Month: March 2012

Now that I’ve talked in circles for a few days, I thought it would be nice, as a change of pace, to circle back to looking at books as visual maps.

In this next iteration of the interface, you can type in (or highlight by dragging with your mouse) whatever text you want to search for throughout the book. I’ve discovered all sorts of interesting things in the short time I’ve been playing with it.

I’m trying to build up to a more general capability, where you can search for larger patterns, or find relationships between different parts of a book.

But one step at a time.

As Barak suggested in his comment, the perceptible sound quality I was actually referring to in yesterday’s post is sound pressure (the actual pressure of a sound against the ear), rather than loudness, which is a convenient logarithmic scale of perception. Barak, thanks for pointing that out.

Sound pressure — the external phenomenon our ears can perceive — falls off as 1/distance, rather than 1/distance². Why is that?

Well, sound waves consist of regions of air that are acting against each other like little springs. Each air particle is actually spinning in a kind of circle. One dimension of this circle is how rarified or compressed the air is. The other dimension of the circle is how fast each air particle is moving.

When you clap your hands together with four times as much energy, the pressure variation becomes twice as big, and each air particle also moves twice as fast: 2 × 2 = 4. You can think of the energy as the area of this pressure/velocity circle: As particles spin around in this circle with a given frequency, the area of the circle (proportional to pressure × velocity) tells us how much energy the particle has.

By the way, how may times per second each particle spins around in its energy circle is indeed the frequency of the sound. For example, the sound from a standard A-pitch tuning fork makes the air particles spin around in their little pressure/velocity circles exactly 440 times per second.

The energy you cause by clapping your hands indeed drops off as 1/distance², but our ears can only perceive one dimension of this spinning energy — the pressure axis. The other dimension consists of velocity — how fast the air moves as it rhythmically expands and contracts. Our ears cannot perceive this velocity dimension (just as, if you get hit by a baseball, you don’t feel how fast it was going, only how hard it has hit you). The part we can perceive — the pressure variation — which varies with the diameter of the energy circle, not with its area — drops off as 1/distance.

Once you understand that all vibrational energy is actually circular motion, then a number of otherwise mysterious things start to become very simple.

For example, the other day a friend posed the following puzzler: We all know that when you make a short burst of sound (say, by clapping your hands together), the sound impulse radiates out into an ever expanding sphere — since the sound goes in all directions. Logically it would seem that the loudness of that sound, when heard from some particular distance, should drop off as 1 / distance², since the surface of the expanding sphere increases as distance², so the sound should be diffused by that much. For example, you’d think that a clap from twice as far away would sound only one fourth as loud.

But that’s not what happens. In fact, the loudness of the sound drops off only as 1 / distance. For example, a clap from twice as far away sounds half as loud.

What’s going on here? It happens this way because a sound wave is actually a kind of circular motion. I’ll explain the rest tomorrow.

In order to understand the power of Euler’s identity e^iπ + 1 = 0, you need to unlearn some pretty fundamental things you were taught in school.

We are all taught that light and sound travels in waves, that alternating current is described by sine waves, as well as tides, water waves, guitar strings, bouncing springs, pendulums and anything other kind of regular periodic movement.

The problem with this mental model is that it misses the most important thing: None of these things are best described by something moving in a sine wave. Rather, they are best described by something moving in a circle. Deep down, they are all pretty much equivalent to tying a rock to a rope and swinging it over your head in a circle.

I know this seems counterintuitive. After all, we can actually see the bob of the pendulum move left and right. The movement doesn’t look anything like a circle. The problem is that the circle isn’t visible — but even so, it is essential.

In the case of a pendulum, one axis of the circle is displacement and the other axis is speed. Any time you see periodic movement, you are actually looking at something that holds energy by spinning in a circle. Unfortunately one or more of the dimensions of that circle is often invisible to the naked eye.

Once you absorb this idea, Euler’s identity takes on much more significance. It’s the key to understanding light, sound, vibrating strings, bouncing springs, and pretty much everything that sends information around the Universe.

March 14 is Pi Day, and March 15 the infamous Ides of March. Today, March 17, is St. Patrick’s Day.

But what about the orphaned day between — March 16? It seems a shame that poor 3/16 receives no love.

I propose we create a new special day, which combines attributes of the other three days. Maybe we could call it Pi And Irish Day After Ides. Or PAI DAI (pronounced “pay day”). Because everybody likes a pay day!

We could celebrate March 16 by serving Caesar Salad and Irish Pie. It would be truly multicultural!!

One of my favorite of all mathematical statements (and many people agree with me on this) is Euler’s identity:

e^iπ + 1 = 0

 

It’s frustrating that I cannot explain to my “friends who are not into math” just why this simple statement is so extraordinary and beautiful. Basically, it opens the door to understanding why many things in the universe which at first seem complicated are actually very simple and elegant.

In my mind this statement is a kind of litmus test: Once someone gets to the point of understanding the profound idea that Euler’s identity embodies, and why that idea is so profound, simple and lovely all at once, then so much other understanding of the beauty of the universe opens up.

It would be interesting to design a fun and engaging course for “non-mathematicians” that specifically leads to this place of understanding.

Pi — a disk’s circumference, divided by diameter —
Quite likely, said Vi Hart, in iambic pentameter,
Contains all Shakespeare’s works. Now, just to be formal,
This simply means that Pi is perfectly normal.
If so, within its digits, in every variation,
We’d find the whole of Hamlet, in the speech of any nation.
She showed the prince of Denmark as a dog, which is revealing.
Well OK, a little strange, but certainly appealing.
Just one thing about “Dog Hamlet” has been gnawing at my brain:
I wish Vi’s canine hero had been played by a Great Dane.

Today, as some of you know, is PI Day — the day of the year, March 14, when the calendar reaches the digit representation of PI (3.14159…).

In yesterday’s post I mentioned Vi Hart. And today I am pleased to refer to Vi’s yearly PI Day offering — a disquisition on the question of whether PI is a “normal” number — that is, a number which contains every possible string of digits (and therefore not only every literary work ever written but that ever could be written).

I actually wrote on this topic four years ago, in my post All the songs ever written, and one song. But Vi has gone further, by creating a lovely philosophical video essay in which a certain Danish canine asks some deep existential questions, entitled Are Shakespeare’s Plays Encoded within Pi?.

The other day I was describing my little novel visualizing tool to a colleague, and I told him that, like the old game Asteroids, it’s on the surface of a torus. That is, in world that is topologically equivalent to a donut. Or, if you are from New York, a bagel.

In Asteroids, when something goes off the right side of the screen, it reappears on the left side. And when something goes off the top edge of the screen, it reappears from the bottom edge. In other words, the screen wraps both ways.

“Doesn’t that mean it’s on the surface of a sphere?” he said.

I needed a really easy way to explain that it couldn’t be on a sphere, so I came up with this: “If you start in the middle of the screen and draw a horizontal line across the screen, you get a loop, because the left and the right ends of the line connect. Also, if you start at that same point in the middle of the screen and draw a vertical line from top to bottom, you get another loop, because the top and bottom ends of that line also connect.

So far so good. Then I said “the two lines only cross once (at that starting point in the middle of the screen). If Asteroids were on a sphere, the two lines would have to cross twice.”

My colleague (who is very quick) got the point immediately.

The next day I was describing this little exchange to Vi Hart, who said to me “Yes, that’s actually the definition of a torus.”

Huh. When you think about it, that makes a lot of sense.

They have houses here
So much room!! Yet the ocean
Is on the wrong side