Two Identical Dissections

Everybody enjoying their holidays? I'm keeping the blog lighter than usual for the moment, so here's a short puzzler for you all. Have a happy new year!


Cut each polygon above into two congruent pieces. "Congruent" means exactly the same, but they may be rotated or reflected.

You may specify the coordinates of your cut (taking the lower left dot as origin).

ETA: I have posted the solutions here.

Our Christmas Traditions

After talking to the internet a bit, I've decided to talk a bit about what I do for the holidays. See, I've always held the conviction, since starting this blog, that my life is fundamentally uninteresting to anyone but myself. To talk about myself all the time would be solely self-serving, and thus be completely out of place on a blog. But after talking a bit with the internet, I've found that there is much interest in the question of "How do you celebrate the holidays?" Despite the strong aspect of tradition in the holidays, there is a surprising amount of variation in how people celebrate. Even though my own family's traditions seem absolutely normal to me, I suppose they are still worth sharing so that we all might compare our different visions of normal.

It shouldn’t be too surprising that I celebrate Christmas. Contrary to popular myth, leaving religion does not mean that you lose your Christmas or any holiday. That is, unless you don’t like celebrating holidays, then no one is forcing you. In any case, the rest of my family celebrates Christmas. For us, it is a very family-oriented holiday. Christmas involves two or three family gatherings (and then there’s another on New Year’s Eve).

Family gatherings are pretty fun. They always have dinners of the sort that leave excess food. And there’s also plenty of time to talk to extended family, watch movies, play board games with cousins, and everything. Sometimes I play some Christmas music on the flute, although privately I am not a fan (an entire month out of every year is dedicated to a relatively small set of music, so of course it annoys me!) It’s actually not particularly different from any of the other family gatherings we have for other holidays. Except, of course, for the presents.

There always seems to be an abundance of presents. Honestly, I don’t know how the rest of the family does it. I don’t think I could ever pick out a gift or two for each of my many cousins, aunts, and uncles, and yet my mother seems to do just that. She’s a very good selector of gifts too. As for me, I seem to have a phobia of shopping.

When I was younger, I used to get a few presents which were marked “From: Santa”. That was the extent to which my family pushed Santa. There were only a few presents from Santa, and they were not clearly any different from the other presents. I do not recall a time when I believed in Santa. I thought of Santa as something I was supposed to pretend I believed in. Because that’s what all the adults are clearly doing. But I’ve always been a bit incredulous that anyone could really believe in Santa. I’m almost inclined to think that true Santa belief only exists in the movies. I’m even more incredulous that any parents would go so far as to dress up in a Santa suit in order to fool their kids. Don’t they only do that in the movies? The whole “fool your kids about Santa” tradition seems rather alien and excessively theatrical to me.

Another “tradition” that I think only really exists in movies and Christmas TV specials, is the practice of all the kids waking up early Christmas morning, and running down the stairs to find lots of new presents under the Christmas tree. We do have a staircase, but there’s hardly any sense in running down excitedly, unless you want to be overly dramatic about it. Most of those presents have been sitting, wrapped, under the Christmas tree (always a real one, covered with a ridiculous assortment of ornaments) for the last week or so, and most of them we open at family gatherings, not at home. We do have stockings though, even if we have no chimney to hang them over.

So tell me readers, was this as boring as I thought it was (be honest), or was it fascinating because of how differently you celebrate the holidays?

Imaginary paradox

Following my proof of the obvious, I thought it might be fun to prove the impossible. Math paradoxes!

Well, of course you all know about the proof of 1=0? It's included below for completeness.

Let a = b
a*a = b*a
a*a - b*b = b*a - b*b
(a+b)(a-b) = b(a-b)
a+b = b
a = 0
Let a = 1
1 = 0

The experienced algebraist quickly spots the error. One of the steps in the proof is an illegal move.

The following proof also makes an illegal move somewhere, but it breaks a rule that most people haven't ever heard of.

We start with Euler's formula:

e^ix = cos(x) + i*sin(x)

Here, "i" is used to denote the square root of -1. Euler's formula is far from a mere mathematical curiosity. I do not exaggerate when I say we use it all the time in physics. Euler's formula is the main reason that imaginary numbers are of any use at all. But I digress.

Using Euler's formula, we know...

e^iπ = -1
(e^iπ)ⁱ = (-1)ⁱ
(-1)ⁱ = e^iπi = e^-π

But then...

e^i3π = -1
(e^i3π)ⁱ = (-1)ⁱ
(-1)ⁱ = e^i3πi = e^-3π

Therefore, e^-π = e^-3π. But clearly this is wrong. Therefore, our proof is wrong. But where?

"Doubt" trailer

This movie trailer intrigues me. I think maybe I want to see this movie now.



This not a comment specific to this movie, but I was just thinking about how much hype The Golden Compass got because it was, shocker of shockers, anti-religious. Or, supposedly it was. The first book had a rather anti-dogmatic sentiment, and the later books were much more explicit about it, but the movie itself was just too crappy to convey any of that. The point is that everyone got so worked up about a movie just because it supposedly touched on religion, and not in a positive way. Before you know it, Christians are boycotting it, and atheists want to watch it just to spite Bill Donahue.

Everyone was entirely lacking perspective. Lots of films and other fictional media touch upon religion. Has anyone here seen Chocolat? Fiddler on the Roof? Contact? Come on, I hardly watch any movies, so I'm sure a typical person can think of plenty more. Does a film really need to be explicit about it for people to take notice? What's so much better about a movie that clearly comes down against religion? Please. Have some taste.

Same goes for music. No need to get all excited about some guy just because he *cough* raps against religion.

Axial Tilt: The Milankovitch Cycles

 [Note: This is not an original image, but the website I was crediting now appears defunct]

Last year, I explained how, exactly, axial tilt causes seasons. This year, I will explain how axial tilt changes over time in what we call the Milankovitch Cycles. The Earth's orbit and spin do not stay constant forever, but change over thousands of years. These changes are much too slow to cause seasons, but they can cause much larger climate changes like ice ages. There are three Milankovitch Cycles:

Precession

I've previously discussed the precession of the Earth, but here is the shorter rehash. Although the Earth's axial tilt is always about 23.5 degrees offset from the orbital plane, the direction of the tilt moves around in a circle every 25,700 years. The cause of this change is the gravity of the sun and moon acting upon Earth's equatorial bulge. Got it?

To understand how this affects climate, we're going to have to understand different kinds of years. Isn't there only one kind of year, you ask? No, there are actually many, many different kinds of years with slightly different definitions and lengths. What do we mean by "year" anyway? If we mean the time it takes for the Earth to complete a full orbit, then what we want is the sidereal year. However, that is not the kind of year that our calendar is based on! Our calendar is based on the time it takes for Earth to complete four seasons, the tropical year. Every tropical year, there is exactly one summer solstice, when the Earth's rotation axis is tilted towards the sun. But because precession changes the direction of Earth's tilt, the summer solstice actually occurs at a slightly different location of Earth's orbit every year. The tropical year is shorter than the sidereal year by about 20 minutes.

But when we're talking about long term climate changes, we're also interested in a third type of year. The Earth's orbit is not a perfect circle, and there exists a point in the Earth's orbit when it is closest to the sun. This closest point is called the perihelion, and it occurs around January 3rd. The gravity from other planets causes the perihelion to occur at a slightly different point in Earth's orbit every year. The time it takes to get from perihelion to perihelion is called the anomalistic year. The anomalistic year is longer than the sidereal year by about 5 minutes.

The reason precession affects climate has to do with the relative location of the summer solstice and perihelion. Currently, the summer solstice in the northern hemisphere is six months away from the perihelion. This makes for milder summers, since Earth is actually a little further away from the sun during the summer. Likewise, it makes for milder winters, since the Earth is closest to the sun during the summer. Incidentally, it also makes for longer summers, because the Earth orbits more slowly when it is further from the sun. However, because the anomalistic year is longer than the tropical year, there is a 21,000 year cycle, in which the perihelion moves from winter to summer and then back again. Therefore, seasons will grow stronger, and then milder again every 21,000 years.

Milder seasons favor ice ages because the summer isn't strong enough to completely melt the ice left over from the previous winter. If the ice never melts, it reflects more light from the sun, cooling Earth and starting a feedback loop which ultimately leads to an ice age. Of course, we could just as easily argue that a milder winter is less likely to start the ice cycle. Ultimately, it comes down to a more quantitative analysis along with experimental observation, and the current evidence says says that when the precession cycle favors milder seasons, it favors ice ages.

Of course, in the southern hemisphere, winter is in June, and summer is in December. When the northern hemisphere has milder seasons, the southern hemisphere has stronger seasons, and vice versa. Why would the 21,000 year cycle affect global climate if there's always one hemisphere with stronger seasons? I don't know the details, but basically, the northern hemisphere is more important (sorry South Africa!) because that's where the majority of the land mass is. Therefore, our current place in the precession cycle favors an ice age, but obviously its effect is being outweighed by something else, possibly the other Milankovitch cycles.

Obliquity

Earth's axial tilt is currently 23.5 degrees, but in fact this number changes slightly over time. Roughly every 41,000 years, the tilt cycles between 22.1 degrees and 24.5 degrees. The cause of this so called obliquity variation is, again, the sun, moon, and planets all tugging on Earth's equatorial bulge. Because axial tilt is the reason for the season, greater axial tilt will cause stronger seasons, and smaller axial tilt will cause milder seasons. Right now, we're near the middle of the cycle, and axial tilt is decreasing. Current arguments say that smaller tilt favors ice ages.

Interestingly, it has been shown that if the moon didn't exist, the Earth's axial tilt would change chaotically from 0 to 60 degrees, causing climate changes that would possibly be fatal to life. This is often used to argue that we're pretty damn lucky to have a moon. On the other hand, current theories say that a mars-sized object crashed into early Earth, and the resulting ejecta coalesced into the moon. If that collision had never occurred, the Earth would be spinning much faster now, and its axial tilt would be stable as a result.

Eccentricity

As I mentioned before, the Earth is not exactly circular. Its orbit is actually in the shape of an ellipse, with the sun placed at one focus of the ellipse. The "focus" is basically a mathematical point in an ellipse, slightly offset from the center. The ratio between the focus's distance from the center and the perihelion's distance from the center is called the eccentricity. An eccentricity near zero means a more circular orbit, and an eccentricity near one means a more elliptical orbit. Earth's eccentricity is about 0.017, meaning it is nearly a perfect circle.

For the same reasons that the perihelion changes its location in Earth's orbit over time, so eccentricity too changes over time. Because of complicated interactions with other planets, the eccentricity varies from 0 to 0.06 in not one but two cycles which last 100,000 years and 400,000 years. Our current eccentricity is a little below a maximum of the 100,000 year cycle, and will get lower over the next 30,000 years. A complicated math calculation shows that the maximum eccentricity causes up to 0.2% more sunlight than the minimum eccentricity, but that's a rather small effect. Perhaps more importantly, a higher eccentricity will amplify the effects of the precession cycle.

We would expect eccentricity to have the smallest effect of the Milankovitch cycles, but it's an unexplained observation that it in fact has the largest effect.

As an aside, we have a rather interesting way of measuring Earth's temperature over long periods of time. See, when marine plankton die, they leave their skeletons on the ocean floor. When the ocean is colder, their skeletons tend to preferentially incorporate the 18-oxygen isotope, which is basically a less common, but heavier version of the oxygen atom. Furthermore, in colder climates, 16-oxygen gets preferentially removed from the ocean and incorporated into the polar ice caps. Thus, during colder climates, the ocean floor sedimentary deposits tend to have a higher percentage of 18-oxygen isotopes. By digging into ocean sediments, we can use this to determine the Earth's temperature for the past several million years.

What the ocean sediments show is that before one million years ago, the biggest cycle in global climate had a period of about 41,000 years, suggesting that obliquity had the biggest effect. However, about a million years ago, something fundamentally changed about the Earth's climate system, and its biggest climate cycle now has a period of 100,000 years, with ice ages slightly lagging the times of low eccentricity. What changed? Why does only the 100,000 year cycle have an effect, while the effect of the 400,000 year cycle remains small? Obviously, there is still science to be done. The current best explanation seems to be that there are "complicated" interactions and feedback mechanisms which amplify the 100,000 year cycle, but obviously the devil is in the details.

So... Seasons: pretty important? Let's celebrate!

[This being a very information-heavy post, I should probably cite my main source: The Earth System, 2nd Ed. by Kump, Kasting, and Crane. Anyways, no one should be looking to my blog as a serious research resource.]

Guess the Meaning II

What phrase is suggested by this photo?

Man, I just realized how much cell phone cameras suck compared to real cameras.

No one smart and beautiful

As I was browsing the TV Tropes wiki, I suddenly got an idea. Maybe there's a perfectly good reason why nerds are stereotypically portrayed as ugly, socially inept, or crazy eccentric.

It could be because nerds in fact do display these qualities, and the stereotypes merely exaggerate them. But we don't like that answer, 'cause that would have negative implications about ourselves. Cognitive dissonance!

But anyways, in my sudden moment of what might be called "inspiration" by excessively optimistic folks on a good day, I had a different idea. There are few perfectly smart, well-adjusted, attractive individuals in fiction because that would fall into the Mary Sue archetype. The Mary Sue is a character that has everything going for her in unrealistic quantities. Everything in the story centers around Mary Sue, and the conflicts only exist so that she can overcome them. Though the Mary Sue is obviously the author's favorite, she often comes to be disliked by the audience. Why? Because she is unrealistic, we can't relate to her, and she has only one dimension: perfection.

The Mary Sue is frequently the result of the author placing him or herself into the story. That's why authors have trouble understanding why everyone else dislikes the character. "How can you not like her? She's perfect (like me)!" It's a little egotistical, but hey, we're all a little egotistical. Except me. I'm a paragon of humility.

But back to the original idea. Why is no one smart and beautiful in fiction? More generally, why is every smart person in fiction either: a) lacking common sense b) totally awkward c) ugly and weak or d) crazy eccentric? Have they never met a genius (like me)? If they had, they'd know that smart people are completely normal, except better in every respect. If I ever had the chance to write fiction, I would portray nerds realistically: we're practically oracles with both our earthly and unearthly wisdom, we dominate every party we set foot into, and there is no problem in the world that can't be solved by a bunch of people like us. Everyone will love us, and want to be us, as we achieve ultimate cultural power!

What about that one circle?

An example of bad reasoning:

"Science has explained most instances of X. Many turned out to be hoaxes. Others had completely natural explanations. But what about the unexplained instances of X? You can't explain them all away!"

If we were playing "Name That Fallacy", I would call this "remembering the hits, and forgetting the misses". But the interesting thing is that the argument acknowledges that there have been plenty of misses, as if acknowledging them would make them go away.

The first concrete example I can think of are crop circles.

Image taken from CircleMakers

Crop circles are a phenomenon that became popular in the 1980s. At first, they were simple circles, but in more recent times, much more complicated patterns have appeared like the one above. It's practically an art form nowadays. I think they look cool, don't you?

According to the mythology, crop circles are created by UFOs, which are always saucer shaped, of course. The UFOs land on a crop and make a circle. There were other theories too, such as whirlwinds or other weather. Now, I would have looked at this and immediately thought that they were man-made. Of course hindsight's 20/20 and I'm too young to understand the 1980s mindset. In any case, there's little point worrying about back then, because the current evidence is very clear. In 1991, two men confessed to making some of the earliest crop circles in England. Well, perhaps "confession" is the wrong word--more like letting the rest of us in on the joke. One of them is interviewed here (god I love this interview).

If that weren't enough, there's even a website "CircleMakers" for people who make these crop circles. Oh look, there's even a field guide. You could be making "unfakeable" crop circles in no time!

Anyways, while I'm sure this evidence convinces most people, there still exist UFOlogists who claim that at least some, if not all crop circles are made by UFOs. You see, even though some of the circles are explained by pranksters, this small group of people couldn't possibly have made every single pattern in the entire world. And even if there were enough circle makers, there exist some circles which are supposedly too complicated to be made by dedicated pranksters.

You can easily see the problem with this reasoning. We already have a sufficient explanation. Adding a second one is just unnecessary, and unlikely to be true. For every circle maker that has confessed, there are going to be other circle makers who chose not to. Just because not every circle has a known creator does not mean that our explanation is insufficient.

Going back to my main point, this is because our investigative ability is limited. We cannot thoroughly investigate every single instance of a phenomenon. Sometimes there are things that just make full investigation too difficult to be feasible. Sometimes there is simply no budget for it. Sometimes all evidence has already disappeared, lost to entropy, leaving the phenomenon permanently unexplained. Sometimes an investigation will even show a false positive, for a variety of reasons. For example, some people heard strange sounds at night which they thought were related to crop circles. Further investigation determined that it was a kind of bird, but what might we have thought if no one had ever figured it out?

But this is not to say that scientific investigation is useless. If done correctly, by the scientific method, tests will correctly discern truth most of the time, if it manages to discern anything at all. But if you ignore most of the results just to look at the few unexplained cases, you've just done away with any validity your analysis might have had. Sometimes, those few unexplained cases are indicative of a new paradigm, but more often than not, they're related to our limited methodology. We need a more compelling argument than the existence of a few unsolved cases.

Selected Putnam Problems

At least one reader (Susan!) expressed interest in seeing the Putnam problems.

For those not in the know, the William Lowell Putnam Competition is a national math competition intended for college undergrads such as myself. The yearly contest consists of twelve problems (rigorous proof required) and six hours' time. The problems range from moderate to very difficult. Well that's kind of subjective. A more enlightening description: a few thousand students participate each year, and the median (not the mean!) is roughly zero.

I have it on good authority that it's safe to talk about them now. You can refer to the Art of Problem Solving forums for a more complete discussion of the entire 2008 contest.

So here are three of the problems. I picked out easy ones with more of an "aha!" feel to them.

A1: Let f: ℝ² → ℝ be a function such that f(x,y) + f(y,z) + f(z,x) = 0 for all real numbers x, y, and z. Prove that there exists a function g: ℝ→ ℝ such that f(x,y) = g(x) - g(y) for all real numbers x and y.

A2: Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008x2008 array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

B1: What is the maximum number of rational points that can lie on a circle in ℝ² whose center is not a rational point? (A rational point is a point both of whose coordinates are rational numbers.)

How did I do, you ask? I think I got four correct, but I hear the graders are particularly harsh, so you never know. I didn't get A2 and B1, but they sure seemed obvious afterwards. If you're still curious, ask me in the comments.

2+2=4: a proof

In case you had any doubts, here is a proof that 2+2=4.

Of course, if by "2" we mean "apple" and by "4" we mean "orange", then the statement is false. It should be clear that "2+2=4" has a specific meaning, and if we change any of its meaning, we've changed the statement. Natural numbers, such as 2 or 4, have specific meanings. They are things which obey the Peano axioms. If they don't obey the Peano axioms, they are not really natural numbers, and we might as well be talking about apples and oranges.

The Peano axioms thoroughly logical and simple to state. But I'm not going to cover it in detail, since you can just peruse the Wikipedia article for more.

For every natural number n, the Peano axioms define the "successor of n", or S(n). Every natural number, except zero, is the successor of another natural number. All natural numbers can be expressed this way:

0
S(0)
S(S(0))
S(S(S(0)))
S(S(S(S(0))))
...

We have names for each of these numbers: 0, 1, 2, 3, 4, ...

And so, by "2+2=4", we really mean this:

S(S(0)) + S(S(0)) = S(S(S(S(0))))

Not only do natural numbers have a specific meaning, but the symbol "+" has a specific meaning. It is defined with the following two axioms:

n + 0 = n
n + S(m) = S(n + m)

So here's the rest of the proof:

S(S(0)) + S(S(0)) = S( S(S(0)) + S(0) )
= S( S( S(S(0)) + 0 ) )
= S(S(S(S(0))))

Fairly simple, eh? But, hey, maybe if you find a way to tap into the power of the other 90% of your brain, you will prove the impossible. Either that or your dreams will be crushed and the resulting cynicism will negatively affect the rest of your life.

A harder problem would be to prove that n + m = m + n. I think you might even have to use the axiom of induction for that one.

In other news, I'm taking the Putnam tomorrow! Also, I'm sure this will come as shocking news: I'm going to minor in math! Yay!

Fractal Results

These are the results of the requests I got for Newton's fractals. Each function generates a fractal that, in principle, covers the entire plane, but I only show a small window of it. When I talk about the "range" of the fractal, I am referring to the location and dimensions of the window I chose.

Susan asked for the hyperbolic trig functions. Actually, they look more or less the same as the regular trig functions, but that's okay because the trig functions turn out well.

This is the function cosh(z) in the range -.5 to .5 on the real (horizontal) axis, and -.5 to .5 on the imaginary (vertical) axis. Note that only the first six roots get unique colors--the rest are all black.


And this is the function tanh(z) in the range -3 to 3 in the real axis and -2 to 2 in the imaginary axis.

An anonymous commenter asked for the function e^-(ixcosx)+e^(xsinx). This is a complicated one, graphed from 0 to 2 in the real and imaginary axes. I suspect those black comb-shaped things are actually artifacts of my program, but it took such a long time to generate the fractal that I wasn't going to try to figure out how to get rid of them. Besides, they look cool.

Why I'm not a humanist

There is a certain virtue in avoiding "positive" labels. On the one hand, you want to present yourself positively. On the other hand, you don’t want to present yourself as better than everyone else.

I don't consider myself to be a humanist. What exactly does that tell you about me? Does it mean that I don't view the good of humanity to be the highest good? Does it mean that I don't believe in any sort of human dignity? Does it mean that I don't value rational human inquiry? No, silly. It just means that I don't like the word "humanist". Maybe I technically qualify as a humanist, but I never call myself one.

In the atheosphere, I nearly always see the word "humanism" in only one context. Humanism is meant to be the positive counterpart to atheism. Atheism tells you what we don't believe in, and humanism tells you what we do believe in. Atheism is just one aspect of a person, while humanism is a complete philosophy. For a word that's supposed to be all-encompassing, I find it odd that I only ever see it in one place.

I just don't ever see the necessity to use the word "humanist". For one thing, no one ever asks me, "What do you believe in, if not God?" except in my dreams (dreams I attribute to my large ego). And if someone did ask, I'd probably just say, "Life? Goodness?" (I might also add "induction" on account of being a fanatical inductionist.) If I responded, "Humanism," who would know what that means? Most importantly, I don't know what it means. I only know the many things that have been told to me. My sources are a little vague about the details, but whatever humanism is, I know it's good! A bit of liberal politics here, a bit of the-good-of-the-human-race there, a dose of church-state separation, a rejection of the supernatural, along with a compatibility with religion. I figure that if it's good, it must be somewhere in the mix (and guess where that leaves the non-humanist).

I have no patience for any of that. If I wanted to say in detail what I believe in, I'll deliver it in plain words that everyone understands, not in a mystery package that not even I understand. For all those humanists out there, maybe you understand what humanism is, but does everyone else understand it the same way you do?

Of course, to be fair, I myself go with the "skeptic" label, which is also a "positive" philosophy. And though I understand what I mean by skepticism, not everyone immediately understands it the same way I do. Call me a hypocrite if you will. But I understand these things, that people can have aversions to labels even if they agree with the ideas represented by the labels. Just because a person doesn't go by a label doesn't mean they go against everything the label stands for. This applies to all labels.