems pretty good to me.

How does the USA do here?  Do we have a “bloated, huge, nanny state?”  Our spending is 41.6% of GDP.  This makes us rank 47th in this indicator, so about the 72-percentile.  We have a “bigger” government than about 72% of the countries on the list.

Who’s ahead of us?  The supposed “socialist” states are (Norway, Sweden, Findland, Denmark) of course.  Everyone “knows” they are entirely nanny states.  Also: France, the UK, Germany.  The usual suspects.  Liberal, bloated, big government monstrosities.

But also: Cuba!?  Libya!?  Bosnia!?  Iraq!?  Malta!?

Maybe Cuba fits well into the narrative.  Cuba is a Communist country, so of course the commies have huge governments.  (I personally think Cuba is an outlier, since its GDP is pitifully small).  But what about other Communist states?

Hmm.  The narrative is starting to break down.  China’s government spending is about 23.9% of its GDP, almost half the size of the US.  The commies in red China are spending half of what we spend.  Vietnam spends about 30.9% of its GDP.

The talking point that the USA has become a “bloated nanny state” doesn’t hold water.  We’re in the top one-third of spenders, it’s true; but our defense budget is Brobdingnagian to say the least; if you plotted “non-defense government spending as % of GDP” our rank would be much lower.  (Note: I lack the skills to do this…feel free to do so yourself.)

But all this is distraction.  Ultimately, I don’t even know if big government is inherently bad or good, and more importantly, you don’t either.  Admit it.  You’re just guessing.

But we don’t have to guess.  We can try to understand the data a little bit more.

It didn’t take long for me to stumble on the cute Where-to-be-born index, a kind of “happiness index” which takes multiple factors into account like quality of life, health, economics, and so on, in order to rank countries based on where you’d prefer to be born.  (Admit it: you’d rather be born in Australia than Bangladesh.)  So hey, I know how to use Excel: let’s plot Size of Government vs. Where-to-be-born and see if there’s a correlation!

Correlations

Firstly: there’s not much of a correlation (the R^2 value is only around 0.17).  This is not surprising; the size of government has little to do with anything.  (It certainly shouldn’t be the entire frakking basis for a political movement.)  But what correlation there is, is positive, meaning that as governments get bigger, people tend to get happier.  All the Viking countries are in the upper right.  And those Scandanavians are doing well, dammit!

Some countries stand out.  Russia and China are lower and to the left of the USA, meaning they have smaller governments (is that surprising?) but are also more miserable.  Cuba, though, is a huge government spender…why is that?  Also, what’s the deal with Singapore?  They’re happy and (apparently) almost an anarchist state.  Tea Baggers take note: emulate Singapore.

The main idea I want you to get out of the graph is this:  there’s a lot of noise there.  You can’t really draw any conclusions.  If you want to say that “Big Government Bad!” in the same way that Tarzan says “Fire Bad!” you’d better have some data to back up your claim.  And finding such data is, well, hard work.  Good luck.

And now I’d better get back to studying something I know about: physics.

 

 

 

 

 

Theology?

Picture1

grrm2

grrm3