The Twelve Days of Christmas: Fun with Algebra

If you're like me, after hearing the song "The Twelve Days of Christmas" for the umpteenth time, you start wondering how all those gifts stack up. How many lords-a-leaping do you end up with at the end, anyway? What do you have the most of? And what's the total?

This problem can be solved by a little bit of math no more complicated than multiplication, but this being my blog I'm going to complicate things unnecessarily for fun. Let's call an arbitrary day of Christmas n (where \(1\leq n \leq12\)). The number of gifts given for the first time on that day is then also n (one partridge in a pear tree on the first day, two turtle doves on the second day, etc.). Then the total number of each gift is simply n times the number of times that gift is given. A little thought shows that this number is simply \(13-n\) (a partridge in a pear tree will end up being given on all twelve days, while the twelve drummers will end up being given only once on the twelfth day).

Putting these facts together, we arrive at a function (I'll call it "g" for "gifts") that will give us the total number of gifts of a particular type received, given a day of Christmas as input. Symbolically:
\[\text{g}(n)=n\cdot(13-n)=-n^2+13n\]This is a simple quadratic equation (a parabola, to be exact); I've marked the locations of integer value inputs in the plot below with annotations to show what each one is.

Looking at this plot, we see that the number of each gift begins at a minimum of twelve, rises to a maximum of forty-two for days 6 & 7, and drops off again to twelve on the twelfth day.

This leaves us with the question of how many total gifts you would receive from all this. Luckily, this is very simple: since the number of gifts is symmetrical after the sixth day, we can simply evaluation the following equation:
\[\text{Total gifts}=2\cdot(12+22+30+36+40+42)=364\]which, I think we can agree, is a whole lot of gifts. A hui hou!

Edit (1/16/2016): Of course, what escaped me at the time is that the song is actually about discrete gifts, not continuous ones, and thus trying to represent it as a continuous function as I did above makes pretty much zero sense.

I'd actually originally planned to do some calculus and integrate under the curve where the gray shaded part is and show how it came out to 364 as well…except it doesn't. I couldn't figure out why at the time and thus just sort of ignored it while leaving the talk leading up to the subject intact, leading to a somewhat disjointed blog post. Sometime later I realized that it's because this is a situation where you can't describe something with a continuous function; instead of a parabola, each point should be connected by a straight line—and then the area underneath that collections of points and lines should add up to 364.

Endings and Beginnings

It's been a little quiet here on the blog front lately as I've been dealing with the passing of my sister Esther a week before Thanksgiving.

Esther had a very rare genetic condition known as Aicardi Syndrome which left her severely developmentally handicapped from birth, and given the severity with which she had it it was really only a question of time (though that's true of all of us, in the end). I don't want to dwell on it as my family is dealing with it and continuing on with life, but I do think an explanation is warranted for this longest-yet break between posts.

Being a naturally cheerful person, I don't like ending a post on a downer note, so here's something I found funny: while back home for the memorial service I picked up one of our new Polish roosters to show a visiting uncle and aunt. My mom happened to have a hair band on hand and decided to put his head feathers up in a pony tail, and the resulting picture was just too funny not to share.

(And to explain the "beginnings" part of the title, I was privileged to be able to attend the wedding of one of my cousins this past weekend.) 

Visiting Lake Waiau

Two weeks ago some friends from work and I took a trip to the summit of Mauna Kea to see Lake Waiau. It's been three and a half years since the last time I saw it, back in February 2011. It was interesting to go back and see my pictures from that trips, since the lake was a bit smaller back then. I've put two pictures from the two trips together below for comparison:

Lake Waiau in February 2011.

Lake Waiau in November 2014 (sorry for the poor quality, only put this up for the comparison).

Interestingly, in between these two photos, the lake pretty much dried up over the summer of 2014. I regrettably never got around to seeing it in that state, but I heard from multiple people that it was little more than a puddle. Mauna Kea got a lot of rain in the last few months, though, due to a few winter storms (and Hurricane Iselle, which dropped a lot of rain).

One thing I noticed while skimming my post from the last trip was that I mentioned the hypothesis I'd read about that Lake Waiau was fed by melting permafrost. Given how the lake dried up over the summer (coinciding with the culmination of a long period of very dry weather both for Mauna Kea and the island of Hawaiʻi), I think we can consider that hypothesis pretty much busted. Looks like it does depend on precipitation after all!

This next picture is a panorama taken from down closer to the lake. It was really peaceful when we got down there, down in the bowl of Puʻu Waiau. It was a beautiful day, blue skies, scattered high clouds, bright sun, gently rippling lake…very idyllic. The peak in the background is Puʻu Poliʻahu, which currently has no telescopes but was the site of the very first telescope on Mauna Kea all the way back in 1964, fifty years ago.

Can you spot the two tourists visible in this picture? (On top of the rim at the far right of the picture.)

(Edit 8/5/18: I've replaced the original panorama with one created with Hugin, but you can see the original by mousing over the image above!)

I also got a picture of my friend Graham getting his own photos of the lake. I like the composition of this picture.

And a close-up picture of the lake's rippling surface just because I like it.

On the way up the mountain we stopped at the Visitor Information Station to acclimate (as you should), and I took the opportunity to visit the silversword enclosure to see how they were doing. Sometime in the interval between the last time I visited and now a bunch of new seedlings were planted, which were visible all though the enclosure. Some of the mature plants I remembered from last time had grown even larger, and looked even more amazing. Here's one, with a close-up of one of the rosettes below:

A mature silversword plant, with a whole bunch of rosettes.

Close up of one of the rosettes.

They really are amazing plants. Wish I could grow one as a houseplant, but c’est la vie. A hui hou!

An Abundance of Lei

Well, by “abundance,” I mean “six,” because that's how many complete crochet lei I have lying around at the moment. I'm now finishing one about every two weeks or so on average.

Some of them are fairly simple, such as the “Rosebud” one on the top right, creating a sparse appearance with the center “stem” easily visible. Others, like the “Hibiscus” one at lower left, have densely arranged petals that hide the stem. It doesn't show up well in this picture, but the two pink ones are actually slightly different shades of pink. The two orange ones used the same color, but are different patterns; the one on the top is a bit sparser than the one on the bottom, though again it doesn't show up well.

I realize, looking at this picture, that some of these lei look rather similar. This is partly because, when I first started crocheting them, I only bought a few colors of yarn that showed up in multiple patterns so I could easily reuse them. I've since bought more yarn in a wider range of colors (the red and pink lei on the bottom were both products of that trip, actually), and I'm planning on doing some with a wider variety of colors in the future. I'm working on one called “ ‘Ola‘a Beauty” at the moment which has dark purple petals and a yellow stem.

Most of the designs are really quite simple in practice, being nothing more than a simple repeated motif which is twisted around the stem to create a (mostly) radially symmetric pattern. The lei in the middle on the top, however, is a different design, and much trickier to pull off.

The book I'm using calls the flower “Mauna-loa,” and instead of radial symmetry it requires a sort of bilateral symmetry which gives it two markedly different sides.

This photo's not the greatest, but you can see how the matched sets of petals emerge on opposite sides of the stem, which itself required a novel, tricky technique to make. Instead of being a simple chain it's more like a flat ribbon, and is the source of the different look of this lei. There's another lei using the same pattern for the stem which also looks pretty interesting, and I plan to get around to it sooner or later.

Since people keep asking me what I'm going to do with all these lei, I want to mention that I've already given away the first three I made while I was visiting family this summer. I also gave away another I'd just finished to my boss, who served as the Associate Director of JCMT for the past two years on loan from the National Research Council of Canada but whose secondment ended at the end of September and had to return to Canada. I'm sure I'll find opportunities to bestow more of them in the future, too. A hui hou!

Slice of the Sun

Wow, has it really been nearly a month since I posted last? Part of the reason for this is that the project I've got for this post took me quite a while a do. Two years ago, on a whim, I decided to create a scale model of the earth's interior for myself, and posted it. I got a lot of positive responses to it, and this month I decided to undertake a similar project I've had in mind for some time: doing the same thing, but with the sun.

When I first looked up the sun to do research for this project, I noticed the serendipitous fact that the sun's radius is about 109 times larger than the earth's. This immediately suggested that I decrease the scale by a hundred times, so that one pixel would represent one hundred kilometers instead of a single one. Once again, I've broken the image up into pieces so Blogger would accept them (you can paste them back together seamlessly if you want to), and written most of my commentary in the body of the image itself. Anyway, without further ado, here it is:

I hope you found that as interesting a trip as I did. I'm sorry it took so long for me to get this out there; I was actually all ready to release it over a week ago, only to discover that I'd accidentally used the sun's radius as its diameter and made the whole thing half as large as it should be. Expanding it and finding more facts to fill up all the new space took some time, but I think it's been worth it. Anyway, now that's finally done I can move on to some other projects I have in mind. A hui hou!

Twenty months.

It's interesting to see what can happen in two years. Well, more like twenty months in my case. When I accepted the offer of my current job at the Joint Astronomy Centre (JAC) at the end of November 2012, this day, September 30, 2014, was as long as I was guaranteed employment due to the funding agencies in the UK and Canada ending their funding of the JAC to focus on other projects.

Thankfully, a lot can happen in twenty months. Funding from the Science and Technology Facilities Council (STFC) in the UK, the majority funding partner, has been extended through January, when the East Asian Observatory (EAO) is poised to take over management of the James Clerk Maxwell Telescope from the JAC as the JAC officially ends as an organization after three decades of astronomy on Mauna Kea.

I know there's a lot of acronyms for terms that I haven't necessarily discussed much before in those two paragraphs, for which I apologize. I haven't talked about the transfer process much because for much of it there were a lot of unknowns, such as whether it was actually going to happen or not, which meant that it was sort of hush-hush. The good news is that it's now virtually certain that it will happen, and as a result, I continue to have a job with the JAC through at least the end of the year, as well as a job with the EAO when the JAC finally comes to an end.

So yeah. A lot can happen in twenty months. Thankfully, I still have a job that I enjoy, and a continuation of that job even as the telescope I work for changes hands. At the very least, it'll be interesting to see where the next twenty months takes me.

PSA: “Shellshock”

PSA: If you use any form of *nix as your operating system – some form of Unix, Linux, or Mac OS X – and you have the Bash shell installed, you may want to see about updating it. This is because of a newly-disclosed critical vulnerability in it (dubbed by some “Shellshock”) that could potentially allow an attacker to execute arbitrary code with a specially formed attack. (If you're on Windows you're probably ok, unless [possibly] you're using something like cygwin, in which case you're probably tech-savvy enough to look up how to fix it.)

It's quite easy to check if your version of Bash is vulnerable: open up a terminal window and run:

> env x='() { :;}; echo vulnerable' bash -c "echo this is a test"

Should Bash be vulnerable, it'll return:

> env x='() { :;}; echo vulnerable' bash -c "echo this is a test"
> vulnerable
> this is a test

If it's not vulnerable, you should just get back “this is a test” and possibly a warning about Bash ignoring an attempted function definition. If it does return vulnerable, you'll want to see about updating your version of it as soon as possible (most [all?] of the major Linux vendors have released patches, and Apple is expected to do so soon). Basically, instead of a harmless “echo vulnerable” (which just prints the word “vulnerable” to the terminal), an attacker could potentially put a much more harmful command in there, which in the right circumstances could do a lot of damage.

One noteworthy feature of this bug is that it's apparently been around for something like twenty years (Bash is fairly old as software goes), so it's expected to be found in as incredible number of devices, including a large number of the servers that host the World Wide Web.

It's still early yet to see what will come out of this, but it's likely to be a serious security issue for quite some time due to the difficulty of getting everything patched.

(I was feeling creative tonight, though I may have been a bit too literal.)

Edit (9/27/14): I wasn't entirely happy with that version so I went back and played around with it some more and made a few more versions (and learned some useful lessons about Inkscape in the process). Here's the version I currently like the most:

Photo Fun with the Tropical Sun

One fun thing about living in the tropics is how there are days when the sun is directly overhead. When it happens is a function of your latitude, but if you live between the Tropic of Cancer and the Tropic of Capricorn, two days of the year the sun shines straight down around noon (exact time dependent on your longitude).

In Hawaii there's a local term (of comparatively modern origin) for it, Lahaina Noon. Lā hainā means “cruel sun” in Hawaiian (it's also the name of a town on Maui named for its famously dry and hot climate). There's a colorful ancient Hawaiian phrase for the time of noon generally, kau ka lā i ka lolo, an idiom that translated literally means “the sun rests on the brains.”

I unfortunately never seem to get around to actually going out and looking when it happens (in Hilo, the Lahaina Noon days are in May and July), but even now in September the sun still stands very nearly straight up in the sky around noon. I happened to notice the neat shadows it produced on the sign at the Smithsonian Astrophysical Observatory building across the street one day, and decided it was worth a picture:

According to the timestamp, this picture was taken at 12:41:44 PM on September 5th, 2014. In case you can't tell from the picture the letters are small metal shapes protruding out from the wall on little supports, which allows this neat interplay of light and shadow. Just another fun feature of living in the tropics. A hui hou!

Hiking Pololū Valley

Well, as you can see this post is a few days later than I'd originally planned, for which I apologize. Upon arriving at Pololū Valley, we were greeted with this lovely panoramic view:

Pololū Valley.

(Edit 27/7/18: I've replaced the original panorama with one made by Hugin, but mouse over it to see the old one.)

In this picture, Pololū Valley is on the right. We're now looking back down the coast towards Waipi‘o Valley, although it's around the bend in the coast and can't be seen from here (you'll notice there are some tiny rocky islands further down the coast that couldn't be seen from the Waipiʻo Valley overlook). Here's a smaller picture that focuses on the sea end of the valley:

You can see the flank of Mauna Kea sticking out behind those sea rocks.

As we descended the trail down the cliff face, we'd occasionally get glimpses further back up the valley. I usually avoid putting myself in pictures, but I had Graham take this one:

Hark! A valley! By this time, clouds had come in and were shrouding the head of the valley.

Thankfully the path was well-shaded for most of its length by the verdant overgrowth that hugged the steep walls of the valley like some overgrown moss carpet. From a lower switchback I got this vantage point of the beach:

For some reason this picture looks “cool” to me (as in “not warm”). I think it's all the blue.

And just because I love the view along the Kohala coast so much, here's a zoomed-in picture of it:

Again, that's Mauna Kea peeking out from behind the sea-cliffs there.

I think I can count four more valleys down the coast in that picture, but I'm not sure. A little further down the trail, a great vantage point and lucky break in the clouds gave us this sight:

Looking towards the head of the valley.

Finally, after perhaps half an hour's hike, we reached the beach, made up of moderately-sized rocks further back and gray sand at the water's edge:

Pololū Valley beach. Dangerous to swim at according to the signs, but nice.

You can see the curvature of the horizon! Probably just my lens, actually.

Wandering a few dozen meters inland we found the large pond we'd seen from above. I think this connects with the ocean at high tide, so I suspect it's brackish water. The ancient Hawaiians used to catch fish and raise them in ponds, and they may have done so here.

You can clearly see the clouds hovering at the head of the valley here.

Graham, inspired by the same sight.

Back beyond the beach there was a small forest of some kind of pine tree (at least, it had needles like conifers – I don't know what kind of tree they actually were). They were scattered thinly, and it was quite pleasant to walk in their dappled shade.

Eventually we reached our goal for the day, the southern wall of the valley where the rock once again rapidly rises to tower five hundred feet or more above the valley floor. Some of the cliffs further along the coast are even higher, I believe (among the highest in the world).

From there, along the curve of the beach, we could look back up the coast, which was quite pretty:

Looking back along the beach and up the coast.

That house in the picture at the top of the cliff sits at the trail head near where I took the panorama at the beginning of this post, to give you an idea of the positioning.

All in all, it was a really great hike, and I'm glad I went. It took us a while to drive there and back, but we weren't in a big hurry and it was perfectly doable in a day. The trail in and out was really quite reasonable given the steepness of the cliff face it was descending. And the weather was pretty much as perfect as it could be. Hopefully I'll be able to make my way out there again sometime. A hui hou!

Getting to Pololū Valley

For Labor Day, me and my friend Graham decided to head to the north part of Hawaiʻi island and hike the trail into Pololū Valley. Pololū (meaning "long spear" in Hawaiian) is the northern-most of the seven major valleys that cut the northeast flank of the extinct volcano Kohala, which sits north of Mauna Kea. On our way, we stopped at Waipiʻo Valley, the southern-most of the seven valleys. These valleys all have very steep walls; there's a 4WD-only road down into Waipiʻo (the only way in other than by sea), but no vehicle roads into any of the other valleys.

Waipiʻo Valley. (Mouse over for original handmade panorama.)

This is a panorama I put together from a series of pictures taken at the Waipiʻo Valley overlook, at the end of the valley on its southern side. (Edit 26/7/18: I've now replaced the original handmade version with one made using Hugin.) After stopping there we continued on to the town of Waimea, where we stopped for lunch at a local burger joint called The Village Burger (excellent food!), before continuing up along the road below the ridge of Kohala. The region of Kohala and the northern flank of Mauna Kea have a lot of cowboy culture, and the parking lot had some rather amusing signs:

Driving north from Waimea along the upper highway that runs below and parallel to the Kohala ridge, there's a pretty dramatic demonstration of the rain-shadow effect. Kohala is an old, extinct volcano, the north-east side of which collapsed in a land-slide that left it with a peak ridge running north-west to south-east and created the dramatic cliffs that end so abruptly at the sea as you saw in the first picture.

The prevailing winds in the Hawaiian archipelago come from the north-east, so most of the humid air condenses into rain as it encounters the north-east side of Kohala; very little makes it over to the west side (which is also why Hilo on the east side is wet and Kona on the west is dry). The road runs mostly parallel to the ridge and the line between wet and dry, as you can see in these pictures. As you drive north, on your right you have this view, looking up towards the ridge:

And on your left, running down to the sea on the west side, you see this:

(That's Kawaihae harbor on the shore there. I'm guessing the extreme haziness in the air on this side is vog from Kilauea being blown up around the west side of the island.)

Another cool sight you get to see on a clear day like we had is the vast bulk of Maui rising in front of you. I was actually quite surprised at how different Mauai appears from the summit of Mauna Kea, as that's usually the only place I see it from. The ʻAlenuihāhā channel between Hawaiʻi and Maui is only about 30 miles across, so it's not that far away.

This actually turned out to be a lot more photos of the drive up than I thought, so I'm going to put the photos from when we actually got to Pololū Valley into their post, hopefully up tomorrow. A hui hou!