Friday, 28 December 2007

Mathematics of Christmas

Post-Christmas, actually. The backside of the holiday, when all that glitters, sparkles, blinks, and sings or dances at the flip of a switch, must be packed away until next year. I always aim to repack things exactly as I found them, but inevitably I fail miserably. Luckily though, in my house we have a system that has been in place for as long as I can remember for many of the ornaments, including some original 1970's Bloomingdale's packaging! You can't beat that.

You know what else you can't beat? The tendency of Christmas lights to tie themselves in knots no matter how hard you try to outsmart them.


Each year, you open the box to find a tangled mess, right? And after each Christmas, possessed with renewed determination, you scheme to store them more carefully this year, convinced you have finally come up with a fool proof system. So maybe you roll each strand in a loop around your elbow and then lay it carefully in a box, or you return each rolled strand to it's original box, or perhaps you wrap each one in tissue paper, secure them with a rubber band, twist tie, or god forbid, even a scrunchy. Once all the holiday accoutrements have been put away, you relax, rest assured that this time you have bested the Christmas lights and their confounding ways. But then next Christmas rolls around, and you retrieve the box full of lights only to open them and discover a twisted, tangled mess, taunting you.

Good news!...sort of. You can stop blaming yourself for this frustrating phenomenon as researchers at the University of San Diego have recently concluded that it's a mathematical and physical certainty. Blame the universe instead!

There are a couple reasons for this argument. The first being the Second Law of Thermodynamics (Code name: Entropy). Order tends to disorder, and your carefully packed strands obey by doing their part. The second more complicated reason is explained in great detail by UCSD physicists, Douglas Smith and Dorian Raymer, in their October 16th paper: "Spontaneous Knotting of an Agitated String."

Basically, mathematicians have studied knots forever and developed all sorts of theories and classifications of their variations, but physicists have only recently began to explore what equations govern their formation. To look into this, Smith and Raymer built a very simple experimental apparatus consisting of a clear plastic box and a motor. They put one piece of string in the box at a time and spun it around and around, then took it out and documented its final state - knotted or not. They did this 3,415 times varying: length of string, rotation speed, number of rotations, and size of the box.

Why 3,415 times, you may ask? I quote: "The scientific answer is that 3,415 was around the point where we had statistically compelling results," Smith said. "The human answer is that 3,415 times was about as much as we could stand." ('A string theory just in time for Christmas', Scott LaFee, Bend Weekly)

They concluded that with a minimum length of string (18.124 inches) and sufficient space for the string to shift around in it's container, knots formed fairly quickly, often within the first few seconds. Inputting these results into a computer model, they even managed to create a program which could identify the 'Jones polynomial' for each resulting knot, a mathematical property based on parameters such as the number of string crossings.

So think about the length of your Christmas tree lights (undoubtedly more than 18.124 inches), how much space they have to move around in their storage boxes (a fair amount I would presume), and how often they might be jostled when moved to and from their yearly resting place (not exactly a smooth ride up those attic stairs). In all likelihood, a knot has formed while the box is till in your hands. I guess you can still blame yourself! 

Another and better, in-depth write-up of this labyrinthine marvel can be found here.

Thrill-a-minute, edge-of-your-seat video lets you see the fun first hand!


Think you are immune to knots? They go deeper into your life than Christmas. Think about all the electrical cords in your house and the iPod headphones in your bag. Deeper still, knots are required for the DNA molecules in your cells to fit into the tiny spaces they are allotted. And even more fundamental than that, knots in the fabric of space-time could explain the distribution of matter we see today. ESA is even launching a satellite next year to look for them.

And...we're back to blaming the universe.

(photo credit: Eduardo Contreras, Union-Tribune)

Thursday, 27 December 2007

Set Your Tree Free

Here are two amazing fun facts for you:
  1. There is a National Christmas Tree Association.
  2. Your Christmas tree is recyclable.
Click here to be green and find your nearest tree recycling center.

Tuesday, 25 December 2007

Christmas in Cambridge

My first Christmas sans family, but never fear, I had a member of my astrophysics family to lean on. My friend Stas and I are keeping misery company as we spend the holidays locked in the Cavendish in front of our respective computers. Our topics actually overlap so we are helping to push each other along up the dreaded and steep hill of completion. (In fact, watch this space for our upcoming publication on the life cycle of black holes!) In the meantime though, we took a brief break to cook an outrageous amount of food and stuff ourselves silly.

Just another day by English weather standards.


The Christmas spread...Yum!


My modest first servings.


Stas's not-so-modest first servings.
Boys have such an unbelievable capacity for food!


Stas post-dinner. He went over capacity.


Attempted Christmas self-portrait of me, Stas and the Christmas tree. (Apparently the camera was more interested in the Eiffel Tower.)

Monday, 24 December 2007

Rudolph the Red Planet

Mars overload? Never.


The previous Mars posts were in reference the close pass between Earth and Mars which occurred on Monday night. However, orbital mechanics gives us something else to celebrate this holiday season. Earth, Mars, and the Sun will all align on Christmas Eve!

This particular configuration is called "opposition" as Mars will be directly opposite from the Sun with respect to Earth, or more simply Earth is directly between the Sun and Mars. For those of you who are uber keen, "conjunction" is the other time this alignment occurs, but with Mars on the far side of the Sun or the Sun directly between Earth and Mars. The diagram in Wednesday's post is actually more pertinent to this post as it illustrates the oppositions of Earth and Mars rather than their closest encounters. Jump back there for a look and then proceed on.

It's not easy to see in the diagram, but the elliptical orbits of the planets actually allow for their closest encounter and their opposition to occur separately. If all orbits were perfectly circular, the two events would be simultaneous and inseparable.

Let's step back a second, and discuss the geometry of the Solar System.

Each planet in the Solar System orbits the Sun in an elliptical orbit, as discovered by Johannes Kepler back in the day. As opposed to a circle, and ellipse has two 'centers' - termed foci - instead of one. For our Solar System, the elliptical orbits of the planets all have the Sun at one focus. But the orbit of each planet is elliptical in a different way; ellipses come in varying degrees of eccentricity, or distortion from circular. A circle is defined by an eccentricity of 0, while ellipses can have a range of eccentricities greater than 0 and less than 1. The more eccentric the orbit, the farther apart the two foci are. At an eccentricity of 1, the closed path of an ellipse breaks and becomes a parabola instead with the second focus at a distance of infinity. But that's a lesson for another day, back to Earth and Mars.

Earth's orbit is less eccentric than Mars' orbit, so the second focus of our elliptical path is closer to the Sun than the second focus for Mars. In the diagram this exaggerated to the point where Earth's orbit looks almost circular by comparison. This, combined with the oppositions plotted on the diagram, make it harder to see how the Earth can really be closer to Mars when the two aren't directly in line with the Sun, but this can be the case. Only once in much less than a blue moon will the two events occur at the same time. More often they will happen in succession as they do this year, with the closest encounter on Monday, December 18th and opposition on Monday, December 24th.

If you feel you are sinking into a geometrical quagmire, just take my work for it.

Whew, all that academic lead-up for what is hopefully still a worthwhile payoff...

The reason for this point being that in honor of this Christmas Eve spectacular, The New York Times ran an AP piece last week that proposed a cover version of "Rudolph the Red-Nosed Reindeer", with Mars as Santa's new best friend. Check out the lyrics in their entirety below and the full article here. Feel free to sing along!

Mars is a red-tinged planet
With a very shiny glow
And if you look to see it
You will find the moon in tow.

All of the other Yuletides
Santa would have at his side
The shiny nose of Rudolph
Acting as his big sleigh's guide

But this very Christmas Eve
Santa came to say:
''Rudolph, now with Mars so bright,
You can stay at home tonight.''

Then all the reindeer teased him.
And they shouted out with glee:
''Rudolph, the red-nosed reindeer
Outsourced to astronomy.''