Monday, July 27, 2015

Diagramming Noun Adjunct-Heavy Sentences

I learned to diagram sentences from my mother (a Linguistics major in college), but didn't appreciate it much at the time (to be fair, my mother didn't appreciate it either until she began teaching me and my sister). Until, that is, I ran across a sentence in the local newspaper that made me want to diagram it.

The sentence in question was “This is not an administrative license revocation matter.” (It has to do with the ongoing investigation into whether the mayor of Hilo inappropriately used government funds for personal benefit.) It's not particularly long—no complicated compound sentence or anything—it was merely the noun phrase in the predicate that attracted me. It took me quite a bit of research, though, to figure out what all those nouns near the end are.

The ending noun phrase “administrative license revocation matter” has what looks like a couple of nested noun adjuncts, which is where a noun modifies another noun the way an adjective normally would. “Administrative” is an adjective modifying “license,” which is itself serving as a noun adjunct modifying “revocation,” which entire phrase is modifying “matter” at the end.

And yet, nowhere could I find how to diagram something like that. One website helpfully told me that noun adjuncts are diagrammed like adjectives—on a diagonal line beneath the noun they modify—but I couldn't see any way to chain multiple noun adjuncts together.

After some time pondering the matter and scratching my head, I decided to pull a mathematician trick reformulate the sentence into something with a prepositional phrase: “This is not a matter of administrative license revocation.” Written this way, I can just about diagram it, though I'm not 100% sure about “administrative.” I ended up diagramming it as you would an adverb modifying an adjective because it looked nice and seemed similar, but I've no idea if that's correct. Anyway, here's what I eventually ended up with:

If any of you out there are more familiar with sentence diagramming and would like to point out how it should actually be done, feel free to sound off in the comments! A hui hou!

Thursday, July 16, 2015

Why New Horizons Can See Pluto, and Hubble Can't

About a day ago the New Horizons space probe finally reached the end of its nine-year journey through space and accomplished its mission: a fly-by of Pluto, marking the first time humanity has gotten to see the surface of this mysterious minor planet.

And what a surface it is!

Pluto, as imaged by New Horizons. Credit: NASA/JHUAPL/SWRI
Just look at all those surface features! There's a large icy vaguely heart-shaped region in the middle (which reminds me of Antarctica for some reason). It's flanked on two sides by extremely dark patches. On the right side of the picture long shadows betray the presence of fierce mountain ranges jutting from the smooth plains around them. What looks like a long canyon sits on the left side of the image, while vast smooth plains fill the top half. Speaking of which, there's a noticeable dearth of obvious impact craters—I can spot a few, but it's nothing like, say, Mercury, or our Moon.

When I was growing up in the 90's, from as early as I could remember I was fascinated by other planets. This was the beginning of my lifelong journey to become an astronomer, as I devoured every bit of reading material I could get my hands on pertaining to the solar system. This was right after the two Voyager probes had completed their missions to the outer planets (Voyager 2 flew by Neptune the year I was born, 1989), so there was an eclectic mixture of information in the books I read, depending on how old they were and how up-to-date their information was. (Looking back, I realize this was excellent training for my young self in sifting multiple conflicting sources of information and piecing together a coherent narrative from them. Huh.)

The newer books had pictures of the outer planets and their moons from the Voyager probes that were of resoundingly better quality than the ones before it. Those two probes taught us so much about the planets that we simply couldn't see from our vantage point on Earth. The point to this rather rambling divergence is that I know now what people must have felt like when those first pictures of each new planet were coming back. If you're not familiar with our previous best images of Pluto, let me show you one (courtesy of the Hubble Space Telescope):

Credit: NASA/STScI
To be clear, the actual photos of Pluto are those two small pictures at the top; the larger ones are computer models extrapolating from those pictures. These were among the best images of Pluto we had until yesterday. And yes, that's a photo from 1996, but we didn't really get any better ones in the intervening time period; here's another one from 2012:

Credit: NASA/STScI
The letters WFC3 at the top of this image stand for Wide Field Camera 3, the last and most technologically advanced camera installed on the Hubble Space Telescope, so this is as good it's possible for Hubble to get. With that in mind it's easier to appreciate just how amazing the pictures from New Horizons are.

“But hang on,” you may be saying, “why can't Hubble get better pictures of Pluto? It gets all those amazing pictures of galaxies, and they're a lot further away than Pluto is!”

If you're asking this, then you're in luck, because I asked myself the same thing driving home from work today. The apparent discrepancy comes about due to us humans not having a good intuitive sense about sizes and distances so far outside our everyday experiences. To really get a feel for why things are the way they are, we need to use math.

My idea for this was find the diameters and distances to Pluto and a nice galaxy that Hubble had photographed, take their ratios, and see just how much bigger the galaxy would appear on the sky. Then while researching these bits of information in order to write this post I discovered that an astronomer named Emily Lakdawalla had already done exactly that. So rather than write up another post that would say pretty much the exact same thing, you get to go read her blog post. (She also already has an excellent image showing the relative sizes of a lot of Pluto-sized bodies in the solar system using the newest images of Pluto and Charon!)

I had an idea to take a picture of a galaxy and a picture of Pluto and shrink the Pluto picture down and stick it on the galaxy picture to see how they compare, but I did a quick back-of-the-envelope calculation with a galaxy picture I picked out and discovered that Pluto would be about two pixels across (which agrees quite well with the conclusion in Emily's blog post that Pluto would theoretically cover less than two pixels of Hubble's WFC3). I tried sticking a little 2×2 bright green square into the image, and could barely make it out at 100% resolution even knowing where to look. So I figured it wouldn't be especially interesting to show given that putting the picture up on this blog would further shrink it. Sorry.

But to come back to the point I was trying to convey originally, this is a historic day (well, yesterday technically) for planetary science, unmanned space probes, and Pluto. If you come across any of the doubtlessly many more images to come back from New Horizons I hope you now better appreciate them for just what a huge leap forward they represent for our understanding of this fascinating little ice-and-rock-ball out on the outskirts of our solar system. A hui hou!

P.S. Also, New Horizons' mission isn't quite as over as made it sound in the opening sentence. It will continue to observe Pluto and its moons for about another month or so as it whips on past, and will probably continue to send back observations about anything else it can see way out there for a long time to come after that. Exciting!

Tuesday, July 14, 2015

Close Encounters with the Katydid Kind

Today at work, during our morning tea time, I spotted something in the hall which at first glance I somehow thought was a giant green spider, but which on a second look turned out to be a fairly normal-sized katydid.

This katydid, in fact. Let's call her “Katy” for simplicity. That's my arm, for scale.

I've never seen a katydid here in Hawaii before, so I had to check with my co-workers that it wasn't some invasive species that should be killed on sight. Once I'd ascertained that it was not, I undertook to return “Katy” to the outdoors. I don't really have any experience with katydids; I only really know what to expect from their relatives, grasshoppers and crickets. I wasn't sure if she would prove flighty and try to jump away, so I very carefully and slowly put one hand in front and one hand behind her to see if I could carry her out slowly without the entire affair devolving into me frantically trying to catch her as she hopped equally frantically away.

To my surprise, she took a calculating look at my approaching hand and calmly and sedately climbed aboard with basically no encouragement on my part. Katydids are herbivorous and harmless to humans, so while she was on my hand and so obligingly demure I took the chance to study her up close as I took her back through the break room (and my coworkers there) on my way outside.

Katy seemed to be about as interested in me as I was in her, returning my curious stare with a frank and thoughtful one. She looked as if she were mulling something over as her miniature mouth parts and antennae slowly waved to and fro. Thus I was caught completely unprepared when she suddenly leaped directly at my face!

I'm pretty sure I let out a startled squawk – I know I jumped, to the great amusement of my coworkers I'm sure. My eyes had reflexively slammed shut due to, y'know, something jumping at my face, and as I carefully pried them open again it was to find Katy calmly hanging out on my nose.

Now, if you read this blog you're probably aware that I don't take pictures of myself lightly or often, but I couldn't pass up the golden opportunity to see what I looked like with a katydid on my nose. To my great sorrow, while I was pulling out my camera Katy decided she didn't like my nose after all and jumped down onto my arm, where I had to be content with the picture above. After a few more pictures (most of poor quality sadly) I took her outside and let her off on one of bushes near the door.

Katy, back in her natural habitat.
All in all, it was fun. It's not everyday you get startled half to death at an office job by the insect equivalent of a cow, after all. A hui hou!

Wednesday, July 1, 2015

Belated Tau Day

I realize I missed it by two days, but happy belated Tau Day everyone! What's Tau Day, you ask? It's like Pi Day (3/14), but for Tau (6/28). "But what's tau," you're probably asking.

Simply put, tau is twice pi, or \(\tau=2\pi\). Like pi, tau is irrational and transcendental. But what's so special about \(2\pi\)? Why not \(3\pi\), or \(4\pi\), or any other integer multiple?

The reason has to do with circles, geometry, and the ancient Greeks. Everyone who's taken geometry is familiar with the formula for the circumference of a circle, \(C=2\pi r\), where C is the circumference and r is the radius. This, however, is the modern formulation. The ancient Greek geometers who first came up with the concept were thinking in slightly different terms: namely, that two times the radius of a circle equals its diameter. Thus, to them, the formula would be expressed as \(C=\pi d\) where d is the diameter (\(d=2r\)).

That formulation looks nice and compact, just four symbols total (although that itself is a modern artifact, as symbolic algebra as we know it wasn't invented until the Middle Ages). However, it's not quite as fundamental as it could be, because the diameter of a circle is a less fundamental concept than its radius. A circle can be—indeed is—defined as the set of all points whose distance from a single central point is exactly equal to the radius. There's no such simple way to define a circle using a diameter other than to cut it in two and use the radius anyway. Hence, the modern formulation uses radius rather than diameter.

Unfortunately, using the radius introduces that factor of two into the picture. You see, the modern formulation goes only halfway. The ancient Greeks liked their version because it was clean and simple: take any line segment to be the diameter of a circle, multiply it by this one single, simple constant (\(\pi\)), and boom, you have the circumference. And you can see the simplicity there that all mathematicians strive for.

So let's say we want to remake the modern formulation to get rid of that ugly extra 2 in there. We can simply multiply pi by it and get tau, to make our new formula \(C=\tau r\). It's simple, elegant, and—importantly—more fundamental than using pi. Take any line segment as the radius of a circle, multiply by tau, and bam—you've got your circumference.

Pi was a good first attempt at getting a very important number in math and physics, the circle constant that shows up everywhere cyclical processes are involved. It's just the wrong constant. It's the ratio of the circumference to the diameter, when the more fundamental constant is the ratio of the circumference to the radius—tau.

It's getting late and I should wrap this post up, so I will end by providing a link to The Tau Manifesto for anyone who's still interested. It's where I first encountered the idea of using tau instead of pi, and shows in detail how tau simply makes more sense to use in many areas of math and physics. I was reflexively against the idea when I first read it, but over time I've come to see how much sense it makes—to the point I'm even writing about it here on my blog. Funny how these things happen. A hui hou!