Measuring Boston coastlines with increasingly tiny measuring sticks…

OSH_in_BostonAs I toured Boston this past week, I seemed to be drawn inexorably towards the Old State House.  Historically the Old State House was built at the center of civic activity. The Old State House “was prominently sited at the juncture of King Street, Boston’s primary commercial thoroughfare, and Orange Street, the peninsula’s sole overland route to Roxbury and interior settlements,” says the Boston Landmarks Commission. What this means is that if you stood in front of the Old State House in 1700, you could see the bay only a few blocks away. Now, the bay is a ½ mile down the road and nowhere in sight.

Here’s two maps of Boston. Notice how the map on the left of Old Boston is almost an island (you can click on it to make it bigger)- Present-day Boston, on the right, does not look like that anymore. But in both cases, the Old State House is basically in the middle.

Boston,_1775bsmallBoston_google_earth

A lot has changed since 1630. Crazy!  Humans leveled the hills that existed in Boston to expand the shoreline. They did this without construction vehicles or steam engines. This blows my mind! Then, much later, Bostonians used landfill to fill more space in.  Here is a map of how things have changed over time.

growth-boston-shore-lines1

This map is from all-that-is-interesting.com who referenced maps.bpl.org which is amazing, but doesn’t seem to have the image on their site any longer.

Boston is a maritime city with lots of boat traffic. Because of this I would love to know how the length shoreline has changed over time. Can we fit more or less boats in Boston harbor than we used to? Did we get more shoreline out of this land expansion? Or less? Since we have a map of Boston, this should be an easy task right?

Shorelines are notoriously hard to measure. This is because shorelines are fractal-like. This is a mathematical feature which (loosely) means that at each level of zooming in there are lots of bumps and rough edges. You can zoom into a coast line over and over again and not know how close to the coast you are! How does this manifest itself when you try to measure the coast?

The length of your measuring stick directly affects the length of the coastline. This is called the Coastline Paradox. Here is an amazingly awesome video about measuring a coastline from science presenter Steve Mould.

But if you don’t have 8 minutes, that’s okay. The long and the short of it is

Stick_Measurements

Thus we can get a shoreline of infinite length for any point over the history of Boston. If we picked a specific measuring stick length, we could compare the shoreline from one time to another. But we can never truly know the length of the shoreline of Boston at any time.

Despite the disappointing state of affairs of the shoreline, there does seem to be a link between the shoreline of Boston and the placement of the Old State House.  Somehow, I believe, it has continued to be the centroid of the peninsula. That is to say it is the center in some way. It is (approximately) the location of the weighted mean of all the points in the peninsula. Which seems pretty cool to me.

I believe this because of my limited experience in Boston. As I traveled back and forth across the peninsula, my trajectory through the city always included the Old State House. It was though the historic placement of the building drew me to it just as it drew travelers towards it 300 years ago. For travelers then and now, the Old State House holds a place of prominence even though Boston isn’t really an island and the shoreline is way further away. Not to mention that there is now a T-stop in the basement…

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Why do Hipsters look alike?

It was a hot day as I stepped into the air conditioned mall. It is 1995 or maybe 1996; my mall has slightly opaque sky lights so you could still see the bright midsummer sunlight outside. I would have preferred darkness. I wound my way around the teens who aren’t enlightened enough to make their own unique fashion choices as I made my way to the coolest store in the mall.

goth_pants

I may or may not have owned this exact pair of pants. I’m not admitting anything!

I got a lot of weird looks, probably because of my strap-covered purple pants and thick black eyeliner. I didn’t conform to the grey t-shirt and khaki cargo pants “uniform” of my peers. I was unique. I was a rebel. I confidently walked into Hot Topic knowing that I was anticonformist  and I didn’t think like the rest of the lemming-esk populous.

But on that hot day, I had a thought which would forever change the way I understood fashion: Why can the US have a nation wide store which sells anti-mainstream clothing to goths? Doesn’t this mean that my awesome pants aren’t actually that unique? Aren’t I actually following some trend by being anti-society and “goth” in my appearance?

Looking back at this time, I still have to laugh at my righteous beliefs of nonconformity. To add further irony, 1996 is the year that Hot Topic went public as a company. My realization couldn’t have been more true. How could there be a large enough, and predictable enough, section of the public to buy that many cookie-cutter goth styles? I think there will always be a cyclic behavior of people trying to be unique or trying to average which causes the population to always have people who will buy clothing at Hot Topic. My theory was recently supported by the work of Dr. Jonathan Touboul.

Dr. Jonathan Touboul, who, like me, is a mathematician interested in social constructions and mathematics. He and I actually have a lot in common because he also studies dynamical systems and his WordPress blog uses a theme that I used for years. (coincidence? I think NOT!).  He just posted a draft of his newest article on arXiv: The hipster effect: When anticonformists all look the same. The Washington Post recently covered the article as well. The Washington Post article does a nice job of explaining the overall choice algorithm of the model. At each moment in time, a single entity will stay with  her style or switch depending on if she wants to be in the majority or minority.

Dr. Touboul says:

Beyond the choice of the best suit to wear this winter, this study may have important implications in understanding dynamics of inhibitory networks of the brain or investment strategies finance, or the understanding of emergent dynamics in social science, domains in which delays of communication and the geometry of the systems are prominent.

Translated from I’m-writting-a-paper language, I believe that Touboul is saying that we may use this type of modeling to understand social decisions. By modeling our social interactions, we might be able predict if and when the hipster fashion will change. While we have used models for years to try to predict the stock market, he is saying that it may also be possible to predict the style choices of the anti-conformists at the same time as you predict the styles of the conformists. There will be some stable ratio and, if you can identify the important features (cut, sleeve length, fit, etc), then you could predict which styles will sell the biggest. Which is why it’s possible to make a graphic like this one:

Image Credit: The Guardian

Okay, let’s look at the math for a couple of minutes because it’s really cool and it’s alarmingly similar to the work that I did in my graduate research. Here’s a clipping from his paper where is shows the resonance that the hipster system has with the conformist system. An oscillation develops and is maintained after a certain point in time.

Hopf_Bifurcation_Hipster_System

From The Hipster Effect: When anti conformists all look the same by Dr. Jonathan Touboul.

Touboul adds a delay in the system as well as some noise to simulate reality. He also uses a dynamic Hopf bifurcation. My dissertation focused on dynamic Hopf bifurcations in externally forced systems.  (Do we need any more evidence that he and I are secretly on the same intellectual wavelength. It’s almost creepy!) One can think of his hipster equation (red) as a forced oscillator. I show in my dissertation that if you add any noise to a dynamic Hopf bifurcation it is extremely likely that the system will lose any relationship with it’s initial condition and tend to latch onto the nearest stable trajectory within the solution space. Thus, it is not surprising to me that the anti-conformists are correlated and/or phase locked to the conformists. If you define your sense of self based of of someone else, even if your definition is “whatever they are not”, then you are dooming yourself to be forever influenced by their choices.

Thus, it is may not be so surprising that Hot Topic can predict the next biggest thing for goths to wear. In a similar way, American Apparel can predict hipster trends (Does American Apparel even count as hipster?). Maybe I shouldn’t be surprised that all hipsters look similar. But what does that mean about my style? I haven’t been in a Hot Topic in years. And I have made peace with the fact that I blend in more than I stand out. But I’m not sure that I can believe that mathematics will ever understand the nuances of fashion enough to predict what next year’s biggest thing is.

05-2015-silly_hats

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Rate of Change of Change

This week Bloomberg published a great article about the change of change in the US. I couldn’t pass up on sharing this article with you all because it’s about math and about social. This article focuses on the bottoms up approach to government. Some big decisions are made at a local and state level before the central government weighs in on the issue.

The article highlights the beautiful connection between the emotion-driven social situations and the rationally-driven analysis/desire to understand. The thing I love most about this article is that the analytical results so clearly reconnect with emotion. This article gives me hope and makes me optimistic about the rate of emotional change of US citizens.

Compared to issues in the past, we are changing our minds so much faster than before. We no longer live in a wait-long-enough-and-everyone-who-disagrees-will-be-dead scenario. We are more pro-active and more responsive than ever before.  So, with that build up, I really hope you’ll go read about America’s rate of change, “This Is How Fast America Changes Its Mind.Rate_of_change_Bloomberg

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Popcorn Analytics

I’m sitting at my dining room table trying to discern the difference in texture, squeakiness, and flavor of the food in my mouth. Across from me, several of my friends are thoughtfully chewing on popcorn and making notes. We all survey the colors and various sizes of kernels in the bowls in front of us while we sip on our wine.  Then we determine that we don’t have enough data points and we need to run the blind taste test again with different numerical assignments. This must be done to determine if our judgments are reproducible. We are pretty sure our judgments are subjective and varying- so we probably won’t get a single best popcorn brand. This means that the only consistency we can hope for is that each person’s opinions are reproducible from one taste test to the next.

On an evening when there wasn’t much of a plan, we realized that we haBowl_of_popd 5 different kinds of popcorn in the apartment. Some expensive, and some not so. There have been lots of experiments about the quality and flavor of wine. When I ask Google, the first few results are impressive: Forbes, Wired, and BuzzFeed.  On the other hand, when I ask Google about taste tests for expensive popcorn I find none that are really about popcorn taste tests. (notable exception seen here.) But who needs Google when you can do the tests yourself?

5 Brands. 2 taste tests. 5 taste testers.

The contenders: Ancient Heirloom, PopSecret, Jolly Pop - White, Jolly Pop - Yellow, and Jack Rabbit.

The contenders: Ancient Heirloom, Pop-Secret, Jolly Pop: White, Jolly Pop: Yellow, and Jack Rabbit. Listed left to right by price- most expensive on the left to the convenience store brand on the right.

Each person gets two votes for each brand (one for each blind taste test we did). Only 3 out of 5 of us picked the same brand as our favorite from one test to the next. Thus, we are only 60% reproducible. Pretty dismal! But our sample size is small. So maybe reproducible isn’t the ideal. Perhaps we are smarter as a group than we are individually? When we compile the votes, something surprising happens. Here’s the big table of ranked votes by brand.

Ancient Heirloom PopSecret Jolly Pop – White Jolly Pop – Yellow Jack Rabbit
First 5 3 1 1
Second 3 3 3 1
Third 4 3 1 2
Fourth 1 2 2 6
Fifth 2 2 1 3 1

As you can see, most brands got lots of different scores. We should take a weighted average of each brand to understand the overall story. We’ll give 5 points to each 1st place vote, 4 points for a 2nd place, 3 for 3rd and so on. Then we divide by 10 to average the weighing. 

Ancient Heirloom PopSecret Jolly Pop – White Jolly Pop – Yellow Jack Rabbit
Weighted Average 3.9 3.1 3.1 2.7 2.3

We, as a collective, ranked the popcorn by cost! That’s kind of impressive and surprising. Based on this small study, I boldly propose that popcorn brand preference is more correlated by price than wine is. This is a great feature of popcorn because, unlike wine, the price difference between the cheapest and most expensive is a few bucks! Cheers to that!

By the way, did I mention that popcorn is my favorite food? At work last week, I had to describe how popcorn represents my personality: Explosive! Comforting… and Boring? I love popcorn so much that I even make cupcakes that look like popcorn for parties.

I made these. Well, not these exactly. But I used this recipe to make popcorn cupcakes that looked almost exactly like this. It’s an equivalence relation of popcorn!

My friends are fabulous. On a similar evening not too long ago, we were making measuring tape circlets as we were trying to determine whose head was larger. Or that other time when we were measuring shoulder width to try to decide how that effects rock climbing techniques… There are so many wonderful things in this world which can be seen more clearly through the lens of mathematics and data. I will happily do analysis on almost anything.  And apparently my friends will too!

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Monty Hall

Monty Hall

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Pitfalls and Traps: Research

Pitfalls_and_Flames_Math

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Happy Easter!

Happy Bunny Day from Social Mathematics!    

  

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