(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:
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:
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.
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:
Now here is the same graph when viewed from directly above, so that the tiger is easier to make out:
Here’s another example of a great 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.
Below is the graph when viewed from directly above, just as I did for the tiger. Pretty cool, huh?
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.
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:There are, of course, many other examples of graphs:
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.
If that’s true, then there’s only one thing we can conclude…
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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)