Calvin And Hobbes And i

Hobbes 2I so miss Calvin and Hobbes, the wondrous, joyful comic strip that cartoonist Bill Watterson gave us between 1985 and 1995.  Hobbes was a stuffed toy tiger — except that 6-year-old Calvin saw him as a walking, talking man-sized tiger with a sarcastic sense of humor.

So many things in life and physics are like Hobbes — they depend on how you look at them.  As we saw earlier, a fictitious force disappears when viewed from the right frame of reference.  There’s that particle/wave duality thing that Duc de Broglie “blessed” us with.  And polarized light.

In an earlier post I mentioned that light is polar, in the sense that a single photon’s electric field acts to vibrate an electron (pole-to-pole) within a single plane.
wavesIn this video, orange, green and blue electromagnetic fields shine in from one side of the box onto its floor.  Each color’s field is polar because it “lives” in only one plane.  However, the beam as a whole is unpolarized because different components of the total field direct recipient electrons into different planes giving zero net polarization.  The Sun and most other familiar light sources emit unpolarized light.

When sunlight bounces at a low angle off a surface, say paint on a car body or water at the beach, energy in a field that is directed perpendicular to the surface is absorbed and turned into heat energy.  (Yeah, I’m skipping over a semester’s-worth of Optics class, but bear with me.)  In the video, that’s the orange wave.

At the same time, fields parallel to the surface are reflected.  That’s what happens to the blue wave.

Suppose a wave is somewhere in between parallel and perpendicular, like the green wave.  No surprise, the vertical part of its energy is absorbed and the horizontal part adds to the reflection intensity.  That’s why the video shows the outgoing blue wave with a wider swing than its incoming precursor had.

The net effect of all this is that low-angle reflected light is polarized and generally more intense than the incident light that induced it.  We call that “glare.”  Polarizing sunglasses can help by selectively blocking horizontally-polarized electric fields reflected from water, streets, and that *@%*# car in front of me.

Wave_Polarisation
David Jessop’s brilliant depiction of plane and circularly polarized light

Things can get more complicated. The waves in the first video are all in synch — their peaks and valleys match up (mostly). But suppose an x-directed field and a y-directed field are headed along the same course.  Depending on how they match up, the two can combine to produce a field driving electrons along the x-direction, the y-direction, or in clockwise or counterclockwise circles.  Check the red line in this video — RHC and LHC depict the circularly polarized light that sci-fi writers sometimes invoke when they need a gimmick.

Physicists have several ways to describe such a situation mathematically.  I’ve already used the first, which goes back 380 years to René Descartes and the Cartesian x, y,… coordinate system he planted the seed for.  We’ve become so familiar with it that reading a graph is like reading words.  Sometimes easier.

In Cartesian coordinates we write x– and y-coordinates as separate functions of time t:
x = f1(t)
y = f2(t)
where each f could be something like 0.7·t2-1.3·t+π/4 or whatever.  Then for each t-value we graph a point where the vertical line at the calculated x intersects the horizontal line at the calculated y.

But we can simplify that with a couple of conventions.  Write √(-1) as i, and say that i-numbers run along the y-axis.  With those conventions we can write our two functions in a single line:
x + i y = f1(t) + i f2(t)
One line is better than two when you’re trying to keep track of a big calculation.

But people have a long-running hang-up that’s part theory and part psychology.  When Bombelli introduced these complex numbers back in the 16th century, mathematicians complained that you can’t pile up i thingies.  Descartes and others simply couldn’t accept the notion, called the numbers “imaginary,” and the term stuck.

Which is why Hobbes the way Calvin sees him is on the imaginary axis.

~~ Rich Olcott

Prime years and such

I’ve liked 4s and 6s ever since when, but lately 3s and 7s have been cropping up.  A lot.  And they have a really weird connection with 2016.

For me New Year has always been an opportunity to inspect the upcoming year’s number for interesting properties.

Maybe the easiest way for a number to be interesting is to be prime, that is, not divisible by anything other than itself and one.  My Uncle Harold once proved to me that all odd numbers are prime.

“One’s a prime, and so are three and five.  How about seven?  Seven’s prime.  Nine?  Not a prime but we can throw that one out as experimental error.  Eleven?  Prime.  Thirteen?  Prime.  Case closed.”

They use that logic a lot in politics nowadays.

There are a few prime-ity tests that just need a quick glance.  Take 2015 for example.  Ends with a “5” so it’s got to be divisible by five.  Not a prime.  A number ending with a “0” is like ending with twice five so it’s not prime either.

Take 2016.  Ends in an even digit so it’s divisible by two.  Not a prime.  Moreover, it fails the “nines test” — add up all the digits (2+0+1+6=9).  If the total is nine or divisible by nine then the number itself is divisible by nine (and by three) so it’s non-prime.  2016 is also divisible by seven but that’s not as easy to diagnose.

That’s about it for quickies.  Beyond those tests you have to slog through dividing the target by every prime number from three up to the target’s square root.  Why stop there?  Because any factor bigger than the square root will have a partner smaller than the square root.

Remember Party Like It’s 1999 (prime)?  Very popular when the Artist Then Known As Prince produced it in 1982 (not a prime).  Unfortunately, we who were working on Y2K projects were too busy to party that year so we couldn’t celebrate 1999 being prime until it was all over.

Y2K itself, 2000, definitely wasn’t prime.  If you know that 1999 is prime you know 2000 can’t be because after you get past 1-2-3, no two adjacent numbers can be prime — one of them would have to be even.  Next-but-one can work, though: both 1997 and 1999 are prime.  Primes separated by two like that are twin primes.

If 2016 won’t be a prime year, is there another way it can be special?  Hmmm…  2016 isn’t a perfect square, nor is it the sum of two squares.  Neither its square nor its cube are particularly noteworthy, but the square PLUS the cube is kinda cute: their sum is 8,197,604,352 which contains every digit just once.

According to The On-Line Encyclopedia of Integer Sequences, 2016 is a hexagonal number.  Start with a dot.  Make that dot one corner of a hexagon of dots.  Then add a hexagon around that, one more dot per side,  keeping the original dot as a corner (like the plan for a starter motte-and-bailey castle)…Hexagonal numbers Keep going until the outermost hexagon has 32 dots along each edge.  All the hexagons together will have exactly 2016 dots.

The OEIS says that 2016 is a participant in at least 925 more special sequences, so I guess it’s a pretty cool number after all.

Those 3s and 7s?  Here they come….

My nominee for Puzzle King of The World is my good friend Jimmy.  I challenged him once to find the connection between

  • the British Army’s WWII section number (2701) for Alan Turing’s super-secret cryptography unit at Bletchley Park, and
  • Jean Valjean’s prisoner number (24601) in Les Misérables 

Turns out it’s all about the primes.  2701 is the product of two primes: 73×37.  24601 is also the product of two primes 73×337.   Better yet, both of the product expressions are palindromes in their digits (7337, 73337). To put whipped cream on top, I first noticed the connection during my 73rd year.

So then of course I went looking for other 3…7 and 7…3 primes. There aren’t a lot of them. Going all the way out to 1037 I found:

37 73
337 733
 (3,337 is 47×71, not a prime) 7,333
333,337 733,333

Pretty good symmetry there.

OK, back to number 2016. I asked Mathematica®, “How many different pairs of primes, like 1999 and 17, sum to 2016?”

What do you suppose the answer was?  Yup, “73.”

Oh, and the next prime year is 2017.  It’ll be great.

~~ Rich Olcott