Regruntled.com

The music was composed first, and the video added later:

Britta Johnson (video director for Andrew Bird, among others) brings Lusine’s gorgeous new single Two Dots to life, illustrating the songs relationships-as-trigonometry analogy in an intricately animated video. In the clip, a pair of marbles—one blue, one yellow—engage in the timeless dance of seduction on a horizontal plain, mapping the ups and downs of a courtship through pencil-drawn geometric principles. Like Two Dots, Johnson’s video lives in the middle ground between technology and humanity, emotional immediacy and obsessive detail.

• At least one of these statements is true.
• At most one of these statements is false.
• All three of these statements are true.

How many of the above statements are true?

Inspired by Logic puzzle T shirt, I considered logic puzzles of this general category.  (I am easily amused.)  Ayn Rand would call these statements floating abstractions.  There is no ultimate subject matter; the statements have meaning only in relation to each other.  Like the Oakland of Gertrude Stein’s day, there is no there there.  The trick is to find an assignment of truth values that is mutually consistent.

With the T shirt puzzle, there is only one consistent assignment of truth values.  My puzzle has more than one.

How many consistent truth assignments are there?

One is tempted to say that since there is no unique consistent truth value assignment, that all three statements are meaningless, neither true nor false.  But if they’re all meaningless, then they’re all false.  One can only say that “all three statements are meaningless” is not a consistent truth value assignment.

This puzzle might be a bit ambiguous, depending on whether “has 2″ means “has at least 2″ or “has exactly 2″.  I thing common usage favors “has at least 2″, while mathies might assume “has exactly 2″.  The puzzle can be solved under either interpretation, but the solution is different.

The puzzle is more interesting with an odd number of sentences and the “has at least 2″ interpretation.

An interesting idea: replace calculus with statistics.

I saw the above equation on the net. Is this really true? Yes, it’s easy enough to verify. Are there any other oddities like this? This is a much harder question. Sounds like a job for Wolfram Alpha! Alpha is the new and much-hyped “computational knowledge engine”. It’s not always easy to ask Alpha a question that it can answer, but when you do, the results are impressive. In this case, it turns out that 666 / 34780 = (6 * 6 * 6) / (3 * 47 * 80).

This piece by Susan Goldstine at the 2009 Juried Mathematical Fiber Arts Exhibit, illustrates the fact that

32 * 33 = 1² + 4² + 7² + 8² + 9^{2 +} 10² + 14² + 15² + 18²

The intersection of math and fiber arts seems to be a recent thing, and perhaps a result of more women going into math.

Is Pinocchio’s statement true or false?  One possible solution:

I got the idea for this from Buttons for Mouse.  Imagine an X-Y plane from math class, and graph a Z function in the vertical dimension.  In this case:

z = x² + y² + 2x + 2y + 1

For example, if x = 7 and y = 11, then z = 207.  Only instead of having the z value popping out of the screen 207 units towards your eyeballs, imagine that the digits 2, 0 and 7 are stacked on the screen.  The 7 is stuck to the screen at x =7, y = 11.  The 0 is stacked on top of the 7, and the 2 is stacked on the 0.  This image is 500 by 500, so there are 25,000 pixels.  At each pixel, there is a stack of digits.  Got that?  Does your brain hurt yet?  Good.

Now we’re going to slice our mess of digits parallel to the screen and take only the 3rd digit.  So, for 207, we discard the 0 and 7 and keep the 2.  Similarly, for 1432, we keep just the 4.  Now we have a 500-by-500 array of digits, so we can replace each digit with a color.

But wait, there’s more!  I didn’t do the calculations in decimal, I did them in base 5.  Why base 5, you ask?  I tried using decimal, but 10 colors is too many.  The patterns are too busy.  Like Goldilocks, I tried different bases and it turns out that for this particular function, base 5 is not too sweet, not too rancid, but just right.

And why the 3rd digit?  Well.  If I use the 3rd digit in base 5, and my function consists only of additions and multiplications, then the pattern has to repeat every 125 pixels.  5 to the 3rd power is 125.  If you’re familiar with modular arithmetic, this is easy enough to see.  500 divided by 125 is 4, so with a 500-by-500 pixel image, I get 4 repetitions in each direction.  The pattern is big enough to be interesting, but small enough that I can see that it repeats.  Just about right.

As always, there are more questions than answers.  I can try different functions, different bases, different digits, different color schemes.  There’s no end to the experiments that can be done, and no telling what patterns are there to be discovered.