Ayse Naz and Anjuli's Math Blog: Anjuli's Page

The World's Largest Fractal!

What has an infinite perimeter but zero surface area?

A fractal!

Figure 1

A few Saturdays ago, the world’s largest fractal was created in twenty different cities around the globe. Organized by the mathematician-in-residence, Dr. Laura Taalman, at the Museum of Math (MoMath) in New York City, this project, titled MegaMenger was sponsored by Queen Mary’s University in London. Queen Mary’s University paid for tens of thousands of business-card size papers to be printed and shipped to each participating location, one of which was MoMath. Dr, Taalman, a professor on sabbatical, then reached out to her math community and found multiple math-centric groups around New York State willing to contribute to this project, including the leader of my math circle, Dr. Mary O’Keeffe. Dr. O’Keeffe was tasked with getting a group of people together to build a Level 2 Menger sponge and delivering it to MoMath in New York City.

I was thrilled to participate in this, and above is a picture of us finishing it up late Friday night!

Now, you're probably wondering, what is a Menger sponge??

(Hint: it's not an animal, vegetable or mineral)

A Menger sponge is a cube-shaped fractal first described by Austrian mathematician Karl Menger in the 1920’s. When physically building a Menger sponge, there are various levels that one can create. A Level 0 is the foundation, and can be created with pieces of paper similar to business cards. A tutorial on how to create a Level 0 Menger sponge can be found within the first 2 minutes of this video. For scale reference, the picture below shows me holding up a Level 0 that I created, and Dr. O’Keeffe (in the red sweater) holding up a Level 2 that she put together.

A Level 1 Menger sponge is created by connecting twenty Level 0 cubes, a Level 2 is created by connecting twenty Level 1 cubes and it could continue infinitely. It is only twenty, because one side of a Menger sponge looks like a tic-tac-toe board with the center square cut out, so when making a 3-D Menger sponge, instead of the usual 27 mini cubes, you have 6 fewer because you don’t need the center square of each side or the square in the very middle of the cube. Below is a visual representation from Wikipedia of this explanation:

Figure 2

Here are a few pictures of the final assembly of the Level 3 in MoMath!

The cool thing is, that huge cube that's dwarfing the humans is made of 8,000 interlocking, handmade cubes, each about 2x2 in (5x5 cm) which stick together only because of how they were folded into each other!
P.S. If my math is correct, that means that over 64,000 business cards were folded to create this cubic leviathan!

The reason that the creation at MoMath (and every other location) was limited to a Level 3 and there wasn't an attempt to connect them physically, was safety. Though they are made of paper, a Level 3 sponge weighs approximately 150 lbs. (70 kg), so if these Level 3 sponges around the world were connected physically, it would introduce the danger of them falling and crushing someone. However, connected virtually, these Level 3 sponges would have joined to create a Level 4 and the world’s largest fractal!

Extra resources:

-You can follow Laura Taalman on twitter here and check out more pictures and her perspective of this project!

-An excellent article about the Albany area’s part in this project was written by Dr. O’Keeffe’s colleague at Union College. You can find the article right here.

Figure 3

Estimathon

How many handcuffs would it take to create an interlocked chain stretching from Seattle, Washington to New York City?

You’re probably a little sketched out now, and/or wondering if this is from an incredibly weird trivia game...

But actually, this is an estimathon question!

In preparation for a math competition, a group of students from my school and my math circle partook in a mini Estimathon created by our team’s mentor, Ms. Alexandra Schmidt. It was inspired by the Jane Street Estimathon (here is their facebook page) and consisted of 7 Fermi questions which we had to answer in a group in 10 minutes. The question above was actually one from our mini Estimathon, and we had to use our powers of approximation to determine a range of possible values.

So what are Fermi questions, you ask?

Fermi problems require you to approximate various values without having any actual data.

For example, in the question I opened with, there is no way you could actually determine the exact number of handcuffs without getting a visit from your local police, or at least weird looks from your delivery guy. However, by doing “back-of-the-envelope” calculations, you can estimate (cause it’s an ESTIMAThon; oof I suck) a range quite easily.

I’ll walk you through this example, and you will find that this is actually quite a simple concept.

It’s important to stay organized, and write things down.

Even though it’s not as familiar to me, we’re going to work using the metric system, since it’s what the majority of the world uses.

About how long do you think an average pair of handcuffs is? I would estimate it to be about 20 cm long.

I know that the earth is about 40,000 km around, and I’d estimate that the U.S. is about 1/10^th of the Earth’s circumference (based on my limited memory of the world map).

Since Seattle and NYC are the opposite ends of the U.S., I’d say the distance between them is about 4,000 km.

Using simple algebra to convert kilometers into centimeters I find the distance between the two iconic cities to be about 400,000,000 cm. So I divide their distance by the approximate length of a pair of handcuffs (400,000,000/20) and reach the glorious solution: 20,000,000 pairs of handcuffs would be required to span the continental United States.

As you can tell, this is all very approximate, but it's at least within the same order of magnitude as the real answer.

With this particular example, I doubt that anyone actually would create an interlocking chain of handcuffs and stretch it across the continental United States, but with many Fermi problems, the real answers are out there. For example, you could estimate the height of the Empire State building by estimating the number of floors and the height of each floor and could then check your solution against the actual height.

You might be wondering why they are called “Fermi” questions (you can skip to the next paragraph if you weren't wondering that and don’t want to learn about an awesome physicist—this is my passive-aggressive way of telling you to keep reading). This method of estimation requiring the use of dimensional analysis and basic algebra was introduced to us non-physicist Plebs by Enrico Fermi. This Italian physicist was awarded the Nobel Prize in Physics in 1938, worked on the first nuclear reactor in Chicago Pile-1 (part of the Manhattan Project) and is considered one of the “fathers of the atomic bomb.” Mr. Fermi is the namesake for many awards, institutions and concepts (Fermi problems!).

He's a pretty cool dude--here’s his Wikipedia page to spare you the effort of Googling him (I know you wanted to)

This skill is actually more useful than it might seem.

Questions like these which require you to have an organized and clear thought process are quite common in job interviews. By asking you these kinds of questions, interviewers hope to get an inside look into how you work through an (essentially) unsolvable problem with your limited resources. They know you can’t actually get the exact value, but they want to see the way you problem-solve when faced with a challenge.

You can find practice Fermi problems here, or just let your imagination run wild!

Are you a Belieber? Then how many Justin Bieber impersonators (provided that they’re all his height) would it take, lying down, to make a path from your house to his house in Canada?

Are you a Brony? How many Twilight Sparkle dolls would it take to fill up this year’s BronyCon site?

As Hungarian mathematician, George Pólya said, “To be a good mathematician, or a good gambler, or be good at anything, you must be a good guesser.”

So get estimating!

Extra Resources:

-This WSJ article describes how these three guys used the Fermi method to justify why they'd never get a girlfriend (interestingly enough, most of them are now happily married)
-If you're finding those hard to follow, here's someone who wrote about it more simply