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Posts Tagged ‘metaphors’

Every Halloween, a steady stream of trick-or-treaters visits my house.  They’re looking for handouts, of course: Snickers, Pixy Stix, Reese’s, jawbreakers.  A simple mathematical model of the children’s visits can help explain the Doppler effect.

zombieland-princesses

Let’s say the children are dressed as zombies (zombies are all the rage).  Let’s further assume that the children act like zombies—not from premature candy consumption and subsequent hyperglycemia—but from a desire for greater verisimilitude.  Thus, we make the following assumptions:

  • The children move at a uniform speed v.
  • The children are separated by a uniform distance λ.

With what frequency f do the children visit my house?

It’s obvious that a greater speed v means that that children visit more often.  So:

f  v,

meaning that frequency is proportional to speed.  Further, it seems clear that a larger distance λ between children means less visits, so that

f  1/ λ.

It is logical to take these proportionalities and combine them, giving

= v/ λ.

In the study of waves, this is the fundamental relation between frequency and wavelength.  We see that in this analogy, the zombie children are meant to represent successive peaks of a wave, and the distance between the children represents wavelength.

(I haven’t defined what a “wave” actually is.  If you’re curious, a wave is something that is periodic in both space and time.  The space periodicity of the zombie kids is codified by the number λ, since if you travel a distance λ in space, you get another zombie kid just like the first.  The time periodicity is codified by the period T = 1/f, since if you’re at the house and you wait for time T, an identical zombie kid will show up at your door.)

So, what about the Doppler effect?

doppler

Sheldon models the Doppler effect.

We now imagine that I live in a mobile home.  ( I don’t, actually, but I was injured in a tornado in 2011, so I guess I am an “honorary” mobile home denizen.)  What is the effect of me driving the mobile home either towards or away from the zombie kids?

Suppose I drive towards the kids at a gentle 1 m/s.  If they are walking 2 m/s towards me, our relative speed is 3 m/s.  The kids show up at my doorstep more frequently.

If instead I drive away, the kids aren’t visiting as often, since our relative speed is now just 1 m/s.  In fact if I drive away at 2 m/s or greater, the kids never catch me and f drops to zero.

The same thing happens with sound.  Regions of less dense/more dense air propagate from a source to a receiver (presumably, your ears).  Each “pulse” is analogous to a zombie kid, and the frequency with which the pulses jostle your eardrums is interpreted by your brain as a pitch.  Higher frequency, higher pitch.  Now, in air the speed of sound is roughly constant, so the f that you hear depends upon one thing: the wavelength.  The more separation between pulses, the lower the pitch you hear.

You can guess what comes next.  If you run away from a sound source, the pulses can’t hit your eardrums as often; you hear a lower pitch.  Conversely, running towards a sound source makes the pitch sound higher.  The f has increased.

Of course its not just the house (the receiver) that can move; the source of the sound (i.e. the zombie kids) can move as well.  I can implement this idea next October 31 by installing a moving walkway outside my house.  If the kids walk at 2 m/s on my moving walkway, but the walkway itself is set at 1 m/s towards me, then the kids are actually moving at 3 m/s relative to me and will visit more often.  (A moving sound source such as a siren sounds higher in pitch if moving towards you.)  I could likewise set the walkway to move away from me, and have the kids visit less often or not at all.  (A siren traveling away from you sounds lower in pitch.)

This analogy is not perfect by any means.  For one thing, the zombie children (who are actual, flesh-and-blood material objects) represent wave maxima, which are mathematical abstractions.  In the case of sound, if I shout and you hear it, that does not mean that any actual physical object traveled from me to you.  A series of wave pulses traveled, sure; but no individual atoms or molecules went all the way across the room.  In a typical wave, energy travels from A to B but matter does not.

If you have trouble seeing how this can be, recall the wave (also known as the Mexican wave) that appears in large sports stadiums.  Wikipedia says it best: “The result is a wave of standing spectators that travels through the crowd, even though individual spectators never move away from their seats.”  Similarly sound can travel from me to you, even though the individual oxygens and nitrogens don’t really move that far.

Which brings up another idea of mine, which I’d like to patent.  Waves (in football stadiums) are always transverse: people raise their arms and then lower them, in a direction perpendicular to the motion of the wave itself.  But if we really wanted to model sound with a stadium wave, we should instruct the audience to move their arms from side to side.  This would set up a longitudinal wave.  I think it would look a little different.

Next time you’re directing festivities for a crowd of 10,000+, get them to do a longitudinal wave.  You’ll thank me for it.  And if you have kids of an appropriate age, dress them as zombies this Halloween.  At my house, zombie trick-or-treaters get the best candy.

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Laser_squirrel_hd

An image from Zynga’s Mafia Wars, obviously.

One way to say it is…

Let’s say we have a bunch of squirrels congregating underneath a tree.  We can startle the squirrels, and they jump up into the branches of the tree.  But these are ground-loving squirrels, so they very quickly jump back down—sometimes to a lower branch, sometimes all the way down to the ground.

An interesting fact: the squirrels scream as they fall.  This isn’t because of fear, so there’s no need to alert PETA; they’re just emitting squeals of delight as they plummet.  And the frequency of their squeals (and hence the musical note produced) depends upon how far they fall.  A larger drop, a higher frequency.

My goal is to have a whole bunch of squirrels scream with the exact same frequency, all at once.  How can this be done?

Clearly, I need a bunch of squirrels all on the same branch, and I have to hope they all jump off that branch all at the same time.  But this is trickier than it sounds.

It turns out that our tree, a northern elm (Ne for short) has lots of branches, most of them slippery.  In most cases a squirrel will jump off such a branch almost immediately.  However—and this is lucky for us—the 8th branch from the bottom is not-so-slippery.  Squirrels actually like to hang out on this branch for a little while.  Squirrels, being squirrels, do succumb to peer pressure, though, so when one squirrel eventually jumps to the next-lowest branch, the rest of the squirrels follow suit—all screaming in unison—producing a nice, loud, resonant scream of 4.7 x 1014 Hz.

Here’s the problem: when we initially scare the squirrels into the tree, they don’t all jump up to that 8th branch.  Why would they?  They jump up at random, and only a fraction land on the branch we want.  Even if a sizeable number then jump down all at once, producing the desired sound, it is drowned out by all the other screams and squeals of all the other jumping squirrels on all the other branches.  This isn’t what we want.

But maybe we can be super clever.  Let’s get another tree, let’s say a hemlock (He for short), and place it next to the Ne tree.  Why hemlock?  The cool thing about hemlock is that there’s basically only one branch that squirrels can reach (the other branches are just too high).  And here’s the luckiest coincidence of all: this single branch of the He tree is at almost the exact same height as the 8th branch of the Ne tree (the branch that the squirrels kind of like).

So here’s what we do.  We scare a bunch of squirrels beneath the He tree, and they all jump up to that lone branch (they don’t have a choice.)  We then slide the He tree next to the Ne tree.  The naturally curious squirrels climb over to the Ne tree, because the branches are at the same height.  And guess what—the conditions are now just right for our squirrels-screaming-in-unison trick!  We have a large population of squirrels on a not-so-slippery branch, and when the peer pressure clicks in—eeeeeeeekkkk!

Another way to say it is…

Let’s say we have a bunch of electrons in atoms in a gas.  We can excite the electrons with an electric field, and they jump up into higher atomic energy levels.  But being electrons, they very quickly jump back down—sometimes to a lower energy level, sometimes all the way down to the ground state.

An interesting fact: the electrons emit photons as they fall.  And the frequency of these photons depends upon how far they fall.  A larger drop, a higher frequency.

My goal is to have a whole bunch of photons emitted with the exact same frequency, all at once and in phase.  How can this be done?

Clearly, I need a bunch of electrons all on the same energy level, and I have to hope they all jump off that level all at the same time.  But this is trickier than it sounds.

It turns out that our gas, neon (Ne for short) has lots of energy levels, most of them “slippery”.  In most cases an electron will jump off such a level almost immediately.  However—and this is lucky for us—the 8th branch from the bottom (the 5s energy level) is metastable.  Electrons actually like to hang out on this level for a little while.  Electrons, being electrons, do succumb to peer pressure, though, so when one electron eventually jumps to the next-lowest branch, the rest of the electrons follow suit—in a process called stimulated emission—producing a nice, intense, in-phase cascade of photons with f=4.7 x 1014 Hz (about 633 nm, which is ruby red).

Here’s the problem: when we initially excite the electrons in Ne, they don’t all jump up to that 5s level.  Why would they?  They jump up at random, and only a fraction land on the level we want.  Even if a sizeable number then jump down all at once, producing the desired lasing frequency, it is drowned out by all the other photons emitted by all the other jumping electrons on all the other levels.  This isn’t what we want.

But maybe we can be super clever.  Let’s get another gas, let’s say helium (He for short), and place it in the same container as the Ne.  Why helium?  The cool thing about helium is that there’s basically only one energy level that electrons can reach (the 1s2s level).  And here’s the luckiest coincidence of all: this single energy level of He is almost the exact same energy as the 5s state of Ne.

So here’s what we do.  We excite a bunch of electrons in He, and they all jump up to the 1s2s level  (they don’t have a choice.)  We then mix the He with Ne.  Through collisions, many of the electrons in the 1s2s level state of He are transferred to the 5s state of Ne.  And guess what—the conditions are now just right for our cascading electrons trick!  We have a large population of electrons in a metastable state (a population inversion), and when stimulated emission clicks in—we get coherent laser light!

Whether or not those electrons raid your bird feeders for sunflower seeds is another issue entirely.

[Note: my book Why Is There Anything? is now available for download on the Kindle!]

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Somewhere in the wilderness of Toelek is a store.  Inside the store are clerks.  The clerks sell bags of Wavy Lay’s potato chips:

Wavy Lays Original

Now, the thing is, if you give money to clerks to buy some chips, the clerks never give any money back: they’re greedy.  What’s more, they leave the store immediately with whatever money they have left over from the transaction.  However, the clerks don’t always get very far, because out behind the store is a bridge that the clerks have to cross, and the bridge is guarded by a troll named Voltar.  The troll is greedy, too.  He demands a toll, and if a clerk can’t pay up then he can’t cross the bridge.

Toelek is a weird place.  Its citizens have a very rigid society, and people are required to wear differently colored clothing depending upon how much money they have in their pocket.  The money is always in multiple of 50 cents.  For example, if you carry $0.50 then you must wear red; if you carry $1.00 you must wear orange; if you carry $1.50 you must wear yellow.

Look, over there—I see a line of people approaching the store.  They’re all wearing red.  They enter the store…but I don’t see any clerks exiting out the back, and consequently no clerks cross the bridge.  More and more reds go into the store, and a faster and faster rate, but it doesn’t matter.  There are no clerks coming out.  The troll doesn’t get any business.

I conclude that a bag of chips costs more than $0.50.

Later, I see a line of oranges go into the store.  I now observe some clerks coming out, but none of these can cross the bridge.  I conclude that chips cost $1.00, but since the clerks have no money left over, they can’t pay the troll’s toll.

Even later, I see a line of yellows go into the store.  Clerks are coming out, and these can cross the bridge.  The troll must be demanding a toll of $0.50 or less.

There are many quantities which are important in analyzing this situation: the amount of money a person has before entering the store (use E for entering), the price of a bag of Wavy Lay’s potato chips (let’s call this price W), the toll that Voltar the troll demands (let’s call this V), and the amount of money a clerk has, K (they speak Dutch in Toelek, so the clerks are called klerks) upon exiting the store.  It should be obvious that

K = E – W,

since the amount of money a clerk has upon leaving the store is just the amount of money a person has upon entering the store minus the cost of some chips.  Additionally, for a clerk to cross the bridge, it must be true that

E – W ≥ V

so that the clerk has enough to pay the toll.  If the clerk barely makes it, this inequality is an equality and

E – W = V.

The V at which this happens (for a given E and W) is called the cutoff toll Vo;  if Voltar were to increase the toll by any amount at all, the clerks wouldn’t get to cross the bridge.

It’s interesting to graph the cutoff toll Vo vs. money that customers have upon entering the store E.  You get something like this:

Photoelectric

Notice that the cutoff value of E is $1.00, which is the price of a bag of chips.  At or below this value the troll need not charge any toll at all, since no clerk will have any money to pay him.  That is, when V = Vo = 0,  then E = W.

WHAT’S GOING ON?

The physicists reading this blog have already guessed the game I’m playing: I have presented an analogy for Einstein’s explanation of the photoelectric effect (hence Toelek, from fotoelektrisch).  Make the following transformations:

Customers = photons;

Color of customer = frequency of photon;

Money = energy;

Store = photoelectric material;

Price of chips = work function;

Clerks = electrons;

Clerk’s money = kinetic energy;

Bridge = potential difference;

Voltar’s toll = kinetic energy required to jump the gap.

With these transformations, you can re-write the story as follows:

There is a photoelectric material, a metal such as platinum.  Inside the metal are electrons.  The electrons can be liberated if enough energy is added.

Now, the thing is, if you give energy to electrons to liberate them, the electrons don’t give any energy back: they’re greedy.  What’s more, they leave the metal immediately with whatever energy they have left over from the transaction.  However, the electrons don’t always get very far, because out behind the metal is a potential difference that the electrons have to cross in order for current to be observed.  Jumping this gap requires a certain amount of kinetic energy, without which the electrons don’t produce current.

How do we add energy to the metal?  Well, by shining light on it.  Light energy is quantized, in chunks called photons.  The energy of a single photon is proportional to its frequency, by Einstein’s formula E = hf.

Look, over there—I see some red light (E = 1.9 eV) approaching the metal.  Unfortunately, no electrons exit out the back, and consequently there is no current.  More and more red photons hit the metal, and a faster and faster rate, but it doesn’t matter.  There are no electrons coming out.

I conclude that the amount of energy need to liberate an electron (called the work function) is greater than 1.9 eV.

Later, I see orange light (E = 2.1 eV) go into the metal.  I now observe some electrons coming out, but none of these produce current.  I conclude that the work function is W = 2.1 eV.

Even later, I see yellow light (E = 2.18 eV) go in.  Electrons are coming out, and these can cross the potential difference.  The potential difference must be 0.08 volts or less.

There are many quantities which are important in analyzing this situation: the amount of energy a photon has before entering the metal (E), the work function (W), the voltage that electrons have to jump (V), and the amount of energy an electron has upon exiting the metal, K.  It should be obvious that

K = hf – W,

since the amount of energy an electron has upon exiting the metal is just the amount of energy a photon has upon entering the metal minus the cost of liberating an electron.  Additionally, for an electron (with charge e) to jump the gap, it must be true that

hf – W ≥ eV

so that an electron has enough kinetic energy to overcome the potential difference.  If the electron barely makes it, this inequality is an equality and

hf – W = eV.

The V at which this happens (for a given f and W) is called the stopping potential Vo.

It’s interesting to graph stopping potential Vo vs. energy of incoming photons E.  You get something like this:

Photoelectric2

Notice that the cutoff value of E is 2.1 eV, which is W.  At or below this value there need not be any potential difference at all, since no electron will be liberated.  That is, when V = Vo = 0,  then E = hf= W.

I hope you find this analogy useful.  As for me, I need to go to the store: all this talk of potato chips has made me hungry.

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War is nothing like chess.

War is nothing like chess.

I am a big chess fan.

I can name every chess world champion since Morphy; I could probably name around 17 of the world’s current top 20; I can checkmate a lone king with two bishops and a king; I have a good working knowledge of just about every opening there is.

(And by working knowledge, I don’t just mean I’ve “heard” of the Sicilian defense.  I don’t just mean that I know that 1. e4 c5 2. Nf3 Nc6 3. d4 cxd 4. Nxd4 Nf6 5. Nc3 e5 is the Sveshnikov.  I mean that I am fully aware of the differences between the 9. Nd5 and 9. Bxf6 Sveshnikov, and prefer the former.)

The problem is, I’m just not that good.

Oh, I can beat most casual players…the ones that begin a game by moving a rook pawn (to a4, say) and then move their rooks out vertically (to a3, say).  In USCF rating terms, my rating is around 1800, which (to my own surprise) is about the 85th percentile for tournament players.  So objectively, I am not bad at all.  But I am good enough to be aware of just how much better other players are.  I have a friend Shawn who is a master (here he is drawing a grandmaster).  I am in awe of his tactical strength, and his fine sense of dynamics.  I have beaten him dozens of times in speed chess, but for every game I win, he wins 10.

It has taken me a while to get to the point of this blog post, which is this: I like chess because of its icy logic and its mathematical purity.  For this reason, chess is a horrible metaphor for war, or for life.

Chess is used in books and movies for two basic purposes.  The first is to establish the intelligence of a character.  For example, Lisbeth Salander (in The Girl with the Dragon Tattoo) is an expert at chess.  This was a bad choice on the author’s part: Lisbeth is also an expert hacker and financial genius, has an eidetic memory, and is an incredible detective—why stretch credulity even further?  A lot of great chess players are certainly smart, but the correlation doesn’t go the other way: many smart people are terrible at chess.  Einstein was probably weaker than me.  Oppenheimer was even worse.  Comedian Howard Stern, a player of about my strength, would crush either one.

The other use for chess in books and movies is as metaphor.  In The Seventh Seal, Antonius Block plays a game of chess against Death.  In Harry Potter and the Philosopher’s Stone (the original title; not the dumbed-down American version) someone plays chess with someone else (like I remember?)  In both cases the chess itself is ludicrous.  For example, at one point Death captures Block’s queen; Block says that he “didn’t see that”.  (Really?  Did Block just learn the rules the day before?)  But I don’t want to evaluate the chess in such works per se; rather, I want to see how well chess works as a metaphor.

First, chess as war.  I can’t think of any examples off the top of my head, but there seems to be an assumption that skill at chess somehow equates to skill at war.  But this is ludicrous: in chess, every move is transparent; you can always see what your opponent is doing, and everyone starts on a level playing field.  In terms of game theory, chess is a perfect information game.  I’m no Colonel Dax, but I don’t think war works that way.  There is always a fog of war, and an element of chance, so war is about contingencies, and adaptability, and bluff, and extrapolation.

Second, chess as life.  I have to admit, I don’t really get this metaphor at all.  Is life therefore a game?  A perfect information game?  If chess represents life, does that mean that I struggle throughout my life against an opponent (Satan?  Howard Stern?) who is trying to thwart me at every turn?  And if I play well, but my opponent does too, then am I destined for a draw?  What is a draw, in life?  Is it retiring at 65 to play shuffleboard in Orlando?

As much as I like chess, I think backgammon is a much better metaphor for war or for life.  In backgammon, there is an element of chance, and so the “the best laid schemes o’ mice an’ men” will often go awry.  That is why a good backgammon player will weigh contingencies.  What move leaves me in the best position, based on what dice rolls are possible, and what might happen?  In backgammon, you’re not just playing against an opponent, you’re playing against the fates themselves (in the form of the dice) and this makes the game feel more “real” to me.

People who don’t play backgammon often think that luck is a major part of the game.  This is true, on the level of a single game, but backgammon is played in matches of multiple games, and luck is much less important at that level.  This is because of the doubling cube.  With the doubling cube, a master will almost always defeat a weaker player, in the same way that a Napoleon will almost always win a war against a General Mack, even if an individual battle is lost here or there.

And so, life.  The dice aren’t always going to go your way.  You should plan with that in mind.  Look at your current position, figure out the possible contingencies—the possible ways God might play dice with your universe—and set up your pieces accordingly.  Even if you get gammoned, tomorrow’s another day.

[Note: I subconsciously chose an inept Austrian general to be the foil against Napoleon’s military genius.  But I want to be balanced in my portrayal of Austrians.  So I will remind everyone that Lise Meitner was Austrian, and she was a super-smart physicist.  And strangely, her father was Philipp Meitner, a chess master and part of the immortal draw.]

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