Rambling Thoughts

A study in non-canonical choices

I was at the Royal Observatory in Greenwich last week. There were several exhibits on the work of the observatory, including a large section on their work dealing with longitude and time.

In the 1700s, people were interested the solving the “Longitude Problem”: how does a sailor determine his longitude? Without a good solution, sailors lacked good position data, which could lead to shipwrecks. It turns out that the Longitude Problem is equivalent to determining local time, and one solution was to build accurate and portable clocks.

These issues of longitude and time are fun to think about, but I noticed that they require making a lot of non-canonical choices. Here, by canonical I mean choices that are natural or motivated; non-canonical choices are
arbitrary and would make just as much sense if chosen any other way.

Consider this: There is a (reasonably) canonical place to set 0 degrees latitude; it is natural to define it as the equator, being halfway between the poles and splitting the globe into equal halves. But as there is no “east pole” or “west pole”, there’s no such natural choice for 0 degrees longitude. We currently use the Greenwich meridian as 0 degrees, but there’s no good reason (besides history and random chance) that Greenwich should be special… in fact, the French used the competing Paris meridian for some time. The way we define longitude on the globe is completely arbitrary, and we could just as well use a Svalbard meridian instead.

The measurement of time is similarly non-canonical. Each civilization in history has come up with their own calendar, with its own quirks. The calendar that we use today is just one of those, and there’s no clear reason that it is naturally better than the others. In the end, the everyday world that feels so natural to us is actually based on an accumulation of historical coincidences.

While we’re discussing non-canonical choices, I’ll quickly list a few more that I noticed recently:

• Cars drive on the left or right depending on where you are in the world.
• The UK has 20 pence coins while the US has 25 cent coins.
• In some parts of the world, floors of a building are numbered starting at 1, while in others, floors are numbered starting at 0.
• Electricity might come in 110V or 240V varieties, with numerous different outlet types.

Sleep

I like data. I believe in collecting data and analyzing it to see interesting trends. And it’s always sad to lose data, because then I lose opportunities to analyze said data.

One piece of data that I find interesting is sleep. The amount of sleep that I get each night strongly affects how happy and productive I am, and I want to analyze this data to see if there is anything that I can learn about my sleep patterns. So I’ve been recording my nightly sleep schedule — when I go to sleep and when I wake up. The hope is that I can eventually produce pretty graphs of when I sleep. It pains me to think that some people could be recording their sleep schedules too, but instead fail to do so out of laziness.

I’ve been recording sleep data since 13 March 2012 (excluding naps, e.g. boring classes; all times approximate and rounded to the nearest five minutes), and I can already see some patterns. The full analysis hasn’t happened yet (I plan to wait until I have more data), but here’s some preliminary statistics (as of today, August 26).

Average since March 13: 7:59:21
Maximum: 12:29 on March 29
Minimum: 0:00 on March 24 (thanks to red-eye flight)
Minimum positive time: 2:45 on June 3 (thanks to ARML)
Standard Deviation: 1:27:08

I think I’ve been sleeping more (and more normally!) over the summer:
7-Day Average: 8:41:25
15-Day Average: 8:39:20
30-Day Average: 8:29:52

Let’s hope that this trend continues!

There have been a couple of erratic weeks:
Maximal 7-Day Standard Deviation: 4:14:15 (week ending March 30)
And some very regular weeks:
Minimal 7-Day Stanford Deviation: 0:04:30 (week ending April 15, a week of times between 8:00:00 and 8:10:00)

Conveniently, recording sleep data also makes me realize when I’m not sleeping enough and motivates me to sleep more. Which is a good thing.

2*Pi Day

Today’s date is 6/28, which is $2 \pi$ Day for those of us with 10 fingers (or toes). Some evil people might tell you that they think today is $\tau$ Day, but this kind of propaganda shall not be tolerated; $\pi$ is better (or at least no worse) than $\tau$ in every imaginable way.

Coincidentally, 6 and 28 are both perfect numbers, so I declare today to be Perfect Number Day. This all goes to show the deep connection that $\pi$ has with perfection. Or maybe it just demonstrates that sufficient numerology can be made to prove anything.

2012 Pi Day

Happy Pi Day!

Here’s a movie named after this most wondrous number: Pi.

Disclaimer: I am not responsible for any mental harm resulting from viewing this movie. You have been warned.

The movie can be entertaining, but it is also somewhat disturbing. The mathematical inaccuracies are also rather unfortunate.

2012

Today marks the start of a new year: 2012. I’d like to take this opportunity to say a few things about this number.

• $2012 = 2 \cdot 2 \cdot 503$
• $2012, 2013, 2014, 2015$ have the same number of primes in their prime factorizations (counting multiplicities). The smallest numbers $n$ such that $n, n+1, n+2, n+3$ have the same number of primes in their prime factorizations are $602, 603, 1083, 2012$, so this is actually fairly rare.

And some more really rare properties:

• 2012 is not odd.
• 2012 is not prime.
• 2012 is not a square.
• 2012 is not squarefree.
• 2012 is not a perfect number.

I know I haven’t posted much recently. It’s a poor excuse, but I’ve been busy. I do have a number of things that I want to write about, however, and I’ll get to them eventually.

Too much typing

I spend a lot of time typing mathematics in LaTeX. Here’s a summary of everything that I typed for my math courses this quarter:

Algebraic number theory:
Notes: 22 pages
Homework: 49 pages

Functional analysis:
Notes: 60 pages
Homework: 42 pages

Representation theory:
Notes: 42 pages
Homework: 15 pages

That’s a total of 230 pages over ten weeks. I hope my computer is not getting too tired.

Update:
I also typed papers for my writing class (PWR) in LaTeX. That comprised a total of approximately 12 double-spaced pages. This means that for every page written for my writing class, I wrote approximately 19 pages of math.

I need more work?

I spent all of spring quarter feeling somewhat overworked. It felt as if I was trying to juggle too many things: three math classes, another class that was a waste of time, SUMO events, ARML coaching, and everything else that happened over the quarter: launching rockets, finishing up SMT stuff, various miscellaneous meetings… I didn’t have time to do anything as well as I would like, and work was often deferred until it become no longer relevant.

Now that the quarter is over, I suddenly have a lot more free time. There are no more constant deadlines, and I can actually look more than a day or two into the future without recoiling in horror at mountains of work. I’m returning to a regular sleep schedule, and everything is good — except for one minor problem. I’ve become accustomed to always having lots to do, and having free time feels rather foreign.

I’m still trying to keep myself busy, of course. My summer to do list is long, longer than what could possibly be finished in one summer. But the lack of firm deadlines makes all of that seem distant, maybe even optional. The clear lesson here is that I need more work.

Surely, I’ll stop feeling this way fairly soon. By the fall, having any work at all will probably feel odious and unpleasant. But for now, I’ll continue to ponder my long to do list, procrastinate on most of it, and wonder why I have so little work.

Dreams

One day, a few weeks ago, I was working on a representation theory problem set, and at some point, I curled up on a couch in order to take a nap. I dreamed that I was in a forest, holding a harpoon gun, hunting representations. The representations were little LaTeX symbols with legs, $\chi_U$ and $\rho(g)$ and character tables that kept running away into the bushes. I dreamed that I was walking softly through the forest, stalking my prey. I can’t recall if I ended up catching any representations, though I think I woke up from the frustration of watching those representations run away.

Another time, I started to fall asleep in an algebraic number theory class. At the time, fields were being discussed, and as I drifted into sleep, I saw a cow standing by the door at the front of the classroom. The cow appeared to be in a grassy field, and it looked very happy.

I think I should sleep more.

Happy Pi Day!

This is the obligatory pi day post.

Despite my insistence on celebrating this most glorious of holidays, I believe that celebrating Pi Day is silly numerology. Here’s why:

• 3.14 depends on the base 10 number system.
• 3.14 is only an approximation.
• Pi Day suggests that pi is more important than other mathematical constants, such as e or tau.

Of course, celebrating e day is rather difficult because February does not have 71 days. There is actually a movement to celebrate Tau Day, however. I believe that celebrating tau instead of pi is even sillier.

• Tau Day suffers from all of the problems of Pi Day that were listed above.
• What food would we eat on Tau Day?
• Why don’t we celebrate 2 Pi i day instead?
• Einstein was born on Pi Day.
• In particle physics, the pi particle is lighter and therefore more nimble than the tau particle. In addition, the pi particle interacts by the strong force while the tau particle interacts via the weak force.
• Roughly speaking, pions are the particles that bind protons and neutrons together in the nucleus (according to the Yukawa interaction). Without pions, the only element in the world would be hydrogen, which would be rather unfortunate. So pions are very important in everyday life. In contrast the tau is basically a fat electron that decays quickly and doesn’t really do anything, so it’s pretty useless.
• In chemistry, pi bonds are crucial while tau bonds are much less important.
• $\prod$ is the symbol for product, so pi represents productivity.
• “Pirates” starts with pi, and as we should know, the decrease in pirates is linked with an increase in average global temperature.

Numerology is silly. In other news, today is the anniversary of the creation of this blog. Guess which post so far has been the most popular? Ironically, it’s 2011: Numerology.

2011: Numerology

It is time to celebrate the first day of the new year, and I hereby welcome you to the year 2011.

Of course, the timing of the start of the year is an artifact of our calendar system. The choice of the date to be called “January 1” was determined more or less arbitrarily, and in fact, other cultures use different calendar systems with the new year occurring on different dates. As far as I can tell, time passes uniformly and there is no canonical time that deserves to be called the start of the year. In that sense, celebrating the new year is no more sensible than numerology.

In fact, 2011 (like every other number) has interesting numerological properties; given sufficient time and effort, the number of numerological properties are unbounded. Here’s a small sample:

• 2011 is prime
• 2011 is the sum of 11 consecutive primes: 2011 = 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211
• The past few prime years were 1997, 1999, 2003, 2011. Notice that the successive differences of these years are consecutive powers of two. In fact, 2027 is also a prime year, though 2059 = 29 * 71 is not.
• 2011^2 = 4044121. Reversing the digits on each side, we get 1102^2 = 1214404.
• 2011 = 1 + 2 * 3 * (4 * (5 * 6 + 7 * 8) – 9)