Wednesday, January 25, 2012

Watching the Clouds Flow

I've always been fascinated with videography, but growing up I never had access to a video camera. It's partly for this reason that I made several stop-motion animations during my teen years when I got my first DSLR. The video below is another such stop-motion animation. I took the images for it almost two years ago now, back in February 2010, but was stymied because my new, Windows 7-computer (which is only 5 days older than these photos) didn't come with Windows Movie Maker, the only software I've ever been able to find that allows one to assemble stop-motion movies. I kind of forgot about it for, well, almost two years now, until I came across the link for Window Movie Maker last month and downloaded it. It's still the same old clunky piece of software it was before (although it now comes with Microsoft's horrid new ribbon interface), but it sufficed to get a semi-passable video together.

Anyway, I've rambled enough, so I'll just say that is a movie about the inversion layer on Mauna Kea, showing how the clouds come down after sunset. The camera is facing more or less west, and I took an image about every 15 seconds. You can watch as twilight deepens into dusk and the clouds from up above flow down the mountain to lower elevations.

And it's somewhat ironic, I guess, that just as I get the ability to make stop motion movies again, I also get a new phone with a bona-fide video camera built in. Of course, I can hardly wait to see how I'll be able to use that!

Tuesday, January 24, 2012

Yet Another Mauna Kea Panorama (YAMKP)

Well, it's taken me almost two months, but here's a panorama of Mauna Kea I was able to get back in November:

Mauna Kea, with Mauna Loa visible in the mid-right.
Edit (3/6/18): And here's a slightly different version made with Hugin, now with Mauna Kea in the center of the photo. I usually leave the old manual versions around as mouseover images, but in this case the two compositions are different enough that I've left them separate.

And Mauna Loa on the left.
I don't have much to say about it, other than that I think it looks pretty good, although it was a bit more work than usual to stitch together. It does nicely show the sea-of-clouds effect we usually get over the Saddle region between the two volcanoes by mid- to late-afternoon.

Random trivia fact: the first image above is merely 20% the size of the original, which was a whopping 17,542 by 2001 pixels and used a total of 10 different photos.

Sunday, January 22, 2012

Seasons of my Life.

Today I flew to Hawaiʻi for the first time without intending to attend school there. It was a rather strange feeling. A moment's reflection revealed one reason for this: namely, that I've spent a pretty good chunk of my life in college, and it's a little disconcerting not to be doing so (even if this is only a break before I continue on in my education). In fact, if we consider each semester to be 4 months long (give or take), and that as of three days ago I was 22 and 3/4 years old, I've spent roughly
\[\frac{5\times4\ \text{months}}{22.75\times12\ \text{months}}=\frac{5}{22.75\times3}=\frac{5}{68.25}=7.33\%\]
of my life in college.

While that by itself isn't too impressive, if we generalize instead that I've been in college for 4.5 years (which is how I actually phrased the problem to myself), then the fraction becomes
\[\frac{4.5\ \text{years}}{22.75\ \text{years}}=19.78\%\]
\(-\) a much higher proportion! And in all honesty, this is closer to how I think of it. That's how long it took me to get my degrees, even if I wasn't taking classes the entire period. So as you can see, having spent nearly a fifth of my life “in college,” I'm bound to feel slightly disoriented upon taking a break from it. Not that I'm complaining, mind you; au contraire, I'm enjoying it! It just remains slightly strange.

P.S. And I made it here just fine, by the way.

Sunday, January 15, 2012

Fun with Factorials

Edit (1/17/12): Due to a sharp-eyed reader, I've rewritten this post after some reflection to better show what's going on. I thought I was proving that \(0!=1\), when what I'm really doing is showing how defining \(0!=1\) is internally self-consistent with everything else about the factorial function.

Let's talk about the factorial function today. Factorials are a part of mathematics that show up in certain areas of calculus, but are simple enough to understand without it. Factorials are represented (somewhat unusually if you haven't seen it before) by the exclamation mark, like so: n! (Pronounced "n-factorial", not "n!" with a little exclamatory flourish when you say it, although you can do so mentally if you like to spice up your mathematics a bit.) A function just means that it takes an input (in this case, a non-negative integer) and produces an output (also an integer).

There is a formal way to define the factorial function, but we'll start with the "shortcut" way. For any (non-negative) integer n, the factorial is simply the product of the integer times each integer less than it. Thus \[2!=2\times1=2\]\[3!=3\times2\times1=6\] \[4!=4\times3\times2\times1=24\]\[\vdots\]\[8!=8\times7\times\dotsm\times2\times1=40,320\]\[9!=9\times8\times\dotsm\times2\times1=362,880\]\[\vdots\]
You may be wondering where this factorial function shows up in real life. It actually makes an appearance in a fairly mundane setting, namely, the number of ways you can arrange a set number of things into distinct patterns. Basically, if you n objects there are n! ways to arrange them. For instance, if you have two objects 'X' and 'Y', there are two ways you can arrange them (as 'XY' or 'YX'), which equals 2!. If you have 'X', 'Y', and 'Z', you can arrange them six ways, as so:
'XYZ', 'XZY', 'YXZ', 'YZX', 'ZXY', 'ZYX'
which equals 3!. This idea of arranging things in a set is important, and we'll come back to it in a bit.

By now you've probably noticed a pattern with the results of the factorial function. The factorial of an integer is the product of the integer itself, times the factorial of the next smallest integer, for instance \(4!=4\times3!=4\times6=24.\ \) We can write this formally as:
The factorial function is what's formally known as a recursive function, which means that each value is obtained by knowing a previous value. This is the exact opposite of the shortcut I gave above because it starts from small values and works up successively instead of working from the "top down", so to speak. If you follow this chain of reasoning, however, you'll realize that we need to know a value to start the whole recursive line off in the first place. In fact, we need two values, because of the presence of both \(n\) and \(n-1\) in equation (1) above.

Thus in practice, the factorial function can be completely described by equation (1) and two initial values, which for various reasons we choose to be the values of \(0!\) and \(1!\). In a sense we can define these initial values to be anything we want, but there's a catch: we want the factorial function to line up with what I mentioned before about arranging items. What values do we need to choose for \(0!\) and \(1!\) to make that happen?

It's probably not to hard to see that we should choose \(1!=1\). That makes sense, both in light of the shortcut I gave above (\(2!=2\times1\), so \(1!=1\) sounds good), and because if you have one item, you can really only arrange it one way.

The value of \(0!\), however, is a bit more tricky. How many ways can you arrange zero items? Is the question even well-defined? And more importantly, zero times anything is zero, so the value of \(0!\) can't be zero. You'll notice that in the shortcut method I used above, I didn't go all the way to zero, because doing that would result in zero for everything.  Instead, the value is chosen to be 1, so that \(0!=1\).

You might be wondering how this makes any sense, as I first did back in second semester calculus. While I could maybe buy the argument that zero items can be arranged one way, it still seems kind of nebulous. Is there anyway to show that choosing \(0!=1\) is consistent with what I said above about the factorial funtion?

As it turns out, there is. Let's start by taking another look the definition I gave above in equation (1): \(n!=n\cdot(n-1!).\) We can rearrange this by dividing by n on both sides to get \[(n-1)!=\frac{n!}{n}\tag{2}\]
Now, let's substitute \(n=1\) and see what we get.
Upon reflection I now hesitate to call this a proof, and consider it more of a demonstration of the internal self-consistency of choosing \(0!\) to be 1. The point is, having defined the values of \(0!\) and \(1!\) we can now go about using equation (1) to work our way up from zero:
And now you know about the factorial function.

Friday, January 13, 2012

A New Phone for the New Year!

So, I got a new phone as a graduation present from my parents before Christmas and I'm just now getting around to reviewing it. This is partly because I was waiting to have used it for a bit to get a better idea of its nature, and partly because I got sick (I'm now firmly on the mend, though).

This phone is only the third one I've ever gotten, and only the second one I've used for any appreciable length of time. I got my very first phone when I was 18, a month or two before I began college. It was a sturdy little clam-shell phone, and I kept it for the two years necessary to get an upgrade, which I did because I wanted something with a few more features (primarily the ability to transfer files between the phone and a computer and the ability to serve as a music player, but some other stuff as well). Unfortunately, I made the mistake of picking my second phone from the ones available at the store on the spot without taking time to review my options. This was (and is) contrary to my nature, which runs much more to exhaustive analysis and comparison between choices, and it soon proved to be a poor move. My new phone was even less capable than my old one (it turned out to be some sort of "simplified phone for senior citizens", which in my defense wasn't immediately obvious on the surface), and within a fortnight I'd moved back to my first phone.

That single phone lasted my entire college career to date, four and a half years, and while I'm glad to have finally updated to a smart phone I do have to admire it for serving faithfully for so long.

But a smart phone is what I got! After researching the options available to me I settled for the Samsung Galaxy S II Skyrocket, the first iteration of the highly-lauded Galaxy S line built for AT&T's new 4th-generation network. How do I like it? Read on to find out!

My new Samsung Galaxy S II Skyrocket.
This is a nice phone. It's got one of the fastest mobile processors on the market (1.5 GHz dual-core) and a stunning amount of RAM (1 GB, if I recall correctly). In fact, it's more powerful than my first computer! The screen is a roomy 4.52-inch (along the diagonal) Super AMOLED Plus display, and is noticeably larger than that of an iPhone. It's not the largest screen on the market, but it's a good size -- large without being unwieldy, and it feels good in my hand. And did I mention it's bright? This thing can really put out some serious light and good contrast, even in fairly strong sunlight, and the detail and resolution is simply stunning. (I would show you pictures of it in action, but for some reason it turns out to be really tricky to get good pictures of the display, and I don't want to show the poor-quality ones I was able to get and have you think they reflect poorly on the phone itself.)

The Skyrocket runs Android version 2.3.5 "Gingerbread" with Samsung's own proprietary TouchWiz 4.0 user interface on top. Both work quite well, and while some bloatware is to be expected on pretty much any phone you buy, the stuff pre-loaded on the Skyrocket isn't too bad (and some of it can be uninstalled). The Skyrocket has also been confirmed to receive Android version 4 "Ice Cream Sandwich" sometime in the coming year, so that'll be cool.

Edit (1/14/12): One thing I forgot to mention in my initial review was battery life. Generally, more power means faster battery drain, and most phones today require frequent recharging. I've been quite pleased in that regard, as my phone manages to hold a charge for a good long while. I've left it uncharged overnight and could probably get two days' normal use out it no problem. Fully charged I should be able to get even moderately heavy use out of it for a good long day. All in all, I'm highly pleased with the battery life (and charging rate) of this thing.

General thoughts: I like this phone. I like it a lot. Part of that is probably due to the fact that it's my first smart phone and makes my previous phone look like a landline in comparison, but I do think that this phone is excellently designed all-around (and several major review sites have agreed, some going so far as to claim the Galaxy S II as "possibly the best smart phone, period" when it first came out last year). I'm sure there will be better models coming out this year, but my philosophy with this phone (as with my computer, watch, etc.) is to get something good, strong, and sturdy that will remain competitive in the years to come, and I've got a hunch I've found it in this phone. This strategy has worked with my watch (which is...around 4 years old) and my computer (almost 2 so far), and I expect this to be a phone I'll be using for years to come, seeing as I've proven highly responsible for phones (er, phone) in the past.

Thursday, January 12, 2012

Conventional Constellations

I haven't been up to posting anything for the last few days due to a raging cold and some exhaustion after my friend's wedding, but recently I came across a list of constellations no longer officially recognized by the International Astronomical Union (IAU), and thought I'd share some of the more amusing ones with you.

To a modern astronomer, the word "constellation" has a very specific meaning: it's a region of the celestial sphere as formalized by the IAU in 1922. Nowadays, there are 88 official constellations, formed so that between them they cover the entire night sky so that any celestial object may unambiguously be found to be located in one or another of them.

Prior to 1922, this was not the case. There was no standard set of constellations, so different authors could (and did) publish star charts with their own favorite constellations on them. Since many of the authors came from different European states at a time when patriotism and national identity were on the rise, you can imagine that there was a bit of one-upping and brinkmanship going on between various star chart publishers. An example of such was Robur Carolinum, Charles' Oak, named in honor of the tree where Charles II of England is said to have hidden from Oliver Cromwell's troops after the Battle of Worcester, which was put on the map by Sir Edmund Halley, of eponymous comet fame. Other examples were Frederici Honores, Frederick's Honors, created to honor Frederick II of Prussia, and Psalterium Georgii, George's Harp, in honor of  George III of Britain (yes, the one who ended up losing the American colonies). See a bit of a trend? Actually, though, these kind of overt advances were fairly rare, and most of the proposed constellations were much more neutral in tone.

Even though they may not all have been national-glory-seeking advances, many of the constellations proposed in the 18th and 19th centuries were mutually incompatible because they incorporated stars used by constellations in other star charts. But to fully understand why people were making up constellations in the first place, it is necessary for me to back up and show how the original constellations came about.

Human beings are wired to look for patterns everywhere, and there have been recorded patterns in the stars for thousands of years. The earliest recorded star catalogs that we know of belonged to the Babylonians; the Three Stars Each catalog appear to have been recorded around the 12th century BC (near the boundary between the Late Bronze Age and the Iron Age, sometime during the period of the Judges in the Biblical chronology). After this, perhaps sometime around 1000 BC, the MUL.APIN catalog was compiled using more accurate observations.

(For the curious, MULAPIN refers to the first constellation of the Babylonian year, "The Plough" [not associated with the asterism known by the same name in Britain, which is usually known as the Big Dipper in North America]. The difference between an asterism and a constellation is fairly subtle. An asterism is just any pattern in the stars. A constellation, as mentioned before, is one of the 88 officially-recognized patches of sky that contains a particular asterism. So the Big Dipper is technically not a constellation but an asterism, located in the constellation of Ursa Major which contains additional stars and whatnot.)

The Babylonians weren't the only ones coming up with patterns in the sky, however. The ancient Chinese came up with their own system of constellations, as did the ancient Indians, the Mayans, and the Polynesians. In the southern hemisphere where the Milky Way is more visually striking, both the Inca and the Aborigines came up with constellations composed not of stars, but of dark nebulae in the Milky Way.

After the conquest of the former Persian empire by Alexander the Great in the 4th century BC, Babylonian astronomical knowledge became available to the Greek-speaking world which adopted it readily. In the 2nd century BC an otherwise little-known individual by the name of Claudius Ptolomy, a Roman-era scholor in Egypt, wrote what ranks right up there among the most influential scientific texts of all time: the Almagest. (The origin of the name is a fascinating aside: the original name in Greek meant "Mathematical Treatise", which became "The Great Treatise" [not undeservedly] and eventually just something like "The Greatest", which sounds like magisti in the Greek of the time, which the Arabs transliterated as al-majisti when they preserved the ancient Greek writings in the Middle Ages, and which was taken into English [finally] as Almagest.  Phew.)

The Almagest is a fascinating piece of work for many reasons (not least of which being that it helped set up the Aristotelian geocentric cosmology that was to reign in astronomy for the next 1800 years), but for our purposes the most interesting part of it is that it formalized a system of 48 constellations. Twenty of these were taken directly from the Babylonian system, while another ten had the same stars but different names. What's interesting is that all but one of these constellations have survived to the present day, as official IAU-sanctioned constellations (and the single one that wasn't, Argo Navis the Argonaut's Ship, was simply broken up into three separate constellations because it was considered "too big". It now exists as Vela, the Sails, Carina, the Keel, and Puppis, the poop deck).

This long, roundabout treatise on the history of constellations is mostly to give you a feel for how they have changed (and how they haven't) through most of history. In the 17th to 19th centuries there was a huge explosion in the number of new constellations being added to star charts. You see, due to his location, Ptolemy couldn't see below a certain southern declination, so he had no constellations for the area around the south celestial pole. Likewise, he didn't bother filling in every area of the sky with a constellation, leading to bare patches even in the northern hemisphere. After the invention of the telescope and the explosion of new objects found in the centuries that followed, astronomers naturally wanted an easy way to roughly locate where something was on the sky, akin to being able to give the state a particular city is located in, rather than having to come up with the exact latitude and longitude all the time. This state of affairs led to people trying their hand at coming up with new constellations to both fill in the gaps left by Ptolomy and to create new constellations where he had never set eye. To make a long story short: some were successful, others not so.

And that's what I originally wanted this much-longer-than-anticipated post to be about: constellations that just didn't make it. The "also rans". They are objects of historical interest, and some of them are just plain funny. Looking over the list, one name in particular stands out as the source of amusing constellations, an English botanist (of all things) by the name of John Hill, who introduced constellations representing the Seahorse, Toad, Leech, Slug, Earthworm, Limpet, Mussel, Rhinoceros Beetle, Eel, Long-legged Spider, Tortoise, Star-Gazer Fish, and two species of extinct mollusk. He even introduced one representing the Pangolin, which just might be my favorite animal, and which I kinda feel bad now isn't an official constellation. Oh, well, not much to do about it. Now that the sky has been officially divided among the 88 official constellations, there's no more room for someone to create their own versions and market them, except for fun. Which isn't bad, really, science needs order and stability to flourish, and an internationally-agreed-upon map of the entire sky has been quite beneficial to astronomy overall.

And with that I think I need to end this rambling post, seeing as it's gone way beyond what I meant when I started, and get some sleep. A hui hou!

Saturday, January 7, 2012

Still Alive

Well, this is now the second time I've been over a week between posts, and for the same reason as last time: I couldn't access the Internet from my computer.

So far, that is the only thing that's ever kept me from updating at least once a week. No matter how tired I am, if the last time I posted was seven days ago, I will get something up, even if it's just a simple "Hey, I'm still, alive" sort of thing. (Which I guess this is, in a way.)

I don't have too much time to write anything else tonight because I need to get my groomsman's attire for one of my best friend's weddings tomorrow (Levi's jeans, in case you're wondering. I hate wearing jeans, but since he's my best friend...). I don't know if I'll be able to write anything tomorrow, but I'll try to get something more interesting up soon. A hui hou!

Edit (1/10/12): Turns out the required shirt was a lot more uncomfortable than the jeans. In fact...well, let's just leave it at that, shall we.