# Monthly Archives: July 2014

## A Pythagorean Pun

There were once 3 kingdoms that bordered the same lake. In the middle of the lake there was an island, and the 3 kingdoms had been fighting over it for years. No one seemed to be able to keep the upper hand for very long and no one had been victorious. The wars over this little island were very costly, but all 3 kingdoms wanted it because of its great beauty and resources.

Finally, the monarchs agreed to a way to settle the matter permanently. Each would send their knights and squires to the island and they would
fight it out. Whoever’s knights and squires won the day would keep the island forever.

One kingdom sent many knights and each knight had a few squires. The night before the battle, the knights polished their armor while the squires readied the weapons. When the armor was finished, the knights sat around the fire drinking.

The second kingdom sent more knights than the first and each knight had several squires. The night before the battle, the knights drank around the fire while the squires scurried about polishing armor and readying weapons.

The third kingdom only sent one knight and he had only one squire. While the squire polished armor and readied the weapons, the knight hung a single pot from the tallest branch of the tree and tied a rope with a loop at the end from another branch. Then the knight sat by the fire and drank while the squire kept working.

The fateful day came and all the squires came out to the battlefield. (The knights had stayed up too long drinking.) The battle was fierce. In the
end, only the lone squire from the third kingdom was left standing. Proving once again, the age old theorem:

The squire of the high pot and noose is equal to the sum of the squires of the other two sides.

The punch line of this joke is a pun on the Pythagorean Theorem, often stated as: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. You may recall from our venture into trigonometry humor that a right triangle has one ninety-degree angle, and that the hypotenuse is the side opposite the right angle (also the longest side in the triangle).

Never argue with this triangle; it is always right.

The Pythagorean Theorem (named after Pythagoras of Samos, but known to people long before his time) states a curious relationship between the lengths of the sides of a right triangle. Imagine extruding the sides of a right triangle outward to form three squares, like this:

According to the Pythagorean Theorem, the area of the square formed from the hypotenuse is equal to the combined areas of the other two squares. In other words, if you could dismantle the squares formed from sides a and b and recombine them into one square, it would be the same size as the square formed from side c.

Mathematically, the Pythagorean Theorem can be expressed as a2 + b2 = c2. It holds true for any right triangle (as long as the triangle is flat; it doesn’t work for triangles printed on curved surfaces). Here’s an animation demonstrating how the Theorem works, courtesy of Wikipedia.

Now about that joke: I dig the pun at the end, but number-wise, it doesn’t quite work. If the third kingdom represents the hypotenuse (or high pot and noose), it should be the largest force of the three kingdoms. Assuming the first kingdom had, say, 300 squires and the second kingdom had 400 squires, the third kingdom would need 500 squires to be their equal, Pythagorealistically speaking. 3002 + 4002 = 5002. I know, I know…it’s just a stupid pun, but I want this blog to be an avenue for learning as well as laughing, so I would be remiss not to mention it.

Advertisements

## On Trig and Cosby

The man on the right is Bill Cosby, an actor and comedian who became a staple of Thursday night television for NBC from the mid-1980’s to the early 1990’s. (More recently, he has become embroiled in controversy because of allegations regarding extremely unsavory actions in his past.  That’s not relevant to this joke, but it probably limits the joke’s “re-tell” value.) His last name is pronounced “KAHZ-bee”, which is important to know if you want to get the joke.

The rest of this image is all about trigonometry. You remember trigonometry, right? Maybe? No? Okay, let’s have a refresher.

Imagine a right triangle. A right triangle has one ninety-degree angle, and two angles that are both less than 90º. Let’s call these angles a, b, and c, with c being the right angle.

Now for the sake of convenience, I’m going to label the sides A, B, and C. Side A will be directly opposite angle a, and so on.

Side C, which is opposite angle c, is the hypotenuse. The hypotenuse of a right triangle is always opposite the 90º angle, and it is always the longest of the three sides.

Sides A and B are known as legs. There is a web of relationships between the measures of angles a, b, and c, and the lengths of sides A, B, and C.

Consider angle b, in the lower left corner of the triangle.

From the perspective of angle b, side B is opposite and side A is adjacent, meaning that side A is one of the legs that forms angle b. Side C is the hypotenuse, as always, and also encloses angle b. Clear as mud? Good.

To fully understand this joke, you need to know about three basic trigonometric functions: sine, cosine, and tangent. These functions are often abbreviated sin, cos, and tan. The sine of an angle is found by dividing the length of the opposite side by the length of the hypotenuse. For angle b, that’s side B divided by side C.

A sine of things to come.

The cosine of angle b is found by dividing the length of the adjacent side (side A) by the length of the hypotenuse (side C).

The tangent of angle b is found by dividing the length of the opposite side (side B) by the length of the adjacent side (side A). It does not involve the hypotenuse at all.

There’s another interesting relationship between sine, cosine, and tangent. The tangent of angle b is equal to sin b divided by cos b. I’ll leave it as an exercise for the reader to prove that.

Now we’re in the home stretch. Multiplying both sides by cos b gives:

And dividing both sides by tan b gives:

Q.E.D.

## Bechdel Jest

Two women walked into a bar and discussed the Bechdel test.

You know, it’s hard to think of another one-liner that is so charmingly self-referential.

The criteria that would eventually be known as the Bechdel test were first described in 1985 in a comic strip called Dykes to Watch Out For, by Alison Bechdel. In one particular strip (“The Rule“) two women are discussing the possibility of seeing a movie. One of the women says that she avoids any movie that fails to satisfy three criteria:

1. The movie must have at least two female characters, who
2. Talk to each other, about
3. Something other than a man.

She then quips that she has been unable to watch any movie since Alien, since the two female characters in Alien talk to each other about the monster.

These criteria have since been adopted for evaluating the gender bias in any sort of entertainment media, including video games, television shows, books, etc. How many films pass the test? According to bechdeltest.com, a user-edited database of more than five thousand movies, roughly 56% of the movies listed pass all three of the test’s criteria (it should be noted that bechdeltest.com requires that both female characters be named).

And now we can add at least one joke that passes the Bechdel test with flying colors. Bravo, anonymous joke author. Bravo.

## Solipsism Silliness

Is it solipsistic in here, or is it just me?

Here’s a bit of philosophical nerdiness for you. Solipsism is the idea that nothing can be known to exist but the individual’s consciousness. When I say “the individual”, I mean you…or me…or whoever. According to a solipsist, everything outside your head must be filtered through your consciousness. In fact, the external world might even be a product of your consciousness. We can only be certain of our own consciousness because we experience it directly. Rene Descartes said it best: I think, therefore I am. I can satisfactorily determine the existence of my own mind, but everything else could be an illusion.

A solipsistic viewpoint is untestable (which is why it falls under the purview of philosophy rather than science). Any test you could devise to determine whether the external world was real would necessarily be filtered through your consciousness. If your own mind was insistent on maintaining the illusion of an external universe, then your test results would be utterly useless, as your mind would mold them into your perceived reality.

While a person may be philosophically solipsistic, most of us behave as empiricists; that is, we act as if we believe in the objective reality of an external world, complete with physical rewards to be won, dangers to be avoided, and consequences for our actions. We do this because it yields consistent results. In most cases, our interactions with the physical world proceed as expected. Many would argue that if the external world (real or illusory) is indistinguishable from an empiricist philosophy, then there’s little point in debating whether it’s real or not. Maybe not, but it’s still kind of interesting (and maybe a bit unsettling) to imagine that reality begins and ends in your own mind, and that the universe as we know it…excuse me, as I know it, may be nothing but a dream.