This question is asked with the intention of understanding basically the decay constant of radiometric dating (although I know the above is not an entirely accurate representation). If there is a group of radioisotopes whose eventual decay is not predictable on the individual level, I do not understand how a decay constant is measurable. I do understand that radioisotope decay is modeled exponentially, and that a majority of this dating technique is centered in probability. The margin for error, as I see it presently, cannot be small.

Physicist: The predictability of large numbers of random events is called the “law of large numbers“.  It causes the margin of error to be essentially zero when the number of random things becomes very large.

If you had a bucket of coins and you threw them up in the air, it would be very strange if they all came down heads.  Most people would be weirded out if 75% heads of the coins came down heads.  This intuition has been taken by mathematicians and carried to its more difficult to understand, and convoluted, but logical extreme.  It turns out that the larger the number of random events, the more the system as a whole will be close to the average you’d expect.  If fact, for very large numbers of coins, atoms, whatever, you’ll find that the probability that the system deviates from the average by any particular amount becomes vanishingly small.

The probability of getting a particular sum for: one die (red), two dice (green).  The probability “bunches up” around the average as more dice are added.

For example, if you roll one die, there’s an even chance that you’ll roll any number between 1 and 6.  The average is 3.5, but the number you get doesn’t really tend to be close to that.  If you roll two dice, however, already the probabilities are starting to bunch up around the average, 7.  This isn’t a mysterious force at work; there are just more ways to get a 7 ({1,6}, {2,5}, {3,4}, {4,3}, {5,2}, {6,1}) than there are to get, say, 3 ({1,2}, {2,1}).  The more dice that are rolled and added together, the more sum will tend to cluster around the average.  The law of large numbers just makes this intuition a bit more mathematically explicit, and extends it to any kind of random thing that’s repeated many times (one might even be tempted to say a large number of times).

The exact same kind of math applies to radioactive decay.  While you certainly can’t predict when an individual atom will decay, you can talk about the half-life of an atom.  If you take a radioactive atom and wait for it to decay, the half-life is how long you’d have to wait for there to be a 50% chance that it will have decayed.  Very radioactive isotopes decay all the time, so their half-life is short (and luckily, that means there won’t be much of it around), and mildly radioactive isotopes have long half-lives.

Now, say the isotope “Awesomium-1″ has a half-life of exactly one hour.  If you start with only 2 atoms, then after an hour there’s a 25% chance that both have decayed, a 25% chance that neither have decayed, and a 50% chance that one has decayed.  So with just a few atoms, there’s not much you can say with certainty.  If you leave for a while, lose track of time, and come back to find that neither atom has decayed, then you can’t say too much about how long it’s been.  Probably less than an hour, but there’s a good chance it’s been more.  However, if you have trillions of trillions of atoms, which is what you’d expect from a sample of Awesomium-1 large enough to see, the law of large numbers kicks in.  Just like the dice, you find that the system as a whole clusters around the average.

If there’s a 50% chance that after an hour each individual atom will have decayed, and if you’ve got a hell of a lot of them, then you can be pretty confident in saying that (by any reasonable measure) exactly half of them have decayed at the end of the hour.

In fact, by the time you’re dealing with a mere one trillion atoms (a sample of atoms too small to see on a regular microscope), the chance that as much as 51% or as little as 49% of the atoms have decayed after one half-life (instead of 50%) is effectively zero.  For the statistics nerds out there (holla!), the standard deviation in this example is 500,000.  So a deviation of 1% is 20,000 standard deviations, which translates to a chance of less than 1 in 10^86858901.  If you were to see a 1% deviation in this situation, take a picture: you’d have just witnessed the least likely thing anyone has ever seen (ever), by a chasmous margin.

Using this exact technique (waiting until exactly half of the sample has decayed and then marking that time as the half-life), won’t work for something like Carbon-14, the isotope most famously used for dating things, since Carbon-14 has a half-life of about 5,700 years.  Luckily, math works.

The amount of radiation a sample puts out is proportional to the number of particles that haven’t decayed.  So, if a sample is 90% as radioactive as a pure sample, then 10% of it has already decayed.  These measurements follow the same rules; if there’s a 10% chance that a particular atom has decayed, and there are a large number of them, then almost exactly 10% will have decayed.

The law of large numbers works so well, that the main source of error in carbon dating comes not from the randomness of the decay of carbon-14, but from the rate at which it is produced.  The vast majority is created by bombarding atmospheric nitrogen with high-energy neutrons from the Sun, which in turn varies slightly in intensity over time.  More recently, the nuclear tests in the 50′s caused a brief spike in carbon-14 production.  However, by creating a “map” of carbon-14 production rates over time we can take these difficulties into account.  Still, the difficulties aren’t to be found in the randomness of decay which are ironed out very effectively by the law of large numbers.

This works in general, by the way.  It’s why, for example, large medical studies and surveys are more trusted than small ones.  The law of large numbers means that the larger your study, the less likely your results will deviate and give you some wacky answer.  Casinos also rely on the law of large numbers.  While the amount won or lost (mostly lost) by each person can vary wildly, the average amount of money that a large casino gains is very predictable.

Answer Gravy: This is a quick mathematical proof of the law of large numbers.  This gravy assumes you’ve seen summations before.

If you have a random thing you can talk about it as a “random variable”.  For example, you could say a 6-side die is represented by X.  Then the probability that X=4 (or any number from 1 to 6) is 1/6.  You’d write this as .

The average is usually written as μ.  I don’t know why.  For a die, .  This can also be written, , and often as E[X].  E[X] is also called the “expectation value”.

There’s a quantity called the “variance”, written “σ²” or “Var(X)”, that describes how spread out a random variable is.  It’s defined as .  So, for a die, 

If you have two random variables and you add them together you get a new random variable (same as rolling two dice instead of one).  The new variance is the sum of the original two.  This property is a big part of why variances are used in the first place.  The average also adds, so if the average of one die is 3.5, the average of two together is 7.  So, if the random variables are X and Y with averages μ_X and μ_Y, then μ=μ_X+μ_Y.  And using expectation value notation (if you’re not familiar look here, or just trust):

You can extend this, so if the variance of one die is Var(X), the variance of N dice is N times Var(X).

The square root of the variance, “σ”, is the standard deviation.  When you hear a statistic like “50 plus or minus 3 percent of people…” that “plus or minus” is σ.  The standard deviation is where the law of large numbers starts becoming apparent.  Since the variance of lots of random variables together adds, , but that means that .  So, while the range over which the sum of random variables can vary increase proportional to N, the standard deviation only increases by the square root of N.  For example, for 1 die the numbers can range from 1 to 6, and the standard deviation is about 1.7.  10 dice can range from 10 to 60 (10 times the range), and the standard deviation is about 5.4 (√10 times 1.7).

What does that matter?  Well, it so happens that a handsome devil named Chebyshev figured out that the probability of being more that kσ from the average, written “P(|X-μ|>kσ)”, is less than 1/k².  Explanations of the steps are below.

i) “The probability that the variable will be more than k standard deviations from the average”.  ii) This is just re-writing.  For example, if you have a die, then P(X>3) = P(4)+P(5)+P(6).  This is a sum over all the X that fit the condition.  iii) Since the only values of n that show up in the sum are those where |n-μ|>kσ, we can say that and squaring both sides, that .  Multiply each term in the sum by something bigger than one, and the sum as a whole certainly gets bigger.  iv) “1/k²σ²” is a constant, and can be pulled out of the sum.  v) If you’re taking a sum and you add more terms, the sum gets bigger.  So removing the restriction and summing over all n increases the sum.  vi) by definition of variance.  vii) Voilà.

So, as you add more coins, dice, atoms, random variables in general, the fraction of the total range that’s within of kσ of the average gets smaller and smaller like .  If the range is R and the standard deviation is σ, then the fraction within kσ is .  At the same time, the probability of being outside of that range is less than 1/k².

So, in general, the probability that you’ll find the sum of lots of random things away from their average gets very small the more random things you have.

Energy beings are an old staple of sci-fi (a good one), but they’re almost certainly impossible, or at least, it’s almost certainly impossible for life (as we know it) to evolve into energy.  Even after billions of years on Earth, life is pretty much the exact same stuff that it’s always been.  Several billion years ago single cells figured out how to metabolize, repair damage, and reproduce.  Everything since then has pretty much just been variations on that theme (sincerest apologies to our evolutionary biologist readers).  The word “evolution” evokes ideas of advancement, and improvement, and ascension, but “in practice” evolution is to accidents as a beach is to grains of sand.

Energy Beings: The ultimate end of evolution, or possibly a dude in a body sock.

Part of that is that there’s no goal that life is evolving toward, or even a path that life is taking.  So, humanity is no more the pinnacle of evolution than every other living thing is.

One of the classic examples of evolution in action is the Peppered Moth.  The Peppered Moth, like many species (including people!), appears in a couple different colors and patterns.  Normally they’re grey (and peppered), but during the industrial revolution the area around London became so nasty and coal-covered that black peppered moths became far more common.  By accident of birth, some moths were black and, by accident of circumstance, they found that they could hide from predators better than their suddenly very visible grey cousins.  That’s evolution; it’s not a matter of being better, or even adapting, it’s just a matter of stumbling forward and whatever happens happens.

It would be great if evolution always made things more advanced, but in general, creatures only become as complex as they minimally need to be.  If group of critters can get along by becoming simpler, then tend to evolve (accidentally be born) into that simpler form.  For example, there are several examples of blind subterranean animals that are descended from species that once had eyes.  Again, it’s not that they purposely evolved to be blind, it’s just that sometimes (by accident) you’re born without eyes, and sometimes it doesn’t matter (because you live in a cave).  It’s a lot easier (more likely) to lose a feature and become simpler than it is to gain a new feature and become more complex.

Nothing to see here.

It is the case that the lifeforms with the greatest complexity will be found later in history rather than earlier, but that’s pretty much because it takes time for things to become complex.  However, for the most part living systems have maintained about the same level of complexity for hundreds of millions of years.  The most successful form of life on Earth (arguably) is still single-celled.  Those little dudes really have it all figured out.

Long story short: evolution isn’t “leading up” to anything, it just drunkenly limps along using the same set of tricks in slightly different orders.

On the physics side of things, “energy life” sounds like a cool idea, it’s not really a possibility (as far as we can say).  Energy generally doesn’t exist on its own without matter, and when it does it’s propagating about at the speed of light (for example: light).  Not experiencing time (which is one of the problems with light speed) seems to go against the idea of “life”.  That is, if something never changes at all, can it really be alive?

The concept we (sci-fi aficionados) usually have of energy beings, as a kind of beneficent glowing ghost or a giant Kirk-harassing cloud, runs contrary to the physical understanding of energy physicists have developed so far.  Despite all the different terms that are used to talk about energy, it only takes a couple forms.  At its most base there’s the “energy of stuff” (like the E=mc² of matter), there’s the “energy of stuff moving” (kinetic energy), and there’s the “potential for stuff to move” (the various forms of potential energy: charged batteries, gasoline, wound clock springs, etc.).  A “ball of energy” that’s independent of matter isn’t really a thing.

It would be cool to think that someday something will “evolve into energy”, but pressed for a prediction, I’d say that evolution will continue to stumble around at random for as long as there are living things around to do the stumbling.  Evolution is a process of accidental baby steps, and turning into an energy being, even assuming it’s possible, is more of a leap.

Side note: When someone says “3+1 dimensions”, what they mean is “3 regular space dimensions, and one time dimension” which is exactly the situation we live in (apologies to our pan-dimensional readers).

Physicist: Right off the bat, more dimensions means more freedom of movement.  One of the more mundane effects of that is that in 4 dimensional space there’s an extra direction you can move and/or fall over in.  So if you want to build a working bar stool you’d need at least 4 legs instead of just 3.  In fact, in D-dimensional space bar stools need at least D legs, or they’ll fall over.  Just one of the subtle economic effects of higher dimensional living.

You’d also find that in 4 or more dimensions, you’d be able to do a lot of tricks impossible in 3 dimensions, like creating Klein bottles or (equivalently) taping the edges of two Möbius strips together.  Sailing knots could take on stunning complexities.  In fact, they’d need too!  All of the knots that work in 3 dimensions fall apart immediately in 4.

In four dimensions you could make this surface without worrying about it intersecting itself.

Most physical laws are already written in a dimension-free form.  For example, in Newton’s second law, , and  are both vectors, but they can be vectors in any number of dimensions.  So you can use  for objects on a line (1-D), on a table-top (2-D), in space (3-D), or whatever (whatever-D).

There are some laws are usually written in a 3-D form, but that’s generally a matter of convenience more than necessity.  For example, we talk about the “angular momentum vector”, which is defined to be perpendicular to the plane of rotation.  It’s convenient because in three dimensions there’s always exactly one perpendicular direction to a plane, whereas in 4 dimensions (for example) there are 2.

In 3-D we can formulate laws about spinning things in terms of the one direction that isn't spinning (h), the "axis of rotation". But we can always formulate laws in terms of the two directions that are spinning, regardless of dimension.

This is pretty easy to fix and generalize, it just becomes a little more difficult to work with.  All that said, while our physical laws can be generalized to any number of dimensions, the manifestation of those laws are wildly different.  So, living in higher dimensions would be pretty alien.

Based on our understanding of gravity (gained from studying this podunk universe), gravitational force should drop by , where D is the dimension and R is the distance between the objects in question.  It so happens that because of the nature of orbits, a stable orbit can only exist in 2 or 3 dimensions.

Orbits can be stable in 2 and 3 dimensions. In all other dimensions planets and moons will always either spiral in or fly away. Shown here is the potential energy from gravity and the centrifugal force combined. If there's a "cup" you can form a "bound orbit" in it.

In 4 or more dimensions orbits are always unstable, and in 1 dimension the idea of an orbit doesn’t even make sense.

Most physicists consider light to be native to only 3 dimensions, because light is an EM wave and it’s direction of propagation is perpendicular to both its Electric and Magnetic fields.  (Fun fact: the direction that light points is called the “Poynting vector“, after John Henry Poynting.  Life’s funny.)  In 4 or more dimensions this direction isn’t unique, and in two dimensions there’s no direction at all.  However, you can express EM waves just in terms of “E” in any dimension without problem.

Assuming light can exist in higher dimensions, it would behave very strangely.  Sound waves too.  In odd dimensions other than 1 (3, 5, 7, …) waves behave the way we normally see and hear things: a wave is formed, it moves out, and it keeps going.  However, in even dimensions, and 1 as well, (1, 2, 4, 6, …) waves “double back” on themselves.  You can see this in ripples on the surface of water (2-D waves).  Ripples are more complex than just a ring; the entire circle within the ripples is disturbed.

In even dimensions (like the 2-D surface of water), waves propagates in a more complex way than we're used to. Instead of a simple pulse, you get an "area filling" wave.

If you set off a firecracker in 3, 5, 7, etc. dimensions, then you’ll see and hear the explosion for a moment, and that’s it.  If you set of a firecracker in 4, 6, 8, etc. dimensions, then you’ll see and hear the explosion intensely for a moment, but will continue to see and hear it for a while.  For light the effect would be fairly subtle, except for extremely long-distance effects, like somebody reflecting a bright light off of the moon.  You probably wouldn’t notice the effect day-to-day.  However, it would ruin the experience of sound.  In 4 dimensional space the firecracker, even in open air, would sound like thunder; loud at first, and leading into a drawn out boom.  It may not even be possible to understand people when they speak.

All the fundamental particles should still exist, but how they interact would be pretty different.  Which elements are stable, and the nature of chemical bonds between them, would be completely rearranged.  Some things would stay the same, like electrons would still have two spins (up or down).  But atomic orbitals, which are determined by spherical harmonics (which in turn are more complicated in higher dimensions), would generally be able to hold more electrons.  As just one example (for our chemistry-nerd readers), you’ll always have 1 S orbital in every energy level, but in 4 dimensions you’ll have 4 P orbitals in each energy level, instead of the paltry 3 that we’re used to.  This messes up a lot of things.  For example, in 4 dimensions Magnesium would be a noble gas instead of a metal.  Every element after helium would adopt weird new properties, and the periodic table would be longer left-right and shorter up-down.

So, while the laws of physics are actually the same, if you lived on a four-dimensional Earth in a four-dimensional universe you’d find that (among other things): your bar stool may need an extra leg, Earth wouldn’t be able to orbit anything, you’d never be able to hear anything crisply, and the periodic table of the elements would be seriously rearranged.

Physicist: It weighs more than nothing, but if you’re at the bottom of a pillar of fire, being crushed should be your second concern.

Fire: bad.

Fires, putting aside details about plasma and chemicals or whatever, is just hot air.  For a given pressure the ideal gas law says that the density of a gas is inversely proportional to temperature, in Kelvin.  You can use this fact, the temperature and density of air (300°K 1.3 kg/m³), and the temperature of your average run-of-the-mill open flame (about 1300°K) to find the density of fire.

For most “everyday” fires, the density of the gas in the flame will be about 1/4 the density of air.  So, since air (at sea level) weighs about 1.3 kg per cubic meter (1.3 grams per liter), fire weighs about 0.3 kg per cubic meter.

One pound of ordinary fire, here on Earth near sea level, would take up a cube about 1.2 meters to a side.  The reason that fires always flow upward is that its density is lower than air.  So, fire rises in air for the same reason that bubbles rise in water: it’s buoyant.  Enterprising individuals sometimes even take advantage of that fact.

Fire: good.

If you were on a planet with no air at all, fire would fall to the ground instead of rise because, like all matter, it’s pulled by gravity.  Also, it would be hard to keep the fire going (what with there being no air).

Physicist: The amount of time that goes into, say, calculating how two electrons bounce off of each other is very humbling.  It’s terribly frustrating that the universe has no hang ups about doing it so fast.

In some sense, basic physical laws are the basis of how all calculations are done.  It’s just that modern computers are “Turing machines“, a very small set of all the possible kinds of computational devices.  Basically, if your calculating machine consists of the manipulation of symbols (which in turn can always be reduced to the manipulation of 1′s and 0′s), then you’re talking about Turing machines.  In the earlier epoch of computer science there was a strong case for analog computers over digital computers.  Analog computers use the properties of circuit elements like capacitors, inductors, or even just the layout of wiring, to do mathematical operations.  In their heyday they were faster than digital computers.  However, they’re difficult to design, not nearly as versatile, and they’re no longer faster.

Nordsieck's Differential Analyzer was an analog computer used for solving differential equations.

Any physical phenomena that represents information in definite, discrete states is doing the same thing a digital computer does, it’s just a question of speed.  To see other kinds of computation it’s necessary to move into non-digital kinds of information.  One beautiful example is the gravity powered square root finder.

Newtonian physics used to find the square root of numbers. Put a marble next to a number, N, (white dots) on the slope, and the marble will land on the ground at a distance proportional to √N.

When you put a marble on a ramp the horizontal distance it will travel before hitting the ground is proportional to the square root of how far up the ramp it started.  Another mechanical calculator, the planimeter, can find the area of any shape just by tracing along the edge.  Admittedly, a computer could do both calculations a heck of a lot faster, but they’re still descent enough examples.

Despite the power of digital computers, it doesn’t take much looking around to find problems that can’t be efficiently done on them, but that can be done using more “natural” devices.  For example, solutions to “harmonic functions with Dirichlet boundary conditions” (soap films) can be fiendishly difficult to calculate in general.  The huge range of possible shapes that the solutions can take mean that often even the most reasonable computer program (capable of running in any reasonable time) will fail to find all the solutions.

Part of Richard Courant's face demonstrating a fancy math calculation using soapy water and wires.

So, rather than burning through miles of chalkboards and a swimming pools of coffee, you can bend wires to fit the boundary conditions, dip them in soapy water, and see what you get.  One of the advantages, not generally mentioned in the literature, is that playing with bubbles is fun.

Today we’re seeing the advent of a new type of computer, the quantum computer, which is kinda-digital/kinda-analog.  Using quantum mechanical properties like super-position and entanglement, quantum computers can (or would, if we can get them off the ground) solve problems that would take even very powerful normal computers a tremendously long time to solve, like integer factorization.  “Long time” here means that the heat death of the universe becomes a concern.  Long time.

Aside from actual computers, you can think of the universe itself, in a… sideways, philosophical sense, as doing simulations of itself that we can use to understand it.  For example, one of the more common questions we get are along the lines of “how do scientists calculate the probability/energy of such-and-such chemical/nuclear reaction”.  There are certainly methods to do the calculations (have Schrödinger equation, will travel), but really, if you want to get it right (and often save time), the best way to do the calculation is to let nature do it.  That is, the best way to calculate atomic spectra, or how hot fire is, or how stable an isotope is, or whatever, is to go to the lab and just measure it.

Even cooler, a lot of optimization problems can be solved by looking at the biological world.  Evolution is, ideally, a process of optimization (though not always).   During the early development of sonar and radar there were (still are) a number of questions about what kind of “ping” would return the greatest amount of information about the target.  After a hell of a lot of effort it was found that the researchers had managed to re-create the sonar ping of several bat species.  Bats are still studied as the results of what the universe has already “calculated” about optimal sonar techniques.

You can usually find a solution through direct computation, but sometimes looking around works just as well.

There are no physical theories that allow for the existence of magic, as described in Harry Potter™ and or other creative ventures of that ilk.  However, if you had an amazing amount of money and energy, you might be able to set up some kind of magnetic system that allowed people, while within the arena, to be propelled around on things the size of broomsticks.

Quidditch: Nope.

Bill Gates could probably get something set up, but he’s too busy curing malaria or something.

I wish I could say something about how awesome superconductors are, or that they might be useful, but all they’d really be useful for is moving the players around as though they were on tracks.  If you really want to control how everything moves around you’d have to use “servo-mechanistic electromagnetic suspension” (that’s a made up phrase, but it is a decent description).  This is how maglev trains today work.

Normally, a pair of magnets will either snap together or fly apart.  To suspend one magnet above another (without snapping or flying), or move one magnet around in a controlled way, requires the use of one or several tightly controlled electromagnets, that vary their strength very quickly to react to the position of the magnet they’re suspending.  This is a “servomechanism“.

It would take stunning buckets of power to create a magnet field the size of an arena, that’s capable of suspending a person (and broom).  But the real difficulty is in coming up with a way of targeting 15 separate objects: two teams of 7 players, and the “snitch”.  In theory, you could put together some kind of extremely sophisticated array of small coils in the ground and in the stands that could target a magnetic field to relatively small areas in space (person sized).  This sort of technique is used for things like “hypersonic sound“, which is a “beam of sound”.

However, there’s a drawback.  In order to get a wave phenomena (sound waves, ocean waves, magnetic, whatever) to stay stay confined (like a beam) and not just go wherever, you need the wavelength to be fairly short.  A good rule of thumb is that the wavelengths need to be about the size of the region in question or shorter.  This is just another incarnation of the uncertainty principle (which is all about waves).  At about the same time that you’re using wavelengths short enough to (theoretically) target individual people and brooms, and not all the objects in the arena, you’ll find that your previously gentle magnetic field is now made up of microwaves (microwaves, and light in general, are just high frequency electromagnetic fields).

So, after years of effort and (let’s say) billions of dollars in R&D, you’ll find that what you’ve really made is the most whimsical death machine ever constructed.  Real-world Quidditch would a very short, but spectacular game.