Monthly Archives: September 2014

Paradoxical Playfulness

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.)

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Musical Mirth

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.

  1. A
  2. A♯ / B♭ (read as “A sharp” or “B flat”)
  3. B
  4. C
  5. C♯ / D♭
  6. D
  7. D♯ / E♭
  8. E
  9. F
  10. F♯ / G♭
  11. G
  12. 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:

  1. Start with C.
  2. One whole step above C (two half-steps) is: D
  3. One whole step above D is: E
  4. One half step above E is: F
  5. One whole step above F is: G
  6. One whole step above G is: A
  7. One whole step above A is: B
  8. 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.

  1. Start with C.
  2. One whole step above C is: D
  3. One half step above D is: E♭
  4. One whole step above E♭ is: F
  5. One whole step above F is: G
  6. One half step above G is: A♭
  7. A step-and-a-half above A♭ is: B
  8. 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.


Grammar Goofiness

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: Whom 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.