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Posts Tagged ‘mathematics’

Math all the way down

What scares you more: that I will talk of death, and injustice, and spiritual annihilation?  Or that I will explain how the equation

affects your life?

Admit it.  You want death.  You want injustice.

You want spiritual annihilation.

I get it, I get it.  Math is an annoyance; math is anathema.  As it did for the learn’d astronomer, math makes you unaccountable tired and sick.

Maybe math scares you.  Or worse—maybe math bores you.  Fear you can take, and anxiety in equal measure; but boredom, never.  It wasn’t time but boredom that sunk Ozymandias into the lone and level sands.  Because—

Because (you say) math is about numbers.  That’s it.  It’s just numbers.  By enumerating, you take away a spark.  That which can be counted, can be dismissed.  A mathematician is a bean counter with a pocket protector, somewhere on the spectrum, digitizing nature, walling off the soul with a wall of 1’s and 0’s.

But it isn’t true.

I could plead that mathematicians don’t usually think of numbers.  They think about patterns, symmetries, interconnectedness.  They see math in the petals of a daisy, and in the predator/prey cycle of lynx and snowshoe hares.  Math is in the strength of nanowires, and the delicacy of hoar frost, and the oomph of an engine, and the whorls of a Spirograph.  

I could plead that math is about connections, structures.  Math is the study of logical systems.  Numbers are beside the point.

Beside the point.

I’m looking right now at the white-board in my office.  Ignore the calendar with a picture of Crater Lake, and ignore the poster of Han in Carbonite, and ignore the Albert Einstein action figure, and the pamphlet which says “Welcome Aboard Marine One.”  Focus on the white-board itself: it’s covered with equations, in red and green and blue, with doodles, starts and stops, arrows and spirals, letters both Roman and Greek.  There are graphs of velocity vs. time.  There’s a derivative, and an integral.  There’s Newton’s 2nd Law, half-erased.

There are no numbers on the board.

I could plead still, but here is what I know: that math is beauty, and that the whole world is math.  Here in my ivory tower, I adhere to the Mathematical Universe Hypothesis, which posits that the multiverse is itself “just” a mathematical structure.  It’s not infinite turtles, but math, all the way down.

Jump if you like: you’ll never hit the bottom.

And what of the equation I gave?  What does it say?  To whom does it speak?

It comes from a paper I wrote, across a gulf of years and disciplines.  It says, in English, that the probability of going from quantum state A to quantum state B is the sum of all the products of closed-loop amplitudes that include A and B.

I am A.

You are B.

To get from me to you, we have to count all the ways we can interact, including ways that go backwards from you to me.

We add up all the ways.

And in the end you don’t have a number, but possibilities.

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Screen Shot 2014-07-22 at 2.00.36 PM

Photo credit: brusspup

Recently I came across the following “optical illusion” on the normally good website I Fucking Love Science:

http://www.iflscience.com/brain/cycloid-optical-illusion-will-boggle-your-mind

I encourage you to read the article and think about the video (otherwise the rest of this post will be less than illuminating).

Supposedly, there is an “optical illusion” in the video because you think there’s a wheel when in “reality” the dots are moving linearly.

This is bullshit.

This is not an illusion. The dots are moving linearly, that’s true. But there is also a wheel. If this causes you cognitive dissonance, so be it. It is not a paradox, however. It is the case that some wheels, when spinning inside other wheels, have points on them which travel linearly.

I admit, this is a field of mathematics close to my heart. Much of my recent work has involved geometric phase, which has connections to Spirographs, epitrochoids and hypotrochoids as mentioned in the IFLS article. I have battered notebooks with over 500 pages of algebra devoted to such things. This is something I know something about.

One way to see that this isn’t an illusion at all is to watch it being drawn. Go to the following Spirograph applet:

http://www.personal.psu.edu/dpl14/java/parametricequations/spirograph/

Play around a little bit. Then create the following specific Spirograph (which is exactly the one in the IFLS “illusion”):
Radius1 = 60
Radius2 = -30
Position = 29
Velocity = 8
If you need to, CLEAR the picture, input the above parameters, and hit DRAW. Hit DRAW again to watch it all over again.

There’s no illusion. A circle is rotating inside another circle. Simultaneously, a particular part of that circle is traversing a straight line. This comes as a surprise to many people: there’s a frisson of incredulity from the idea that your motion can be simultaneously linear, but curved as well. But there’s an easy explanation. Your motion is different in different frames of reference.

Consider an ant on the wheel. With respect to the center of that wheel, he just orbits in a circular manner. But with respect to us, outside of the contraption, he moves back and forth along a single line. This is not an optical illusion. It’s the relativity of geometric shapes.  And I think that’s even cooler than some cognitive trickery.

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