Tag Archives: math

Life, the Universe, and Everything

matrix

(Click here for the PDF version of this presentation.)

Math is everywhere, hidden in places where we don’t even expect to see it. For example, take a look at the following image:

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What do you see?

Most people say “music.” People who have studied the piano might recognize this as a piano score. And a true enthusiast might recognize it as the third movement of Beethoven’s Moonlight Sonata.

What you’ve probably never thought of before, though, is that a musical score is actually a form of graph. It tells the performer what combination of notes to play at a given moment in time. In other words, it shows sound as a function of time.

In the image below, I’ve added labeled axes to draw attention to this:moonlight_sonata_graph

Now consider a photograph. Below is one of the most spectacular images I found when Googling “photograph.” (Thanks to whoever posted it!) I love how it shows the strings of mucus frozen in time.

tiger_photo

Anyway, a photograph itself is also just a type of graph — and not just metaphorically. In fact, even the way images are produced in our brains is just a way of numerically graphing the intensity and frequency of light that falls on different portions of our retinas. In essence, your retina is the x-y plane and the light is the quantity being graphed.

Below is what the photograph looks like when graphed in three dimensions from different angles, with the colors changed to a different color scale:

tiger_photo_graphs_1

Now here is the same graph when viewed from directly above, so that the tiger is easier to make out:
tiger_photo_graphs_2

Here’s another example of a great photo:
frog_photo

And here it is with the same procedure applied to it. This one works a little better than the tiger because it isn’t filled with little white spots that end up looking like noisy spikes in the graph.frog_photo_graphs_1

Below is the graph when viewed from directly above, just as I did for the tiger. Pretty cool, huh?frog_photo_graphs_2

Now consider something that really seems to have nothing to do with math: a piece of literature. Below is the first paragraph from A Tale of Two Cities, by Charles Dickens.
Tale_of_2_cities

It, too, can be considered as a type of graph. It’s a graph that tells the reader what words to speak or think as a function of time:Tale_of_2_cities_graphThere are, of course, many other examples of graphs:

footballstragety

What I’m saying is that anything can be thought of as a kind of graph. Really, though, it’s not just graphs that are so powerful, but numbers themselves. This is because numbers encode information. For example, an entire song can be encoded in a single number. So can a photograph, or even a movie.

What’s particularly fascinating is that physicists now believe that physical reality itself is composed of information. In fact, the universe might even be digital. And since numbers encode information, it is possible that the entire universe could be represented by a single number.

Take a minute to meditate on that.

universe

If that’s true, then there’s only one thing we can conclude…

matrix_drop_math_is_everything

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This post is based on a PowerPoint presentation I made for my math students in an attempt to inspire them. Here it is in PDF form:

Math Is Everything (PDF version)

How to Move the Earth

earth-rotationIf you’re like me — and who isn’t? — then you’ve often wondered how traveling around on the surface of our planet affects its rotation. If you haven’t ever wondered this, then I suggest you go in for a psychiatric evaluation, because no sane person should be able to go for a walk without worrying about the cosmic ramifications of every step he or she takes.

The Earth rotates toward the east; and when you’re standing still with respect to the ground, you’re actually moving along with its surface. If, however, you decide to start walking east, the initial steps you take to get moving will push westward against the Earth, slowing its rotation ever so slightly. You’ll be stealing a bit of the Earth’s angular momentum. If you go west, your feet will push eastward against the Earth, and you will actually speed up its rotation.

The question on my mind — and on yours too, I’m assuming — is how much the Earth’s rotation will be affected if you travel eastward all the way around the world and come to a stop right back where you started. During the trip, the Earth will be rotating at a slower-than-normal rate; and when you come to a stop, it will return to its normal rate of rotation. As a result of this period of slowed rotation, each point on the Earth’s surface will now lag behind where it would have been otherwise. Sunrise will happen a little bit later for everyone.

Well . . . how much later?

This question burns, doesn’t it? Well, relax, because we’re going to answer it right here. We just need to find expressions for the total angular momentum of the system (which consists of you and the Earth) for when you’re standing still and when you’re traveling. By conservation of angular momentum, we can set these two expressions equal to each other and solve for the Earth’s reduced angular velocity during the trip. From this, we can then determine how much lag the Earth will accumulate. Piece of cake!

For the curious, I’ve written out the full solution here, in PDF form. Below, I’ll spare you the calculations and just present the results.

For a 60-kg person making the trip, the amount of time by which sunrise will be delayed is 2.17 attoseconds. In case you’re wondering what the heck an attosecond is, it’s 1/1,000,000,000,000,000,000 of a second, which is roughly how long it takes a beam of light to travel the length of three hydrogen atoms lined up against each other. In other words, it’s a very short time.

One interesting thing about this answer is that it is completely independent of how fast you make the trip. Whether you zip all the way around in less than a second or crawl along over a period of several years, the net effect will still be a delay in the Earth’s rotation of 2.17 attoseconds. It’s your mass, not your speed, that determines how big the resulting delay will be.

If you gained a bit of weight, you would have a bigger impact. We might ask, for example, what your mass would have to be in order to delay the Earth’s rotation by a full second. It turns out you would have to weigh a hefty 27.6 quintillion kilograms, which would require a large number of trips to McDonald’s and is not something you should aim for.

So now we know how the Earth’s rotation is affected each time a person circumnavigates the globe. It’s a small effect; but just to be safe, whenever you take an intercontinental trip, you should probably return the way you came rather than going all the way around the world. Keeping track of time is difficult enough with Daylight Saving Time.