Yet another geneticist’s pick-up line: If I had to choose between RNA and DNA, I’d choose RNA – because RNA has U in it.

Buckle in, folks, because we’re diving once again into the murky molecular world of genetics. We’ll be focusing our attention on DNA (**D**eoxyribo**N**ucleic **A**cid) and RNA (**R**ibo**N**ucleic **A**cid), two indispensable molecules that make us – and by *us* I mean *all living organisms on Earth* – what we are.

We’ve discussed DNA before. DNA is the molecule that contains the instructions for building a living thing. Every one of your cells – except mature red blood cells – stores a complete blueprint for making you. That blueprint is encoded in DNA.

The beautiful thing about this genetic code is its simplicity. Everything in your genes is spelled out using just four letters: A, T, C, and G. These are the initials of four molecules called *adenine*, *thymine*, *cytosine*, and *guanine*. DNA, you may recall, is shaped like a twisted ladder. The rungs of this ladder are made of two of these molecules in complementary pairs. For example, adenine and thymine – A and T – always pair up together in a rung, and so do cytosine and guanine – C and G.

At the risk of digressing from the main topic, that seems pretty incredible! How can just four letters make a *you*? After all, you’re *way* more complex than that, right?

The letters tend to work in triplets called *codons*. So for example, one side of a DNA molecule might read:

TCA ACC AAA TGC GGC GTC GAT.

And its complementary strand would read, in the same direction:

AGT TGG TTT ACG CCG CAG CTA

Each triplet codes for a particular amino acid, and the sequence of amino acids makes up a protein, which is a major structural and functional component of living cells. Working proteins handle every other function the cell needs to live, from manufacturing molecules to digesting nutrients.

Here’s an analogy; if the cell were a city, then DNA would be blueprints for important facilities and machines to build the city. Each machine is fine-tuned for an important function, so it’s important that its blueprints are stored, interpreted, and copied as accurately as possible.

Human cells keep DNA in the nucleus – the membrane-bound central region of the cell. The nucleus is like a vault. It is vital to the safety of these all-important blueprints that DNA never leaves the relative safety of the nucleus. How, then, do the genetic instructions get to the cellular machinery and infrastructure that interpret them?

That’s where RNA comes in. RNA is built in the cellular nucleus, transcribed directly from DNA. DNA unzips, and complementary bases are laid in along the exposed bases. After transcription, RNA is shipped outside the nucleus, where it dictates the instructions for building proteins to special protein-building units called *ribosomes*.

As you can see, RNA is single-stranded, unlike double-stranded DNA. But that’s not the only difference.

RNA does not contain thymine. When the DNA-to-RNA transcribing machinery encounters an *adenine* base on the parent DNA molecule, it adds a different base, *uracil*, U, to the growing RNA strand. So if a DNA strand reads:

TCA ACC AAA TGC GGC GTC GAT,

The complementary RNA strand reads:

AGU UGG UUU ACG CCG CAG CUA

Now here’s the big question: Why? Why does RNA have U in it?

The better question, as we shall see, is why *doesn’t* DNA have U in it?

As with all of life’s problems, the answer has to do with chemistry. All of your cell’s important molecules are bathed in a soup of chemicals with varying degrees of reactivity. Plus, we amble about in an environment awash in radiation; ultraviolet, gamma rays, X rays – you’re sure to absorb a few hits from ionizing radiation in your day-to-day activities. It’s just common sense that your DNA is going to take a beating.

For example, the base cytosine, C, is easily converted to uracil, U. Left unchecked, this simple substitution could wreak havoc on your genetic code. Fortunately, all of your cells have a repair mechanism to hunt down and correct errors just like this one. Your cells “know” that uracil doesn’t belong in DNA, so they convert it back to cytosine whenever they find it.

This is no small problem, by the way. It happens about 100 times per day, per cell. The enzyme that handles this repair, Uracil-DNA glycosylase, or UDG*, has its work cut out for it.

So DNA cannot contain uracil as one of its bases, because if it *did*, then UDG would have no way of knowing which uracils to keep and which to replace with cytosine. That repair pathway would not work.

But hold on, the savvy reader will ask: if U isn’t good enough for DNA, why is it good enough for RNA? Can’t cytosine get converted to uracil in RNA just as it is in DNA?

Yes, but RNA isn’t meant to exist for very long; just long enough to transfer the genetic code from the nucleus to the ribosomes, where proteins are assembled. See, DNA gets copied and transcribed over and over; its code has to be durable. A persistent error in DNA can kill the cell, or worse, lead to cancer. But RNA is a short-lived throw-away molecule. If an RNA molecule suffers a cytosine-to-uracil mutation, the worst that happens is a couple of proteins don’t get made correctly. It doesn’t matter in the long run; there will be hundreds or thousands of RNA molecules that *don’t* get mutated. Business will carry on as usual.

So we know why thymine is preferable to uracil in DNA, but we haven’t discussed why uracil is preferable in RNA.

It’s because uracil is energetically “cheaper” than thymine. It costs less energy and resources to manufacture and use uracil than it does for thymine. So thymine is only used in DNA, whose accuracy is tantamount to the cell’s – maybe even the entire organism’s – survival. And cheap, easy uracil is used in cheap, disposable RNA. It’s true what they say: you really do get what you pay for.

Now that we know how DNA and RNA use thymine and uracil, respectively, is this really an effective pick-up line? If the intended recipient of your woo knows anything about genetics, probably not. In essence, you’re telling him/her that they are cheap and easily replaced. These are not the words of a lover.

*UDG is notable as an initialism in that one of its letters stands for another initialism. Which is a shame, because I think it would be fun to talk about UDNAG.

]]>Q. Two cats are sitting on a roof. Which one slides off first?

A. The one with the smaller mu!

Of course this joke assumes that the cat in question is totally complacent to slide off the roof, making no effort to maintain his position. Strange cat.

Anyway, *“mu”* is pronounced like “*mew*“, as in the sound made by a cat. It is a Greek letter, usually represented by the following symbol: µ. Mu must be the favorite Greek letter of mathematicians and scientists; it pops up in fields as diverse as computer science, number theory, physics, orbital mechanics, chemistry, and pharmacology. In *this* joke, µ is meant to represent the *coefficient of friction*, about which more in a moment.

What is friction? To greatly oversimplify things, friction is a force that *resists relative motion* between two surfaces, or between a surface and a fluid. When you experience resistance while pushing a refrigerator across a tile floor, you’re working against friction. When you rub your hands together to warm them up, friction is your friend. Friction is an even *greater* friend to the skydiver; when she opens her parachute, fluid friction against the atmosphere reduces her speed from a spine-shattering 120 miles per hour to a totally survivable 10 miles per hour.)

Here’s an interesting side note about friction; scientists used to think that the friction was caused by microscopic grooves and bumps that tended to lock surfaces together, requiring extra force to break their grip and get the surfaces sliding past each other. Now, scientists think that friction is caused by chemical bonds forming between the atoms in the adjacent surfaces. That’s a strange thought; merely by touching something, you bond with it. In a way, you become a part of it and it becomes a part of you. Deep, man. Deep.

But I digress. Mathematically, the friction between two surfaces – such as, say, a roof and a cat’s butt – can be expressed using the following formula:

F_{f} = µ * m * g * cosθ

F_{f} represents friction, which is measured in units of force called *newtons*. The letter *m* represents the mass of the cat in kilograms, *g* is the acceleration due to gravity (On Earth, that’s about 9.8 m/s/s) and cosθ is *cosine of angle theta*, where *theta* (another Greek letter strongly favored by the academic elite) is the angle that the roof makes with the ground.

Just to have some numbers to play with, let us assume that the cat’s mass is 3 kilograms, giving her an Earthly weight of about 6.6 pounds. Now let us assume that the roof has a pitch of, say, 30º. To find the friction between the cat’s derriere and the rooftop, we would substitute and multiply:

F_{f} = µ * m * g * cosθ

F_{f} = µ * 3 kg * 9.8 m/s/s * cos(30º)

F_{f} = µ * 25.5 newtons

I have not yet specified the roof-feline coefficient of friction, because frankly, I don’t know what it is. My search of the literature has been fruitless. For the sake of argument, let’s assign a completely arbitrary value of 0.6 to µ, and see what that gets us.

F_{f} = 0.6 * 25.5 newtons

F_{f} = 15.3 newtons (about equal to 3.4 pounds of force)

So there you go; there are 15.3 newtons of friction preventing the cat from sliding down the roof. Whether the cat actually slides or not depends on whether the gravitational component pulling the cat down the roof is greater than the friction holding the cat in place.

But let us assume that the coefficient of friction between the cat and the hot tin roof were smaller, perhaps because the cat had just finished grooming and her fur was unusually even and smooth. Instead of 0.6, let’s say the coefficient of friction were only 0.3, giving the cat a static friction of only about 7.7 newtons. Naturally, with a smaller coefficient of friction – a smaller *mu* – the cat would be less able to hold its position on the roof and more likely to start sliding downward.

So there you have it: the cat with the smaller *mu* is the one that starts sliding first. Next time somebody tells you this joke, they’ll be met with less friction, because you’ll understand it purr-fectly.

Okay, I’ll go now.

]]>Two mathematics professors, Bob and Steve, are talking in a bar. Bob laments to Steve that the average person just doesn’t understand math anymore. Steve thinks Bob is being too hard on people. When Bob excuses himself to use the restroom, Steve walks over to a pretty blonde sitting by herself and says “When my friend comes back from the bathroom, I want you to make some pretense of walking by us. I’m going to stop you and ask you a question, and I want you to say “x squared”. Got that?”

The blonde looks puzzled: “x…squared?”

“Yes, that’s right.”

The blonde repeats: “x squared…okay, I think I got it.”

A moment later, Bob returns from the bathroom. Steve says “Hey, I bet I can prove that the average person knows math. Take this woman, for example,” he says, motioning to the blonde who, right on cue, is walking past them. “Excuse me miss, so sorry to bother you, but I wonder if you can tell me: what’s the integral of 2x?”

“x squared” says the blonde, and she starts to walk away. Bob is flabbergasted.

Then the blonde turns around and yells “*PLUS THE CONSTANT OF INTEGRATION!*“

First, I know it’s been a long time since I’ve updated this blog. When I started writing it, I thought I would be able to dig up enough nerdy jokes to fill a football stadium, but I lost momentum as it became more difficult to find jokes that dealt with topics about which I had not already written.

Still, I’m not ready to call “Nerdy Jokes” a done deal just yet. I’d like to continue to update this blog whenever possible, if for no other reason than for my own education and for the entertainment my two or three loyal readers. Thanks: I love you guys!

Now then…integration is a mathematical process from calculus with applications in physics and other sciences. To teach you how to do integration would be *far* beyond the scope of this blog, so here are some helpful links. Take a few minutes to study these websites, and you should be up to speed on calculus in no time at all!

Integration is the opposite of differentiation, and it’s much easier to understand integration in that context. Plus, understanding what differentiation is will make it easier to comprehend the blonde’s triumphant rejoinder *vis-à-vis* the constant of integration. So let’s start there.

Consider the function f(x) = x^{2}, whose graph is shown below.

This graph is a parabola; in fact, you might say it’s the *archetypal* parabola. Every other parabola you can imagine (and I don’t know about you, but I can imagine quite a few parabolas!) is merely a variation on this theme. Some are wider, some are skinnier, some are upside down, but all parabolas have in common a certain set of characteristics; chief among them is that the *height* of any point on a parabola is directly proportional to the square of the horizontal distance of that point from the parabola’s axis of symmetry.

But pardon me; I digress. Could you find the *slope*, or slantiness, of this parabola? The answer seems to be no; after all, since the parabola is not a straight line, it does not have a constant slope. But what if I asked you to find the slope of a single point on the parabola? Again, this seems like an impossible task; the slope of any line segment can be expressed as *rise* over *run*; a single point, by definition, has neither *rise* nor *run*, and so it cannot have a slope.

Fine, fine, but what if I told you to imagine a straight line lying tangent to the parabola at a certain point, say, *x = 1*. Surely you could find the slope of *that* line, couldn’t you?

And the answer is: *yes, you can find the slope of a straight line lying tangent to a curve at a specific point*. That’s where *differentiation* comes in. When you differentiate a function, or find the derivative, you come up with an expression that gives the slantiness of a line that glances off of a function at a certain point. In the case of f(x) = x^{2}, the derivative is f'(x) = 2x. (You can read the first part as “f-prime of x”. There are many different ways to indicate a derivative.) So if you want to know the slope of a line lying tangent to the graph of f(x) at the point *x = 1*, you simply substitute *1* for *x* in the derivative equation, and that gives you the slope of that particular tangent line:

f'(x) = 2x

f'(1) = 2(1) = 2

The slope of line lying tangent to f(x) at *x = 1* is 2, which means that this line rises 2 units for every 1 unit it runs. A line tangent to any other point of the parabola would, of course, have a different slope.

If you’re still struggling with the concept of differentiation, here’s a real-world analogy that might help. Say you’re sitting at an intersection when the light turns green. You step on the gas pedal, causing your car to accelerate from rest to, say, 100 km/h (roughly 60 mph, for those not inclined to use metric units). If you were to look at the speedometer at any time while speeding up, the reading of the speedometer would be similar to the derivative of a function. It would tell you how fast you were going *right that moment*, not ten seconds ago or three minutes from now.

Integration is the *opposite* of differentiation. Imagine that I gave you the derivative of an equation, and asked you to suss out the original equation. You could use integration for that purpose. If 2x is the *derivative* of x^{2}, then x^{2} is the *integral* of 2x. Right? Riiiiiiight?

Well, almost. See, there are *infinitely* many functions that have the derivative 2x. Besides f(x) = x^{2}, there’s also:

- f(x) = x
^{2}+ 1 - f(x) = x
^{2}– 1 - f(x) = x
^{2}+ 8 - f(x) = x
^{2}– 148 - And so on…

Basically, you can add or subtract any number – called a *constant* – from x^{2} and you’ll get a new function that has the *exact same derivative*.

- If f(x) = x
^{2}, then f'(x) = 2x - If f(x) = x
^{2}+ 1, then f'(x) = 2x - If f(x) = x
^{2}– 148, then f'(x) = 2x - And you get the point…

It’s impossible, given only the derivative of an equation, to integrate and figure out *which* constant to add or subtract. You need additional information. Neophytes learning about integration for the first time are often told to include a *constant of integration* – a variable that stands for any conceivable number that might be added to or subtracted from the function. When asked to integrate 2x, the most appropriate response is:

∫ 2x dx = x^{2} + C

Where *C* is the *constant of integration*. It could stand for *4*, or *-5*, or it could even be *zero*. But we don’t know for sure, so we use *C* as a stand-in. Unless we’re given further information (say, the coordinate at which the function intersects the y-axis), we cannot determine what the value of C is.

The blonde in our joke proved that she was calculus-savvy enough to run with the mathematics professors – perhaps even *more* savvy, since Bob failed to mention the constant of integration in his instructions, and she still managed to stick the landing. Good for her.

Q: What’s a good anagram for Banach-Tarski?

A: Banach-Tarski Banach-Tarski.

If you’re scratching your head about this one, don’t worry. I was too at first. I had to do some research about the Banach-Tarski Paradox before I understood the joke. Now that I get it, I think it’s quite clever; sadly, it has *very* limited re-tell value. (See what I did there?)

Let’s start with the easy part: an anagram is a rearrangement of the letters in a word or phrase, usually in such a way that they make readable new words. For example, you can rearrange the letters in Nerdy Jokes to form Yo, Send Jerk! I’m not sure what that *means*, but there you go.

You normally expect an anagram to have the same total number of letters as the original word or phrase, and that’s sort of where the humor of this jokes comes in. Let’s talk about the Banach-Tarski Paradox, and maybe you’ll see why this joke is funny.

The Banach-Tarski Paradox comes from something called Set Theory: a mathematical theory dealing with the ancient Egyptian storm god, Set (I assume). I’m not really sure how Egyptian mythology ties into the Banach-Tarski Paradox, but hey, I’m not a mathematician!

The Banach-Tarski Paradox starts off like this: Imagine a three-dimensional solid ball. Are you imagining it? See how round it is? Good. Now imagine breaking the ball into a finite number of pieces. Next, rearrange the pieces to form two solid daughter balls from the broken pieces of the original. You can’t stretch or add pieces, you can only move and rotate the existing pieces.

Now…imagine that each daughter ball has the exact same volume as the parent.

If that doesn’t make you cry “Balderdash!” then perhaps you haven’t been paying attention. I’ve just said that you can take the limited pieces of an object and rearrange them in such a way as to make two *exactly identical* copies. That flies in the face of everything we think we understand about the relative permanence of volume. You simply shouldn’t be able to do that.

But mathematically, you can. And furthermore, you can do it endlessly, meaning that a single ball can be replicated endlessly until the universe is lousy with balls.

If you’re the enterprising sort, you’re probably already scheming about a way to get your hands on a gold bar and a hacksaw, perhaps thinking that you can replicate your wealth *ad infinitum*. Well, here comes the part where I rain on your parade by telling you the *caveats* I didn’t mention before.

This only works on *mathematical* objects. A mathematical object is different from a physical object in several important ways: first, a mathematical object is a collection of *points* (not particles), which, in the case of a ball, lie within well-defined parameters. Second, since mathematical objects are not made of atoms, there’s no limit to how often or how finely they can be divided. Third, and perhaps most importantly, the non-granular nature of mathematical objects means that when you *do* divide them into smaller pieces, the *facets* of the cut can be arbitrarily, even infinitely, small.

This potential for infinitely rough cuts makes the Banach-Tarski rearrangement possible. A piece with an infinitely rough surface has an undefinable volume; consequently, the normal rules for *adding* volumes do not apply. In fact, you only need to divide a mathematical sphere into *five* pieces to create two clones, which is somewhat suprising to me. Of course, the whole *idea* is surprising to me, so I guess more surprises shouldn’t be surprising, and I think I’ve just talked myself into another paradox.

So let’s revisit the joke: a normal anagram of Banach-Tarski would yield a word or combination of words with a sum of twelve letters, since there were only 12 letters to start with (Example: Satan Hack Bit). But a *mathematical* rearrangement of Banach-Tarski, in the manner of Banach-Tarski, would yield two identical copies of the original (assuming, of course, that Banach-Tarski were a mathematical collection of points instead of a real-world physical object.) I have been assured that the mathematicians who first described the paradox, Stefan Banach and Alfred Tarski, were in fact physical objects, meaning they were not subject to the rearrangement that bears their name. (Nerdy Jokes would like to strongly discourage its readers from attempting to perform Banach-Tarski rearrangements on living organisms.)

C, E flat, and G walk into a bar. The bartender says “Sorry, no minors.”

Sorry if this joke doesn’t strike a *chord* with you. Some people are a bit *tone* deaf when it comes to musical humor. Okay, enough of that.

Most Western music is constructed from 12 notes. They are named after the first seven letters of the alphabet.

- A
- A♯ / B♭ (read as “A sharp” or “B flat”)
- B
- C
- C♯ / D♭
- D
- D♯ / E♭
- E
- F
- F♯ / G♭
- G
- G♯ / A♭

This set-up bears a brief explanation, so stick with me for a moment. The notes that do not have sharps or flats (A, B, C, etc) are called *naturals*. The space between a natural and its sharp or flat is called a *semitone* (or a *half tone* or *half step*). For example, there is an interval of one semitone between the notes A and A♯. The naturals A and B are separated by a *whole step*, or simply a *tone*. Most pairs of naturals are separated by a whole step, with the exceptions of B and C, and E and F. For reasons we won’t get into right now, there are no sharps or flats between B and C, or between E and F. The note B♯ is the same as C natural, and C♭ is the same as B natural. A similar relationship exists between E and F.

Although there are 12 notes to choose from, a lot of musical pieces only emphasize seven of them. The seven notes featured in a musical piece make up the *key*. Keys come in two varieties: *major* and *minor*. The key helps set the mood of the music; a major key is generally upbeat and happy-sounding, while a minor key can be somber and haunting.

The notes for a key are chosen by a relatively simple formula. For major keys, you choose a starting note, then pick out the notes that fall in the following intervals: whole step, whole step, half step, whole step, whole step, whole step, half step.

If you wanted to build a major key around the note of C, you’d follow the formula to pick out the rest of the notes in the key:

- Start with C.
- One whole step above C (two half-steps) is: D
- One whole step above D is: E
- One half step above E is: F
- One whole step above F is: G
- One whole step above G is: A
- One whole step above A is: B
- One half step above B, and we’re back to: C

The C major scale has seven unique notes, and ends where it began (albeit one octave higher…oh wait; I didn’t explain about octaves? Oh dear.)

If you want to build a *minor* key around C, there’s a slightly different formula to follow: whole step, half step, whole step, whole step, half step, step-and-a-half, half step.

- Start with C.
- One whole step above C is: D
- One half step above D is: E♭
- One whole step above E♭ is: F
- One whole step above F is: G
- One half step above G is: A♭
- A step-and-a-half above A♭ is: B
- One half step above B, and we’re back to C

From the notes in a scale, you can construct chords – combinations of notes played at the same time. Just as there are major and minor keys, there are major and minor chords. One common chord structure used in music is the triad. As the name suggests, a triad is made of three distinct notes. In our opening joke, a triad consisting of C, E♭, and G, walk into a bar. Since those notes represent the first, third, and fifth notes in a C minor key, it is a minor triad.

In case you understand everything about music theory but still don’t get the joke, the word “minor” can also refer to a person who isn’t old enough to legally purchase and/or consume alcohol. Hilarious.

]]>Knock knock.

Who’s there?

To.

To who?

No, to *whom*.

This is one of those grammar quibbles that people often don’t get, and I understand that. In the minds of many, the word *whom* is superfluous. It’s a relic from English’s virtually extinct dative case. There’s no clear grammatical reason to keep it in the language, except that it serves to distinguish the subject of a sentence from the direct object (if that was even in question to begin with). It also makes you sound kind of classy, but only if you use it correctly. There are about a billion grammar-focused websites that will teach you how to use *who* and *whom*, but perhaps none with as much humor as The Oatmeal.

So, to summarize what a billion websites will tell you, use *who* as a stand-in for the subject of a sentence, and *whom* for the direct or indirect object. In other words, if you’re asking about somebody that is doing something, use *who*. Example: *Who* thinks *Nerdy Jokes* is the greatest blog in the Universe? If you’re asking about somebody that is having something done or given to them, use *whom*, as in: W*hom* should I congratulate for writing the fantastic blog Nerdy Jokes?

One oft-suggested trick is to answer the question using the pronoun *he* or *him*. If *he* sounds more correct, then use *who* when asking the question. If *him* sounds more correct, use *whom*.

*Who* thinks Nerdy Jokes is the greatest blog? *He* does. (It would sound wrong to say “*Him* does.”)

*Whom* should I congratulate? Congratulate *him*. (You wouldn’t say “Congratulate *he*.”)

This joke contains the preposition *to*. A preposition is a word that expresses a relationship between things. The old elementary school mnemonic is this: many prepositions can describe the relationship between a caterpillar and an apple. A caterpillar can go *to*, *from*, *around*, *through*, *inside*, *above*, *below*, etc, an apple. The noun that follows a preposition is called the *object of the preposition*, and it follows the same rules as the direct object of a sentence. If you have to decide whether to use *who* or *whom* after a preposition, it’s always *whom*.

** To whom** should I address this correspondence expressing my admiration for the blog Nerdy Jokes?

You get the idea.

]]>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).

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 a^{2} + b^{2} = c^{2}. 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. 300^{2} + 400^{2} = 500^{2}. 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.

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*.

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:

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:

- The movie must have at least two female characters, who
- Talk to each other, about
- 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.

]]>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.