S03E18: The Pants Alternative

There’s more science than meets the eye on The Big Bang Theory. Literally.  Tonight’s viewers were undoubtedly puzzled by the presence of a little non-Euclidean geometry on the white boards.  Why was it there? What is it?

Non-Euclidean geometry on tonight's whiteboards...unobstructed by distracting characters and action. (from CBS promo clip.)

First let us recall that  Euclid himself developed his non-non-Euclidean geometry in 300 B.C.   He started by assuming as little as possible and left it to generations of junior high school students to prove all the rest, that triangles have 180 degrees and so forth.  As is typical in mathematics, the game is to use as few, most elegant, postulates as possible and leave the rest to clever derivation.

Every junior-high school student learns Euclid’s five postulates (briefly listed, with some legalities dropped):

1. Any two points can be joined by exactly one line segment.

2. A line segment can be extended to an infinite line.

3. For any line segment, a circle can be drawn.

4. All right angles are the same.

Now these four sound pretty elementary. It is hard to imagine ever being able to prove or disprove them and so they serve “obvious” as starting points.  But there is a pesky “fifth postulate” Euclid added:

5.  For a point outside a line,  there is exactly one line through the point that never meets the original line.

Now Euclid was a smart guy.  Why not just prove “the fifth postulate” from the first four?   He tried.   For 2000 years, mathematicians tried.  Even Karl Friedrich Gauss.   No dice.  It was not proved.  Modern mathematicians know it could never have been proved from the first four.

So mathematicians made lemons out of lemonade.   In 1826,  Nicolai Lobachevsky said, let me assume there are not one, but many, such parallel lines.  In the 1850’s Bernhard Riemann  said let’s assume there are none.  Chaos ensued.   Without the fifth postulate you cannot even prove all triangles have 180 degrees.   Crazier still, assuming Lobachevsky’s  version of the fifth postulate gives you less than  180 degrees in all triangles.  Riemann’s gives you more than 180. These are some strange triangles.

(As it happens,  it was all worked out decades earlier by Gauss and he stuck it in his desk without publishing.  Note to aspiring mathematicians…whatever you work on, there’s a good chance Gauss already tried it.)

A few decades later, at the turn of the 20th century, Albert Einstein and others realized there was more than abstraction to the fifth postulate, but that it relates to an actual uncertainty we have about our own Universe.  Being unable to prove Euclid’s fifth postulate is equivalent to not knowing if we live in curved space or a flat one.  We don’t really know if extremely large triangles in our Universe have exactly 180 degrees or a little more or a little less.    How can a triangle have more than 180 degrees?  Let’s take a trip:

It is possible, when walking around the globe, to make a triangle with three right angles. This triangle has 270 degrees, not 180.

Let’s walk (and dogsled, and  swim)  from the North Pole along the Greenwich meridian (0 deg. longitude) through England and down to the Equator.  Here make a right turn.  That’s a 90 degree angle.  Go until you hit San Salvador Island, west of Ecuador, located on the Equator at the 90th meridian (+90 degrees) .  Now take another right turn (another 90 degrees) and head back north through St. Louis, back to the North Pole.   So far we’ve taken two right turns, 90+90=180 degrees. But then we hit the North pole again, coming up through Canada.  We find we arrive having completed a triangle.  We took three turns.  But you are coming in at right angles to where we left.   The last angle has 90 degrees.  So we’ve made a triangle containing 90+90+90= 270 degrees, not 180.   Such is life on a curved surface.  Even if you can’t see the curvature while moving around, it is there.

You can do the same in a Lobachevsky space.  Imagine walking around on a Pringle’s potato chip.

Non-Euclidean geometry is the key to Pringles potato chips.

Often it is the case that a piece of mathematical abstraction has a direct bearing on our physical world.  In this case,  it just took over 2000 years to realize it.

In his fascinating 1965 sequel to Flatland, Dionys Burger’s Sphereland chronicle’s the life of the grandson of A. Square,  named A. Hexagon.   A surveyor in Flatland is distressed to find the angles of triangles do not add to 180 degrees.  A. Hexagon, being the progeny of the insightful hero of Flatand, awakens his surveyor friend to the fact that Flatlanders really inhabit a curved space.  Like our trip, triangles their angles do not have angles that add up to 180 degrees.

Now even our own, non-fictional, 3-dimensional space may be curved.  We just don’t know.   Astrophysicists measure the largest triangles they can find looking for small deviations.  They use the rays of the oldest light in the universe, the microwaves left over from just 380,000 years after the Big Bang.  So far, all triangles add up to 180 degrees, but with more precision we may at any time find we live in a curved space.  And that is not a problem.  Everywhere junior high school geometry books would just have to revise Euclid’s fifth postulate.  (That’s good news for the textbook publishers, who are always looking to come out with a new edition to sell.  Even if there is nothing new, I find for my own classes, they will just re-number the questions every 3 years and call it a new edition.)

Fun stuff, but what did this have to do with the show?  Why did we put it on the whiteboard?  Actually it directly references a speech Sheldon had in the script.  Had.  It was an earlier version script.

The scripts go through many revisions throughout the production week.  Every day the actors rehearse and the writers improve the script.   Comedy seems to work like an experimental science.  The writers sometimes find something better.  And for this week,  after I already sent in the equations and diagrams to the set dressers, the writers rewrote  Sheldon’s lines about retreating into a Riemannian space to relax.

So viewers never saw this little piece of science. The writers replaced it with something funnier:   Riemannian space was replaced by Sheldon’s favorite location in Sim City.  But maybe it still relates.  After all, how well has anyone measured the triangles of Sheldonopolis anyway?

S03E17: The Precious Fragmentation

Viewers tonight may have anticipated disappointment, that there was no physics to blog about in this evening’s episode.  But of course physics is everywhere.

Tonight our friends Leonard, Sheldon, Raj and Howard acquire a “Lord of the Rings” ring.   The “One Ring”, we’re told, to bind the Rings of Power of the land of Middle Earth.  At least they had a replica of it.   None of them ever tried the ring on, but had they done so — if it were the real ring — it would have rendered them invisible.  Invisibility is one of the principle powers of the ring.  Oh, and world domination, too.

Physicists today are working on creating invisibility.   Not having access to the Fires of Mordor or Elven blacksmiths, we instead have turned to Maxwell’s Equations.  These are the four elegant equations that describe all of electricity, magnetism, and light.  Because of James Clerk Maxwell and his four equations, we know that light is made from electric and magnetic fields.

The four equations that describe all of classical electricity, magnetism, and light -- Maxwell's Equations -- are simple enough to fit on a T-shirt

Maxwell’s Equations were developed over many decades of the 19th century by physicists studying electricity and magnetism, among them many great minds: Gauss, Ampère, Faraday.  But it was Maxwell who found the linchpin, the term that bound them all, and summarized the result in those four beautiful equations.  At my own university, UCLA, as at most others,  understanding Maxwell’s equations  is the pinnacle of the first-year course in physics.   Starting with balls rolling down an inclined plane and step-by-step understanding more and more physics, undergraduates are able to understand all of Maxwell’s equations after just one year of study, and thus understand perhaps one of the greatest intellectual achievements of all time.

The received wisdom about popular science writing  is that with every equation I will lose half my readers.  So I will skip the “equations” and just describe Maxwell’s  four “rules” that have stood the tests of time by experiments:

#1. Electrons and protons create electric fields.

#2. There are no magnetic charges.

#3. Changing magnetic fields create loops of electric fields.

#4. Moving electric charges OR changing electric fields create loops of magnetic fields.

Now, what do these four rules mean for everyday living? A lot.

Rule #1  describes why you receive a shock in the winter after handling your fleeces.  When rubbing fabric, you build up an excess of electric charge in your body.  By this rule, an electric charge creates an electric field, which  means it creates two points in space with different voltages.   You are now resting at a higher voltage than, say your doorknob.  A voltage difference will drive currents and can do anything from toast bread to run your television.   When you touch the doorknob, current flows and … Zing!   By the way, “voltage” has no meaning on its own.  Only differences in voltages ever matter.  So when you see a sign that says “Danger:  Ten Thousand Volts”, it really should say  “Danger: Ten Thousand Volts Relative to YOU”.  If you touch something 10,000 Volts different than you, current will flow and for a few seconds you become the toaster.

Rule #2 was the subject of the boys’ Arctic expedition.

Rule #3: tells us how electric generators work.  Move a magnet through a metal loop and you will set up a voltage difference that will drive a current, making a generator.    If you reverse the situation, and drive a current in the loop then you can move a magnet and you’ve made a motor.   Every motor is a generator and every generator is a motor.   These basic building blocks of our technology were not invented by someone saying: “We need a way to make electricity” nor by saying “Is there a way I can use an electrical current to move things?”  Rather, they were the by-product of the basic research of the day, into the elemental nature of electricity and magnetism.  Curiosity, not necessity, is the mother of invention.

Rule #4 : The first part tells you how to make a magnet by running a current through a loop of wire.  Even a refrigerator magnet works because electrons in the magnet are moving in circles and making magnetic fields.

Now for the second part of Rule #4: “Changing electric fields make loops of magnetic field”.  Discovery of this last piece is Maxwell’s genius.  Maxwell wondered why, if a changing magnetic field could produce loops of electric field, why could not the converse be true as well?  He guessed that changing electric fields would produce loops of magnetic fields.

Indeed they do.  Very small, almost inperceptably, but they do.   So most remarkably, even in a perfectly empty space, changing electric fields will produce changing magnetic fields that will produce changing electric fields and so on, forever and ever.  Maxwell noticed by combining equations #3 and #4 he could make a wave, a wave that turned out to travel at precisely the speed of light.  The wave is created by changing electric and magnetic fields creating one another forever according to rules #3 and #4.     The fact that he could predict the wave traveled the speed of light led Maxwell to conclude that light is an electromagnetic wave.  (The conclusion was correct, but by accident.  We now know many things, not just light, that travel near or at the speed “of light”.  Had he known about all these other  he could not have concluded so quickly  that his waves were light waves.  Sometimes a little ignorance is bliss.)

Now, over 100 years later engineers are designing materials to interact with the electric and magnetic properties of light.  These materials can cause light to bend in ways that no naturally occuring material can, and are thus called meta-materials.  Light entering a meta-material can be made to bend completely around an object effectively making a cloaking device.

An example of a "metamaterial". Using Maxwell's equations, this new material can bend light around an object, rendering it invisible. (from Science Magazine)

The tricky part is that you must fabricate electrical components  smaller than the size of one wave oscillation.   Pieces less than an inch have been shown to effectively bend radio and microwaves (which are a type of light) around objects.  Visible light would require elements  100,000 times smaller. But techniques are getting closer all the time!

(Easter-egg alert:  Metamaterials featured prominently on the whiteboards of a previous episode: “The Creepy Candy Coating Corollary”, S0305.)

So our heroes don’t really need some  Dark Lord to make them a silly invisibility ring.  To the lab!!

S03E16: The Excelsior Acquisition

Tonight Sheldon wants to ask Stan Lee how the Silver Surfer uses his silver surfboard to accomplish interstellar flight.  As well he should!   Nobody, not even Sheldon, knows how we are going to travel between stars.

The Silver Surfer accomplishes interstellar travel on his silver surfboard. How will we?

Proxima Centauri is our best bet.  It is the closest star to our home orbiting around our own star, Sol.   Proxima Centauri is  an unremarkable red dwarf star named appropriately from the Latin proxima, which is “next to”, as in “proximate”.  It is not so-named because it is close to us, but rather because it is close to the star Alpha Centauri, a star in the constellation Centauri.   Alpha Centauri is the third brightest star in the night sky, but mostly just because it is so close.  We may want try to visit someday.  After all we are neighbors and have yet to bring them so much as a fruit basket.

“Close” is a funny word to use on interstellar distances.   Proxima and Alpha Centauri are so far away it takes light 4.2 years to arrive.  Nothing we know of can allow us to travel faster than light, our ultimate speed limit.  Even the television transmissions of the pilot episode of Big Bang Theory, which have been traveling at the speed of light since late 2007, are only halfway to whoever might inhabit the rocks orbiting those stars.  Not even Hulu.com  in Alpha Centauri has TBBT available yet.  (Life near Alpha Centauri has that in commonwith Earth.)

Alpha Centauri, being so bright, has probably been known to the earliest hominids who bothered to look up.  But Proxima Centauri being so dim was only discovered using powerful telescopes in 1915.  We may not be done yet.  Even dimmer stars known as brown dwarfs may be traveling the galaxy even closer to us than Proxima Centauri.  These stars are so cool, you have to look for them in infrared light.   Finding such nearby stars is one of the key missions of the newly launched WISE satellite.   When I told one of the BBT writers/exec producers we may soon find closer stars than Proxima Centauri he said “The Federation may be closer than we think”.

Proxima Centauri (red star, center) is the closest known star to Earth at 4.2 light years distance. (If you enjoy astronomy pictures such as this one, I highly recommend visiting NASA's "Astronomy Picture of the Day")

Right now our plate is full just with interplanetary travel within our own solar system.  A trip taking astronauts to Mars, as recently imagined by NASA, even at its closest approach will take over half a year.  Proxima Centauri is 750,000 times farther Mars’s closest approach to Earth.  At the same speed, that would take over a quarter million years to get there.  We must invent something faster.

Suppose our human engineers develop a technology that allows us to travel 1% the speed of light on average to Proxima Centari.  The astronauts only need now to spend 400 years on the spacecraft.   (I’m ignoring the tiny  benefit due to time dilation slowing the astronaut’s lifespan as we discussed earlier for the story of Paolo and Vincenzo.) The astronauts won’t survive to get there, but if they keep having children their 16th generation could make it.  I don’t think the intermediate generations will be particularly happy with their forbears for condemning them to a lonely flight through interstellar space.  If one generation rebels, and refuses to procreate the mission will be a failure.   Even if that 16th generation arrived successfully, they would hardly be Earthlings.

I think we can prove that we humans will never attempt interstellar transit until we know how to travel at least 25% the speed of light.  (The mission to Mars discussed above is only 0.001% the speed of light.)   Suppose a mission really was undertaken to travel to Proxima Centauri with a fantastic new technology that would take us there at 1% the speed of light.  It will take 400 years.  Now suppose anytime in the next 200 years, a new technology is developed to increase that average speed to 2% per year.  Given the rate of technological progress that is not a bad bet.  So the spacecraft that launches later would beat the earlier craft.   So not until a technology reaches some reasonable fraction of the maximum speed limit, the speed of light, would anyone bother to take an early flight.   The speed would have to be as large as 25% the speed of light to nearly guarantee this would not be a problem.  At least then the same generation will arrive as left the Earth.  It may not ever be possible, but the argument shows it is unlikely any such mission would be mounted until that is possible.

These are the stars in your neighborhood. Each white ring is about 1.7 light-years appart.

(If some smarty-pants wants to suggest worm-holes or other space-bending technology, keep in mind that these ideas don’t even work on paper.)

This says nothing of the many other technological hurdles must be met.  Traveling even at 1% the speed of light, the spacecraft would suffer terrible damage from interstellar gas and dust.   The rate of cosmic rays, charged particles flying throughout interstellar space, would likely give fatal cancer to anyone who tried this mission and they would arrive long dead.

So it pays to go back and understand what is special about the Silver Surfer’s surfboard that allows interstellar transport.  Often science fiction writers will come up with an idea before engineers and scientists.   Perhaps with the Silver Surfer there is an idea we’ve missed.  A good place to start with any such questions is James Kakalios’s terrific book “The Physics of Superheroes“.  Yet no explanation of Silver Surfer can be found — maybe it is just because Silver Surfer started out as a super-villain, not super-hero.  Fortunately someone actually asked the Silver Surfer’s creator, Jack Kirby, why he uses a surfboard.  To which he explains:

“Because I’m tired of drawing spaceships.”  -Jack Kirby

Excelsior!

David