Fouriest Series

Natural language processing on the first 2016 presidential debate

The first debate in the 2016 presidential race was held on September 26. It’s no secret that Clinton and Trump are running on drastically different platforms, but how do they compare when it comes to their speech patterns and word choice? To quantify this, I dug into the data, using the debate transcript and natural language processing.

I measured the sentiment of Clinton’s and Trump’s responses, and examined how emotional their words were throughout the debate. I also looked at each candidate’s most commonly used adjectives. Building off the work of Alvin Chang at Vox, I was also able to examine how the speech patterns of Clinton and Trump each changed when directly responding to and when skirting the questions.

Sentiment

Using the Google Cloud Natural Language API, I measured the sentiment of each candidate’s answers. The polarity of a response is a measure of how positive or negative it is, and the magnitude indicates how much emotion the words convey. The chart below shows the polarity of each candidate’s responses, weighted by the magnitude.

Trump and Clinton matched each other’s polarity for the first half of the debate, but after his defense of stop-and-frisk around 9:50 PM, Trump’s words became much more negative.

Throughout the rest of the debate — during the questions on birtherism, cyber security, homegrown terrorism, nuclear weapons, and Clinton’s looks and stamina — Clinton became more positive and Trump more negative.

The combination of polarity and magnitude gives us the best understanding of each line’s overall sentiment, and each candidate’s most positive and negative responses are posted here.

Braggadocios, and other adjectives

I was also interested in the adjectives each candidate used most frequently during the debate. Using syntax analysis to extract each word’s part of speech, I identified the most-used adjectives of each candidate.

Answers vs non-answers

As Chang found, the candidates spent a lot of time not answering Holt’s questions — 48% of Clinton’s words and a whopping 69% of Trump’s words were used in non-answers — and using the data Chang compiled, I was able to look at how the candidate's speech patterns differed when answering and not answering the questions.

Sentence subjects (“I alone can fix it”)

Using part-of-speech tagging, I also identified the subjects of each candidate’s sentences. Clinton was more inclusive in her words, but only when directly responding to questions — using the plural “we” more frequently than the singular “I” — and the the opposite was true for her when avoiding a response. Trump, on the other hand, was always more likely to use “I” over “we”.

Non-answer phrases

The words each candidate used when directly answering the questions are all, unsurprisingly, highly related to the questions Holt asked. What’s interesting here are the topics the candidates defaulted to when avoiding a response.

A handful of my findings didn’t make it into this post. If you’re interested in more, there’s some additional analysis, including multiple classification models, in the project’s GitHub repo. The text of this article (excluding this sentence) has polarity -0.4 and magnitude 15.5, so despite my best efforts it’s leaning slightly negative.

R code posted here.

Many thanks to Alvin Chang and Vox for their permission to use their annotated transcript, and to Kelsey Scherer for designing the charts and lead image.

Analysis was performed in R. Plots were generated using ggplot2, and then styled by Scherer using Sketch.

Mapping the frozen yogurt shop closest to each Manhattan apartment

I love frozen yogurt. When I first moved to New York three years ago, I lived only 1/8th of a mile from the closest froyo shop. The convenience of this 4-minute walk is something I neither appreciated nor utilized enough at the time.

After moving to Harlem last year, it’s been harder than ever to satisfy my near-constant craving for this cold candy soup — I’m now a 24-minute walk to the nearest frozen yogurt.

As someone who loves data and has too much time to spare, I decided to find the locations in Manhattan with highest and lowest froyo densitiy. Inspired by Ben Wellington's work on I Quant NY, I calculated the distance from every lot in Manhattan to the nearest froyo shop and mapped it out.

The highest density of froyo is right around West 33rd St. and 8th Ave., with three shops within a 1-block radius. The lowest density is right in Harlem. The red circle on the map shows the location farthest from frozen yogurt. The record belongs to 700 Esplanade Gardens Plaza, a co-op right by the 145th St. stop on the 3-train, with a 51-minute trek across Manhattan to the Pinkberry by Columbia.

The map shows all of the froyo shops in Manhattan, and you can click on any lot to find the distance to the closest shop.

R code posted here.

All distances in the map are measured using great-circle distance (i.e., ”as the crow flies”), according to the law of cosines.

Frozen yogurt locations were found via the Google Places Nearby Search API. The API returned some non-froyo-exclusive shops like Ben and Jerry’s, which I kept in the dataset since they technically serve some frozen yogurt (although we all know these shops don’t really count).

I only included froyo shops that were in Manhattan, so some lots may have a closer shop than the one listed if we include those in other boroughs.

Manhattan lot locations are from PLUTO.

The map was created using CartoDB.

Wave Equation

The wave equation is a partial differential equation that describes the propagation of various types of waves.

The equation appears throughout many fields in physics, including acoustics, fluid dynamics, electromagnetism, and quantum mechanics. With some modifications, it can even describe the spread of traffic jams on busy highways!

The one-dimensional equation was first discovered by d’Alembert in 1746 as he studied how vibrations propagated through a string, and the two- and three-dimensional equations were solved soon after by Euler during his study of acoustics.

The simulations above show the propagation of a disturbance on a two-dimensional surface for two different sets of boundary conditions [1] [2].

Platonic Solids

A Platonic solid is a polyhedron where (1) each face is the same regular polygon, and (2) each vertex joins the same number of faces.

The Platonic solids are highly symmetrical, and, in three dimensions, only five such solids can exist: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

This was first proven in Euclid’s Elements around 300 B.C., and has since been more rigorously proven using the Euler characteristic. The proofs are relatively easy to follow, and if you’re interested you can check them out both here and here.

Mathematica code:

Lonely Runner Conjecture

Imagine n runners on a circular track of length 1. The runners start from the same spot at the same time, and each has a distinct, constant speed. A runner is considered “lonely” whenever it is a distance of at least 1/n from every other runner. The Lonely Runner Conjecture (LRC) states that each runner will eventually, at some point in time, be lonely.

Said differently, the LRC states that for each runner, the spacing around it will eventually be greater than or equal to the spacing it would experience if the all of the runners were equally distributed around the track.

The conjecture has been proven to be true for 7 or fewer runners, but, interestingly enough, has never been proven to work for all cases of 8 or more runners. [In my 8-runner simulation above, I’ve only shown that it works for a specific set of runner speeds — I haven’t proven that it works for all sets of speeds.]

In the GIFs above, an arc appears around a runner whenever the runner is lonely, and the color of a runner fades after it’s been lonely at least once.

Mathematica code posted here.

Modeling Comet 67P

Two weeks ago, the ESA made history by landing a spacecraft on a comet. The spacecraft, named Philae, was carried to Comet 67P by a larger space probe named Rosetta.

Determining where Philae would land was a big step in this mission. Many attributes about the comet, including its topography, were taken into account. To help in this process, the ESA derived a 3D model of the comet's surface, and the data (made up of over 30,000 measurements) was recently released.

The images above use ESA data to model the surface of 67P. By adjusting lighting and orientation, actual photos taken by Rosetta (images 3 and 5) can be reproduced.

You can play around with the model yourself! Either use the code below in Mathematica or on the Wolfram Programming Cloud (with a free account), or play with the model online by clicking here*. * This link will expire on Dec 23, 2014, after which you'll have to upload the .obj file directly to the online viewer.

Mathematica code:

objLink = "http://sci.esa.int/science-e/www/object/doc.cfm?fobjectid=54726"; comet = Import[objLink, "OBJ"]; pts = Import[objLink, {"OBJ", "VertexData"}]; ListSurfacePlot3D[pts, MaxPlotPoints -> 20, Mesh -> All, MeshStyle -> Opacity[0.4]] Show[comet, Background -> Black, Lighting -> {{"Directional", LightGray, {-7, -10, 10}}}]

Harmonographs

A harmonograph is a mechanical device consisting of two or more pendulums attached to a pen. The swinging pendulums control the motion of the pen, tracing out a geometric pattern on a sheet of paper.

Since the system is damped by friction, the pattern spirals in on itself as time progresses.

Each of the GIFs above simulate the output of a 4-pendulum system (modeled after a harmonograph as configured in this video). The different outputs are generated by using different pendulum length ratios in each simulation.

Mathematica code posted here.

Thanks! I used to submit some of my posts on reddit, but I've done so less frequently ever since reddit's self-promotion rules were brought to my attention.

Feel free to post any of my stuff yourself, though! All I ask is that you link back to the actual post (not just to the images), since a lot is lost without the context.

Curves of Constant Width and Odd-Sided Reuleaux Polygons

A curve of constant width is a convex, two-dimensional shape that, when rotated inside a square, always makes contact with all four sides.

A circle is the most obvious (but somewhat trivial) example. Some non-trivial examples are the odd-sided Reuleaux polygons — the first four of which are shown above.

Since they don't have fixed axes of rotation, curves of constant width (except the circle) have few practical applications. One notable use of the Reuleaux triangle, though, is in drilling holes in the shape of a slightly rounded square (watch one of the triangle's vertices and notice the shape it traces out as it spins).

On a less technical note, all curves of constant width are solutions to the brainteaser, "Other than a circle, what shape can you make a manhole cover such that it can't fall through the hole it covers?"

Mathematica code posted here.

Atomic Models

Evidence-based theories on the structure of atoms have been around since the early 1800s. Dalton’s billiard ball model was the first on the map, and with further discoveries and experiments — like Thompson’s discovery of the electron and Rutherford’s gold foil experiment — improved models of atomic structure were introduced.

The first GIF above shows Rutherford’s planetary model, which was proposed in 1911. In his model, negatively-charged electrons orbit an incredibly small, dense nucleus of positive charge. Despite being a completely incorrect model, most people still think this is what atoms really look like*. This is not an atom. It’s physically impossible for electrons to stably orbit like this, and the idea of orbiting electrons was entirely replaced by 1926.

I can’t say what an atom actually looks like, but the most accurate model we have today is governed by the laws of quantum mechanics. The location of an electron is determined by a probability distribution, called an atomic orbital, which tells us the probability of an electron existing in any specific region around a nucleus. The second image shows the surface around a hydrogen nucleus on which an excited electron is most likely to exist.

Mathematica code posted here.

Cops and Robbers (and Zombies and Humans)

Cops and Robbers is a mathematical game in which pursuers (cops) attempt to capture evaders (robbers). The game is one of many pursuit-evasion games, each of which is governed by a different set of rules. The general goal of these problems is to determine the number of pursuers required to capture a given number of evaders.

The GIFs above show two versions of the game. The first is similar to the standard Cops and Robbers rendition, and the second is best described as "Zombies and Humans".

In both versions, an evader moves in the direction that gets it furthest away from the pursuers (focusing more on the closer pursuers), and a pursuer moves in the direction that gets it closest to the evaders (focusing more on the closer evaders).

In the first simulation, members of both groups have a constant speed. In the second simulation, members of a group move more quickly the closer they are to members of the opposite group, and slower when further away.

Mathematica code posted here.

Thanks for following! Here are some things:

I’m Brian. I’m currently working towards my M.S. in applied math and I love making physics and math visualizations.

Suggestions and requests are always welcome. Seriously. If there are any physics or math topics you want to know more about or anything you'd like to see visualized, pleaseeeee contact me.

I also want to note that all of my visualizations and code are free to use and adapt for non-profit-making purposes, as long as you cite me and link back to my original post.

Signal Collection and Parabolic Reflectors

A reflector is a type of antenna that receives and focuses various types of signals. Reflectors have numerous applications, from satellite dishes and telescopes, to long-distance microphones and car headlights. One common feature of these examples is their parabolic shape, giving them the name parabolic reflectors.

It turns out that paraboloids are the perfect shape for focusing signals from distant sources. When pointed directly at the the incoming signal, a parabolic reflector (GIF 1) collects the signal to a single focal point, where a receiver, called a feed horn, is placed to collect the focused transmission.

In many applications, parabolic reflectors are too costly to produce, so spherical reflectors (GIF 2) are used instead. The disadvantage of spherical reflectors is that they have multiple focal points, and therefore produce blurry results.

Mathematica code posted here.

Taylor Series Approximations

A Taylor series is a way to represent a function in terms of polynomials. Since polynomials are usually much easier to work with than complicated functions, Taylor series have numerous applications in both math and physics.

There are many equations in physics — like the one describing the motion of a pendulum — that are impossible to solve in terms of elementary functions. "Approximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problems" solvable for a restricted area of interest [1].

The GIF above shows the five-term Taylor series approximation of a sine wave about x=0.

Mathematica code:

Rotational Stability

Time for an experiment! Find a book and secure it shut using tape or a rubber band. Now experiment with spinning the book while tossing it into the air. You’ll notice that when the book is spun about its longest or shortest axis it rotates stably, but when spun about its intermediate-length axis it quickly wobbles out of control.

Every rigid body has three special, or principal axes about which it can rotate. For a rectangular prism — like the book in our experiment — the principal axes run parallel to the shortest, intermediate-length, and longest edges, each going through the prism’s center of mass. These axes have the highest, intermediate, and lowest moments of inertia, respectively.

When the book is tossed into the air and spun, either about its shortest or longest principal axis, it continues to rotate about that axis forever (or until it hits the floor). For these axes, this indefinite, stable rotation occurs even when the axis of rotation is slightly perturbed.

When spun about its intermediate principal axis, though, the book also continues to rotate about that axis indefinitely, but only if the axis of rotation is exactly in the same direction as the intermediate principal axis. In this case, even the slightest perturbation causes the book to wobble out of control.

The first simulation above shows a rotation about the unstable intermediate axis, where a slight perturbation causes the book to wobble out of control. The second and third simulations show rotations about the two stable axes.

Unfortunately, as far as my understanding goes, there's no intuitive, non-mathematical explanation as to why rotations about the intermediate principal axis are unstable. If you're interested, you can find the stability analysis here.

Mathematica code posted here.

Gabriel’s Horn and the Painter’s Paradox 

Gabriel’s Horn is a three-dimensional horn shape with the counterintuitive property of having a finite volume but an infinite surface area.

This fact results in the Painter’s Paradox — A painter could fill the horn with a finite quantity of paint, “and yet that paint would not be sufficient to coat [the horn’s] inner surface” [1].

If the horn’s bell had, for example, a 6-inch radius, we’d only need about a half gallon of paint to fill the horn all the way up. Even though this half gallon is enough to entirely fill the horn, it’s not enough to even coat a fraction of the inner wall!

The mathematical explanation is a bit confusing if you haven’t taken a first course in calculus, but if you’re interested, you can check it out here.

Mathematica code:

x[u_, v_] := u y[u_, v_] := Cos[v]/u z[u_, v_] := Sin[v]/u Manipulate[ParametricPlot3D[{{x[u, v], y[u, v], z[u, v]}}, {u, 1, umax}, {v, 0, 2*Pi}, PlotRange -> {{0, 20}, {-1, 1}, {-1, 1}}, Mesh -> {Floor[umax], 20}, Axes -> False, Boxed -> False], {{umax, 20}, 1.1, 20}]

Lagrangian Points

The Lagrangian points are the five locations in an orbital system where the combined gravitational force of two large masses is exactly canceled out by the centrifugal force arising from the rotating reference frame.

At these five points, the net force on a third body (of negligible mass) is 0, allowing the third object to be completely stationary relative to the two other masses. That is, when placed at any of these points, the third body stays perfectly still in the rotating frame.

The first image shows the fields due to the first mass, the second mass, and the rotating reference frame. When added together, these fields generate the effective field shown in the second image. The five Lagrangian points are indicated with gray spheres.

The first three Lagrangian points (labeled L1, L2, and L3) lie in line with the two larger bodies and are considered metastable equilibria. L4 and L5 lie 60° ahead of and behind the second body in its orbit and are considered stable equilibria.

Lagrangian points offer unique advantages for space research, and the Lagrangian points of the Sun-Earth system are currently home to four different satellites.

Mathematica code posted here.