It turns out that I couldn’t stay away.

About 15 months ago I decided that it was time to close down this blog.  My reasoning, more or less, was that I needed to be more “serious.”  I had just finished my PhD and started my first postdoc, and I reasoned that if I really hoped to “make it” as a physicist then I couldn’t afford to waste time writing long, rambling posts about physics and semi-physics topics that are outside of my real research.

But I have missed blogging during the last year+.  And I have come to realize that blogging was more than just a fun way to spend idle time.  In fact, I think blogging provided me with something really valuable that I will need going forward.

It seems to be true, at least for me, that the only way to really learn something is to teach it to someone else.  From my perspective, teaching an interesting idea to someone else has three important effects on the teacher:

• First, teaching forces the teacher to sharpen their own thinking: to identify the features of the idea that are most essential, to develop multiple parallel ways of understanding and explaining the idea, and to tie the idea firmly to a wider base of knowledge.
• Second, teaching cements the idea in the teacher’s own memory.  There is no better way to learn a story than to become the storyteller.
• Third, and perhaps most importantly, teaching allows one to reconnect in a personal way with the excitement behind the idea being taught, and to rekindle one’s love for the topic.

In short, this blog has been my outlet for teaching ideas that I love.  And I have realized that such teaching is immensely valuable for me, not just as a hobby, but as a tool for professional and personal development.  I love physics, and I want to make a career as a physicist.  This is a surprisingly daunting goal sometimes, but, in my final analysis, it turns out to be precisely the reason why I need to “waste time” blogging about physics.

So Gravity and Levity is back!  Try to contain your excitement.

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I think the time has also come for me to make a slight shift in policy.  I had originally imagined that Gravity and Levity would serve as a locus for conceptual discussion of ideas in upper-level physics, and for this the idea of preserving my own anonymity (and that of other commenters) was valuable.  Physics students are often quite insecure, after all.

But now I have come to understand that this blog is by necessity a very personal endeavor.  To be simplistic, Gravity and Levity is not really a blog about physics; it is a blog about myself and the way I think and feel about physics.  And so it makes sense to acknowledge that personalness directly, and to explicitly tie this blog to my own identity.  It was never much of a secret anyway.

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So let me introduce myself.

My name is Brian Skinner.  I am a 29-year old postdoc in theoretical condensed matter physics at the University of Minnesota, where I also completed my PhD.  My undergraduate years were spent at Virginia Tech, where I studied physics and mechanical engineering.  My childhood was spent in lots of  different places, since my father was a fighter pilot in the US Air Force and our family moved every year or two.  I have an arthritic right knee and a receding hairline.

To the right is a picture of me looking like I’ve never seen a camera before.

One benefit that will perhaps come of introducing myself directly is that it will allow me to communicate in a more personal way the uncertainty and fear that comes from trying to make a career as a scientist.  My job is one that makes me feel inadequate every day.  More or less every day I feel insufficiently intelligent, insufficiently motivated, and insufficiently hardworking to achieve my goal of becoming a competent physicist and/or physics professor.  And I suspect that many other hopeful scientists feel this way.  I truly don’t know whether (or to what degree) I will “make it” as a physicist, but perhaps some public documentation of my own attempts to do so will provide a bit of catharsis to others who feel similarly inadequate.

Finally, I think the future of this blog will also contain less hesitancy about getting “off topic.”  To whatever readers I may have, be warned that I intend to consider Gravity and Levity as my outlet for discussing and developing any ideas that seem interesting and/or profound to me.  Such ideas will mostly have to do with physics, but I consider myself to have no allegiance to any particular discipline or professional banner.

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I look forward to the future of G&L!  Thank you to all of you who read it.  It is a pleasure to meet you.

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[The quote in the title of this post comes from Terry Pratchett's novel Small Gods, which I have never actually read.]

On that final note, I think it’s time to bring this blog to a close.

Now that I’ve moved on from graduate student to post-doc, my priorities have shifted, and you may have noticed that I haven’t been posting enough to keep this blog respectable.  This seems like more of a permanent change in my work habits than a temporary increase in busyness, so I think that now is the right time to close down Gravity and Levity.

It’s been a tremendous amount of fun, and along the way I somehow attracted a sort of dazzlingly intelligent set of readers leaving intelligent comments.  So thank you.

I leave unfilled a fairly significant list of half-started blog posts and half-developed ideas for future posts.  So if anyone ever wants to invite me to write a guest blog post, I will likely be highly tempted to do so.  If for whatever reason you would like to make such an invitation, feel free to say so in the comments and I will respond by email.

Since this post will sit at the top of Gravity and Levity for the forseeable future, I’ll close with a list of my 16 personal favorite blog posts.  Thanks again, everyone.

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Parenting and the feeling of time: My eight lifetimes

In which I speculate about how we live in logarithmic time.

The fastest possible mile

I search for an asymptote in the progression of the mile word record and come up with 3:39.6.

Finding the hot (and cold) hand at a local gym

With a statistical analysis rebuked, Mrs. G&L and I head to the gym in search of the “hot hand.”  We find instead only evidence for the cold hand.

When Nature plays Skee-ball: the meaning of free energy

I explain free energy by imagining a four year old girl playing Skee-ball.

Braess’s Paradox and the Ewing Theory

An analogy between highway traffic and basketball might explain why your favorite team can get better when its best player is sitting out.

Friedel oscillations: wherein we learn that the electron has a size

Wherein Friedel oscillations are explained using the following sentence: “It’s a bit like letting the richest men in America decide the tax code: it may be right for the guys up front, but it’s too damn much for the people that come later!”

The most important idea in science, and why it’s true

Explaining atomism and the Lennard-Jones law using cheap hand drawings and a youtube video.

Does your culture really affect the gender distribution?

In which I play King Solomon with a suggestion made by science author Matt Ridley.

Your body wasn’t built to last: a lesson from human mortality rates

What we can learn about the human body from the mathematics of mortality.  This post is responsible for more than 1/3 of all web traffic to G&L.

The path integral: calculating the future from an unknown past

Using a life-or-death situation for ants to illustrate the power of path integrals.

A story about quasiparticles on the beach

A good story in science will affect your summer vacation.  In the best possible way.

Being pushed around by empty space: the Casimir effect

Dancing witches produce the Casimir effect.  True story.

“So is the universe made of tiny springs or isn’t it?!”

A memory of being exasperated in quantum field theory class.  Also, Freeman Dyson is much smarter than me.

Feynman’s Ratchet and the perpetual motion gambling scheme

Can you spot a (thermodynamic) scam when you see one?

LeBron James and the Lottery in Babylon

Jorge Luis Borges explains beautifully why we are so drawn to sports.

This is not a story about irony

In which I remember “Paul”, who felt strongly about his calling in life.

In the abstract, there is something remarkably glamorous about being a theorist.

Writ large, my job is to arrive at fundamental truths about the universe just by thinking about them.  I am then supposed to codify these truths in the form of mathematical equations, which (supposedly) state in an absolutely unambiguous way what the new true idea is.

It occurs to me every now and then what an outrageous job description this is.  Saying to someone that “my job is to arrive at fundamental truths just by thinking about them” fills me with something like embarrassment and trepidation, but also with a very real sense of excitement.  “Physicist” is definitely a profession that appeals to the human urge to make a permanent mark on the world during one’s finite life span.

I remember one particular moment near the beginning of graduate school when I was caught up in this kind of ego-driven excitement, thinking that in the near future I would actually derive a new law of nature.  I tried to explain to (the soon-to-be) Mrs. G&L how cool this was, and how it was conceivable that in the near future I would have my own equation.  I think she was somewhat unconvinced, but not wishing to discourage my enthusiasm, she said that when I published my first paper we would celebrate by purchasing a printed copy of the journal.

It wasn’t too long before I did publish my first paper.  But somehow I didn’t feel like celebrating.  It was a fine paper, one that offered a potential explanation of a surprising experimental result.  But somehow I had imagined that a published paper would be something more convincing.  I guess that when your only perspective on physics comes from taking physics classes, you get this sense that a derived equation is some irrefutable piece of truth that you can immediately go out and use to build airplanes or computers or particle accelerators.  But this paper had no such “irrefutable piece of truth”.  It was just a proposed explanation for something, one that was probably just as likely to be non-useful as useful.

At this point, almost 4 years and more than a dozen papers have passed since my first publication, and I still haven’t purchased any printed copies of any journals.  None of my papers have lived up to that lofty initial ideal: an unimpeachable derivation of some new law of nature.  It’s true that, in principle, one of the many equations in one of those papers could receive some terrific experimental confirmation, be adopted in a widespread way, and become known as “Skinner’s law”.  But it’s not particularly likely.

Which, it turns out, is fine.  And it doesn’t mean that I did anything wrong or dishonest.  It just means that creating a new theory is a more human process than I gave it credit for.

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There’s a line from Robert Penn Warren’s All the King’s Men (a beautiful and distinctly American novel) where a crass and skilled politician is discussing his craft:

Man is conceived in sin and born in corruption and he passeth from the stink of the didie to the stench of the shroud. …

If you want goodness, you got to make it out of badness…  And you know why?  Because there isn’t anything else to make it out of.

The protagonist of the novel answers with

If, as you say, there is only the bad to start with, and the good must be made from the bad, then how do you recognize the good?

To which the politician replies:

You just make it up as you go along.

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Sometimes I think about All the King’s Men when I’m doing my job.  Because it turns out that a new theory about the world almost never arrives like a karate chop of enlightenment.  The world is a complex place, and it’s impossible to simultaneously take into account everything that enters a problem.  The best you can do is envision in your mind some cartoon version of the world and then solve that cartoon problem to the best of your ability.  Then you present your idea to others, being careful to acknowledge all the ways in which it is faulty, and making clear all the ways in which your cartoon world is different from the real world.  In most cases, if your theory ends up being a highly accurate and highly useful description of reality, then you should feel more fortunate than intelligent.

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My advisor has a more concise way of (unwittingly?) paraphrasing Robert Penn Warren: “When you make a theory, you’ve got to start with shit, because that’s usually all there is.”

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I don’t think there’s any real problem with the process by which theories are made.  But it has occurred to me lately there is something of a problem with the way the public perceives the process.

Recently, I wrote a paper called “the problem of shot selection in basketball“.  This paper is not a “karate chop” in any sense; the jury is still out as to how directly it can be applied to improving the performance of basketball teams.  It’s very possible that nothing will ever come of it and eventually it will be completely (and rightfully) disregarded.  But it constitutes my attempt to make a theory in a place where there was no theory.  In order to make it I had to cartoonify basketball to the point where I could solve some things, then I solved them, then I tried to discuss where the theory was lacking and where it might be improved.  It was a very fun paper to write, now that I have appropriately lowered my expectations about what a theory is.

The problem is that lots of people, upon hearing of this paper, generally thought about it using the same perspective about “published theories” as I had had 4 years ago.  Consequently, reactions to the paper largely fell into two categories: people who were amazed that I could “solve basketball”, and people who considered it an obvious and stupid fraud that I was perpetuating.  For example, one write-up about the paper had the headline “Hoops dilemma solved by physics whiz?“; on Reddit it was called “imaginative BS“.

I actually object much more to the former characterization than to the latter.

If I’m doing my job correctly, then I will throughout my career produce plenty of pieces of “imaginative BS”.  Imagination is an essential thing for establishing new understanding about the world.  And as long as I acknowledge all the ways in which a proposed theory might be “BS”, then the scientific process is working.

And perhaps this is an important thing that the public should know.  Having the freedom to create BS is an important part of science.  The only sin is pretending to produce absolute truth from a limited skill set and a finite imagination.

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Footnote:  “The problem of shot selection in basketball” is a sort of speculative paper, in the sense that the cartoon world it describes is different from the real world in a couple of very notable ways.  But it’s funny to me how the perception of it is different from similarly speculative papers that I’ve published for a physics audience.  One such paper, for example, got the following review from a referee:

Although the model used seems to be rather artificial, it is very propaedeutic and allows one to catch the main features of the process.

Translation: “The problem the authors are solving is not much like the real world, but you can still learn something from it.”

And yes, I had to look of the word propaedeutic in the dictionary, too.

On Friday I defend my PhD thesis, entitled “Microscopic Theory of Supercapacitors”.

“Supercapacitor” is something of a marketing term (the more technical term is “electric double layer capacitor”), but the word is becoming pretty widely used to describe a particular class of devices.  Searching Google Scholar for “supercapacitor”, for example, gives more than 10,000 hits during the last 4-5 years.

Of course, putting “super” in front of a word is (apparently) enough to generate interest from scientists (e.g.  superconductivity, supernovae, superfluidity, supersymmetry, superstring theory…), but a thesis defense is supposed to be presented to the general public.  So I decided that my thesis topic needed just a little bit of extra pizzazz.  Namely, it needed a great logo:

Coming August 12, 2011

Now that should generate public interest, right?

I should make the logo into a skin-tight, long-sleeved shirt that I wear whenever I’m called to work on supercapacitor theories.

Did anyone call for high-power, infinitely-rechargeable electrical energy storage?

I never intended for Gravity and Levity to have so much talk about basketball.  But it’s becoming increasingly clear that my hobby (nerdy problems in basketball) is much more popular than my real job (nerdy problems in “real” life).  Sadly, my real research has never been written up in Science News.

As it turns out, though, there really are a surprising number of interesting theoretical problems that come up when thinking about basketball.  The answer to the question “what is the best strategy for my team?”, for example, includes elements of network theory and game theory, in addition to good old-fashioned probability theory.  And if you’re a nerd like me as well as a basketball fan, those are pretty exciting things to dabble in.  So basketball continues to be a source of fun for me even though I was never good enough to play it beyond the high school level.  Probably not as fun as actually being a basketball player, but I suppose that in life we each work with what we’re given.

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A month or so ago I was on an airplane and feeling restless, so I started thinking ambitious thoughts about constructing a great optimization theory for basketball.  As I thought more, though, I realized that there were some very basic questions to which I didn’t know the answer that would have to be addressed first.  This process of slowly realizing the depths of my own ineptitude continued, and after multiple rounds of mental simplifying I finally ended up with a comically over-simplified model of decision-making in basketball that I still couldn’t immediately write down the answer to.  Specifically, I came up with the following question:

Suppose that your team runs an offense with exactly one play.  Every time your team has the ball, you run the play, and at the end of the play you arrive at a shot opportunity.  The quality of that opportunity varies: sometimes you end up with an easy layup that is almost certain to go in, and other times you end up with a contested jump shot that has essentially no chance.  At the moment of the shot opportunity, your team has to make a decision: should you take the shot or should you reset the offense and run the play again in hopes of arriving at a higher-percentage opportunity?

Now suppose that the quality of the shot opportunity, as characterized by the probability p for the shot to go in, is randomly distributed between 0 and 1.  How good must be the probability p for the shot to go in before you should take it?

It sounds like a pretty simple question, right?  It’s tempting to say right away that you should shoot whenever $p > 0.5$, which is somehow equivalent to saying “take only above-average shots”.

But the right answer is actually a bit more complicated.  For one thing, the answer depends very much on how much time you have (say, before the shot clock expires).  If you have a lot of time, for example, you can just keep resetting the play whenever it doesn’t give you an easy layup.  Having more time gives you a greater capacity to be selective.

So, generally speaking, suppose that you have enough time to run the play $n$ times.  What is the lower cutoff $f(n)$ for the shot quality such that your team should shoot whenever the shot opportunity has $p > f(n)$?  This function, $f(n)$, is what I call “the shooter’s sequence”.  Posing the problem of $f(n)$ seemed so simple when I first wrote it down on the airplane; I was surprised that I couldn’t immediately say what the answer was.

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n = 1

Actually, if you start at the beginning, it’s not hard to write down some answers right away.  For example, if your team has only enough time to run the play once, then you should take any shot that presents itself.  This means that when $n = 1$, you should shoot whenever $p > 0$.  So the first term in the sequence $f(n)$ is

$f(1) = 0$.

As a consequence, your average shooting percentage will be equal to the average shot quality $1/2$, so that you score half the time.

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n = 2

Now suppose that you have time to run the play twice.  If you don’t like the first shot opportunity that comes up, you can reset and run again.  If you choose to reset, you can expect to score half the time (as explained above).  So, it’s not hard to see that the first shot opportunity should have $p > 1/2$ in order to be worth taking.  This means

$f(2) = 1/2$.

If the team shoots only when $p > 1/2$, then on the first attempt its shooting percentage will be $3/4$ [the average of the interval (1/2, 1)].  Thus, the team’s combined shooting percentage for whole possession is:
(Probability that the team will shoot the first time around) $\times$ (Team’s shooting percentage on the first attempt) $+$ (Probability that the team will hold and wait for the second play)$\times$ (Team’s shooting percentage on the second attempt)  $= 1/2 \times 3/4 + 1/2 \times 1/2 = 5/8$.

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n = 3

Now it should be fairly easy to see that if you have enough time to run the play three times, on the first attempt you should only take shots with $p>5/8$.  Because if the shot attempt on your first time through is worse than that, you can hold and with enough time for two plays you will on average get a shot whose chance of going in is $5/8 = 0.625$.  This means

$f(3) = 5/8$.

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large n

In this way you can building a recursive sequence for finding out when your team should shoot given that there is enough time for $n$ plays.  The general recursion rule looks like this:

$\textrm{(lower cutoff for shot }n\textrm{)} = \textrm{(average scoring rate for }n-1\textrm{ shots)}$

$f(n) = \{1 + [f(n-1)]^2\}/2$.

Along with the condition $f(1) = 0$, this equation defines the “shooter’s sequence”.  Writing down the first few terms is actually kind of surprising:

$f(1) = 0$

$f(2) = 1/2$

$f(3) = 5/8$

$f(4) = 89/128$

$f(5) = 24305/32768$

$f(6) = 1664474849/2147483648$

$...$

$f(\infty) = 1$

It’s a pretty strange sequence of numbers to come out of such a simple problem, right?  And, in fact, there is no analytical expression for the sequence.  It can only be defined recursively, or evaluated approximately for large $n$.

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And, of course, you can expand on the problem by making the problem statement a little less specific.  I ended up calculating some more variations, including the effects of turnovers and different ranges of shot quality, and eventually it became a short paper (although I’m still not sure what kind of journal, if any, would publish it).

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It’s still far from clear how much (if any) practical knowledge can be gained from this “shooter’s sequence”.  But I know that at moments like this I’m glad that I grew up to love nerdy problems in math and physics.  Because if you like this sort of thing, then fun problems are everywhere.

Just look around you:

Why are the players at a sporting event more famous than the halftime show performers?

This question, which was posed in a recent column on ESPN.com (and which features one of my personal favorite performers, the Red Panda Acrobat), sounds a little dumb at first, almost tautological — the halftime show is, by definition, not the main event.

But try this slightly different version of the question: How many professional athletes can you name?  How many circus performers can you name?  Why are those two numbers so different?

If you were observing the human species from afar, you would probably be surprised by the discrepancy.  After all, both categories of people are entertainers.  Both base their livelihood on some elite level of physical skill.  And it’s hard to believe that athletes have more natural talent or devotion to their craft than, say, this guy.  But we (myself included) seem to be much more captivated by athletes than by circus performers.  Why the big difference?

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There are lots of ways to explain why humans love sports so much.  My personal favorite, though, can be found in “The Lottery in Babylon”, a short story written by Jorge Luis Borges in 1941.  I came across this story during high school and was immediately hooked by this killer first paragraph:

Like all the men of Babylon, I have been proconsul; like all, I have been a slave. I have known omnipotence, ignominy, imprisonment. Look here– my right hand has no index finger. Look here–through this gash in my cape you can see on my stomach a crimson tattoo–it is the second letter, Beth. On nights when the moon is full, this symbol gives me power over men with the mark of Gimel, but it subjects me to those with the Aleph, who on nights when there is no moon owe obedience to those marked with the Gimel. In the half-light of dawn, in a cellar, standing before a black altar, I have slit the throats of sacred bulls. Once, for an entire lunar year, I was declared invisible–I would cry out and no one would heed my call, I would steal bread and not be beheaded. I have known that thing the Greeks knew not–uncertainty. In a chamber of brass, as I faced the strangler’s silent scarf, hope did not abandon me; in the river of delights, panic has not failed me. Heraclides Ponticus reports, admiringly, that Pythagoras recalled having been Pyrrhus, and before that, Euphorbus, and before that, some other mortal; in order to recall similar vicissitudes, I have no need of death, nor even of imposture.

I owe that almost monstrous variety to an institution–the Lottery– which is unknown in other nations, or at work in them imperfectly or secretly.

(You can read the full text here.)

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“The Lottery in Babylon” is, like so many Borges stories, an exploration of a mathematical idea in a fantastical setting.  This particular one (it seems to me) is about how we perceive chance and randomness in life.  In the story, The Lottery is an all-encompassing, secretive, cult-like religious institution that dictates almost every facet of people’s lives in a way that is supposedly both random and just.

To me, the most interesting part of the story is the telling of how The Lottery came to be.  The narrator speculates that The Lottery began just like any other simple gambling game: some people bet money and a few won based on the roll of a die.  But:

Naturally, those so-called “lotteries” were a failure. They had no moral force whatsoever; they appealed not to all a man’s faculties, but only to his hopefulness.

In the story, the key breakthrough for the lottery system is the inclusion of both prizes and penalties as the outcomes of lottery drawings.  Knowing that a lottery player risks some kind of serious punishment changes entirely the emotional tone of the lottery.  That is, playing the lottery becomes not just a matter of blind hope for gain, but something that involves courage and one’s sense of justice.

Babylonians flocked to buy tickets. The man who bought none was considered a pusillanimous wretch, a man with no spirit of adventure. In time, this justified contempt found a second target: not just the man who didn’t play, but also the man who lost

The rest of “The Lottery in Babylon” describes how The Lottery grows increasingly out of control: penalties and prizes become more severe and fundamentally non-monetary in nature, all drawings and decisions are conducted in secret, playing the Lottery becomes compulsory, etc.  If you have any nerdy inclinations at all, the story is a pretty good investment of ten minutes of your leisure time.

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But I have been thinking lately about “The Lottery in Babylon”, and about our love of sports.  It seems to me that we love sports for a lot of the same reasons that the Babylonians loved The Lottery: it appeals not just to our sense of hope — as in, “Oh, I hope my favorite team wins” — but also to our sense of morality.  A sporting event is, to a very large extent, a controlled series of random events.  But when fans talk about sports we don’t do it because we love probability theory (okay, maybe a small fraction of us have this motivation on occasion).  We talk about sports because we love to exalt the virtues and decry the vices of the players.  We love to watch our heroes succeed.  We love to watch our villains fail.  We love to see how normal people become heroes and villains through competition.

But the truth is that all these athletes are, to a very large extent, playing a lottery.  Those virtues and vices we talk about are largely things that we project onto the players based on random events.  As Matt at hooptheory said, “The biggest fallacy in sports is that the better team will win.”  In a sense, our athletes are so famous and so well-paid because we pay them to live a life where their glory or ignominy is dictated by random events.  Watching those events unfold and judging the character of the participants is entertaining and moving to us for some profound psychological reason (that I don’t understand).

In contrast, when you go to watch a circus performer, there is no real chance of failure.  Sure, the performer might fall briefly or drop a juggled ball.  But ultimately, you can be pretty assured that their performance will make only one point: you get to be awed by some display of great skill.  Unlike sports, the performance does not appeal to your sense of justice or of character development.  You, as a spectator, have no real opportunity to judge the character of the performer: the show has only (apparent) heroes, and no villains.  The circus, in other words, is like a “lottery” without penalties, and as such it can only appeal to a limited range of emotions.

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These days, I think in particular about LeBron James.  Now that the mighty Miami Heat have fallen in the NBA finals, LeBron is considered something of a “pusillanimous wretch”.  His biggest sin, committed on the biggest stage, was an apparent refusal to play the game as wholeheartedly as we wanted him to.  It seems to me, though, that had a few shots gone differently (say, if Dirk Nowitzki’s layups at the end of Games 2 and 4 had rolled out), we would be talking about LeBron James very differently right now.  So many hundreds of articles decrying Dallas’s heroic nature and Miami’s flawed nature would be completely reversed.  In other words, our sense of the “true character” of the participants in this sporting contest is based very largely on the outcome of two shots.

As a more extreme example, you can notice that Michael Jordan would have lost his most famous game had it not been for two blown shot-clock violation calls earlier in the game.  Without that last-second, out-in-a-blaze-of-glory, second-threepeat-winning shot, the ultra-heroified Michael Jordan might have a significantly different public image.

What if this shot hadn’t mattered at all?

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All this is not to say that we are somehow barbarians or idiots for being sports fans.  I personally love watching sports, and I love the process of evaluating the character of the players involved, and I don’t feel at all guilty for that.  For whatever reason, it’s an endlessly enjoyable process, and for the most part no one is being exploited in the name of my entertainment.  But I do think it’s good to be aware of what it means to be an athlete and what it means to be a spectator.  Namely, that we pay to watch our athletes play a modern-day version of the Lottery in Babylon.

…I would definitely include the following problem:

You live in the dorms and your upstairs neighbor, LeBrian Skinner, is a serious basketball player.  He is about to declare for the NBA draft, but he fears that his merely average height will put him at a disadvantage.  To compensate for his relative shortness, LeBrian decides that he needs to have a vertical jump of at least 36 inches.

In the evening you can hear LeBrian practicing his vertical leap, since he lives directly above you: you hear a loud creak when he first jumps followed by a loud thump when he lands again.  You use a stopwatch to time the interval between the moment he first leaves the floor and the moment when he lands again.  You measure this interval as 0.8 seconds.

Assuming that LeBrian lands with his legs fully extended (in the same position as when he leaves the floor), how high is he jumping?  Is it enough?

For those who are curious, the solution is after the page break.

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