August 14, 2015

A Real-Life Paradox: The Banach-Tarski Burrito

Who knew the Axiom of Choice could help me decide whether to get guacamole for an extra \$1.95?

A couple of weeks ago, the popular YouTube channel Vsauce released a video that tackles what it details as “one of the strangest theorems in modern mathematics”: the Banach-Tarski Paradox.  In the video, Michael Stevens explains how a single sphere can be decomposed into peculiar-looking sets, after which those sets can be recombined to form two spheres, each perfectly identical to the original in every way.  If you haven’t had a chance to watch the video, go ahead and do so here:

Although this seems like a purely theoretical abstraction of mathematics, the video leaves us wondering if perhaps there could be a real-world application of such a bizarre phenomenon.  Stevens asks, “is [the Banach-Tarski paradox] a place where math and physics separate?  We still don’t know … The Banach-Tarski Paradox could actually happen in our real world … some scientists think it may be physically valid.”

Well, my friends, I would like to make the bold claim that I have indeed discovered a physical manifestation of this paradox.

And it happened a few years ago at my local Chipotle.

Let me start off by saying this:  I love Chipotle.  It’s a particularly good day for me when I walk in and get my burrito with brown rice, fajita veggies, steak, hot salsa, cheese, pico de gallo, corn, sour cream, guacamole (yes I know it’s extra, just put it on my burrito already!), and a bit of lettuce.  No chips, Coke, and about a half hour later I’m one happily stuffed math teacher.

The only thing that I don’t like about Chipotle is that the construction of said burritos often ends up failing at the most crucial step – the rolling into one coherent, tasty package.  Given the sheer amount of food that gets crammed into a Chipotle burrito, it’s unsurprising that they eventually lose their structural integrity and burst, somewhat defeating the purpose of ordering a burrito in the first place.

If you have ever felt the pain of seeing your glorious Mexican monstrosity explode with toppings like something out of an Alien movie because of an unlucky burrito-roller, you have probably been offered the opportunity to “double-wrap” your burrito for no extra charge, giving it an extra layer of tortilla to ensure the safe deliverance of guacamole-and-assorted-other-ingredients into your hungry maw.

Now, being a mathematically-minded kind of guy, I asked the employee who made me this generous offer:

“Well, could I just get my ingredients split between two tortillas instead?”

The destroyer-of-burritos gave that look that you always get from anybody who works at a business that bandies about words like “company policy” when they realize they have to deny a customer’s request even in the face of logic, and said:

“If you do that, we’ll have to charge you for two burritos.”

I was dumbfounded.

“Wait … so you’re saying that if you put a second tortilla around my burrito, you’ll charge me for one burrito, but if you rearrange the exact same ingredients, you’ll charge me for two?”

“Yes sir – company policy.”

Utterly defeated, I begrudgingly accepted the offer to give my burrito its extra layer of protection, doing my best to smile at the girl who probably knew as well as I did the sheer absurdity of the words that had come out of her mouth.  I paid the cashier, let out an audible “oof” as I lifted the noticeably heavy paper bag covered with trendy lettering, and exited the store.

When I arrived home, I took what looked like an aluminum foil-wrapped football out of the bag (which was a great source of amusement for my housemates), laid it out on the kitchen table, and decided to dismantle the burrito myself and arrange it into two much more manageable Mexican morsels.  I wondered whether I should have done this juggling of ingredients right there at Chipotle, just to see whether the staff’s heads would explode.

It was in that moment, with my head still throbbing from the madness of the entire experience, that I began to realize what had just happened.  How was it possible that a given mass of food could cost one amount one moment and another amount the next?  I immediately began to deconstruct my burrito, laying out the extra tortilla onto a plate and carefully making sure that precisely one-half of the ingredients – especially the guacamole – found their way into their new home.  As I carefully re-wrapped both tortillas, my suspicions were confirmed.  Sitting right in front of me were two delicious burritos, each identical in price to my original.

I had discovered the Banach-Tarski Burrito.