Ahoy readers,

I've written two new posts on PhysicsForums summarizing some recently published papers on single-molecule biophysics. One is a summary of a paper I wrote, and one is more of a news-style article about another paper that I'm familiar with but wasn't involved in.

https://www.physicsforums.com/insights/new-research-on-untying-knots-in-polymers/

https://www.physicsforums.com/insights/using-the-spaghetti-twist-to-align-dna/

# Post-Doc Ergo Propter Hoc

## Saturday, 18 November 2017

## Sunday, 22 October 2017

### Applications of higher-order derivatives of position

One of the first things we learn in physics is that velocity is the rate of change of position, acceleration is the rate of change of velocity, and how to figure out the quantities you don't know based on the ones you do. Velocity and acceleration are important throughout physics because of velocity's part in momentum and kinetic energy and acceleration's role in Newton's law of motion. What we don't hear much about are the higher-order derivatives. Here, I'll briefly discuss these quantities and what they're useful for. For more detail than in this post, this paper summarizes a lot of the information and applies it to trampolines and rollercoasters.

The rate of change of acceleration with respect to time is called jerk. In highschool, when I was dealing with a lot of acceleration based questions, I imagined that when I got to university I would start seeing jerk-based questions. I was wrong, they never come up. There is a lot of information on the Wikipedia page about jerk, more than I'll get into here.

Most applications of jerk relate to its minimization. Reducing changes in acceleration throughout a trip makes the trip for comfortable, and most engineering for "smoothness" of some sort deals with minimizing jerk. There are a lot of papers on this, many of them having to do with robotics. The challenge is figuring out which joints to move and when in order minimize the jerk of the payload.

There have been some papers published in the American Journal of Physics and related publications about the educational value of studying jerk, and in my paper on minimizing relativistic acceleration, we left jerk minimization as an exercise for the reader.

Whereas jerk tends to be something that is minimized to ensure a smooth trajectory, snap tends to be used for predictive motion. Predicting the motion of wrists for prothetics, of quadcopters for interception, and of cats' eyes when they're watching videos. There was a paper examining the snap of the cosmological scale factor, which may be one of its most fundamental uses.

Crackle is rarely used (based on a literature search), and when it is it again tends to be in predicting human motion. Pop is similar, there is very little written about it (besides "it's called pop!") except for an engineering paper again about predicting motion. Less applied, in the world of physics there is this paper, on integrating the N-body program defines kinematic variables all the way down to pop in its algorithm.

After that, there has been nothing written about the seventh derivative, one paper about using the eighth derivative for satellite orbits, nothing for ninth or tenth, and a 1985 MIT internal memo about AI and eye tracking that mentions the eleventh derivative as an example. I think that is the bottom turtle.

The integral of position is called absement for some reason. If you spend one second standing on a one-meter stool, you will have gained an absement of one meter second. Gas pedals in cars function with absement in mind, your speed depends on how far you depress the pedal and how long you keep it down. There is a musical instrument called a hydraulophone which looks like several adjacent water fountains, and sound is produced when someone pushes down on one of the streams. The tone produced is proportional to how far and how long the stream is depressed. In fact, I would semi-seriously argue that the whole concept of absement just exists to describe hydraulophones and vice versa. There is a big explanation on Wikipedia about how there are higher integrals like "absity and abselleration" and an integral of energy called actergy, but I'm pretty sure someone just made these up on a whim sort of like those weird animal plural names like "a parliament of owls." There are two names, Mann and Janzen, that repeatedly come up when searching these things, so I suspect those are the main promulgators of these words.

**Jerk**-**the third derivative**The rate of change of acceleration with respect to time is called jerk. In highschool, when I was dealing with a lot of acceleration based questions, I imagined that when I got to university I would start seeing jerk-based questions. I was wrong, they never come up. There is a lot of information on the Wikipedia page about jerk, more than I'll get into here.

Most applications of jerk relate to its minimization. Reducing changes in acceleration throughout a trip makes the trip for comfortable, and most engineering for "smoothness" of some sort deals with minimizing jerk. There are a lot of papers on this, many of them having to do with robotics. The challenge is figuring out which joints to move and when in order minimize the jerk of the payload.

There have been some papers published in the American Journal of Physics and related publications about the educational value of studying jerk, and in my paper on minimizing relativistic acceleration, we left jerk minimization as an exercise for the reader.

**Snap, Crackle, and Pop (4th, 5th, and 6th derivatives).****The derivatives of jerk are sometimes called, respectively, snap, crackle, and pop. In searching for the origin of these terms (obviously they're taken from the Rice Crispies characters, but the origin of their use in physics), I found a reference in a 1997 paper that stated:**

These terms were suggested by J. Codner, E. Francis, T. Bartels, J. Glass, and W. Jefferys, respectively, in response to a question posed on the USENET sci.physics newsgroup.Doing some sleuthing and getting into weird old internet stuff, I found the current iteration of that newsgroup, which has a lot of John Baez arguing with crackpots, and the thread being referred to may be here. Doesn't quite explain the origin though. They may have just been invented by John Baez.

Whereas jerk tends to be something that is minimized to ensure a smooth trajectory, snap tends to be used for predictive motion. Predicting the motion of wrists for prothetics, of quadcopters for interception, and of cats' eyes when they're watching videos. There was a paper examining the snap of the cosmological scale factor, which may be one of its most fundamental uses.

Crackle is rarely used (based on a literature search), and when it is it again tends to be in predicting human motion. Pop is similar, there is very little written about it (besides "it's called pop!") except for an engineering paper again about predicting motion. Less applied, in the world of physics there is this paper, on integrating the N-body program defines kinematic variables all the way down to pop in its algorithm.

After that, there has been nothing written about the seventh derivative, one paper about using the eighth derivative for satellite orbits, nothing for ninth or tenth, and a 1985 MIT internal memo about AI and eye tracking that mentions the eleventh derivative as an example. I think that is the bottom turtle.

**Absement: the Integral of Position**The integral of position is called absement for some reason. If you spend one second standing on a one-meter stool, you will have gained an absement of one meter second. Gas pedals in cars function with absement in mind, your speed depends on how far you depress the pedal and how long you keep it down. There is a musical instrument called a hydraulophone which looks like several adjacent water fountains, and sound is produced when someone pushes down on one of the streams. The tone produced is proportional to how far and how long the stream is depressed. In fact, I would semi-seriously argue that the whole concept of absement just exists to describe hydraulophones and vice versa. There is a big explanation on Wikipedia about how there are higher integrals like "absity and abselleration" and an integral of energy called actergy, but I'm pretty sure someone just made these up on a whim sort of like those weird animal plural names like "a parliament of owls." There are two names, Mann and Janzen, that repeatedly come up when searching these things, so I suspect those are the main promulgators of these words.

## Monday, 9 October 2017

### Traversing the Six Gaps of Hell

[This was written in July 2017 for the MIT cycling team blog. The guy I sent it to left MIT shortly after and it was never uploaded, so I am posting it here. Also, it's not about physics]

"Yesterday" I, along with 15 other riders from the MIT cycling club, rode the Six Gaps ride through central Vermont. It was 130 miles long and featured six extremely arduous climbs through mountain passes. It was a great experience and one I hope to never repeat any time soon.

Strava link #1: https://www.strava.com/activities/1096905818

Strava link #2: https://www.strava.com/activities/1096828371

We drove up from Cambridge and stayed at the Swiss Farm Inn in Pittsfield, had a brief pow-pow before going to bed, and woke up at 5:30 to enjoy their proclaimed World's Best Breakfast, then drove to a nearby school to park the cars and head out/up on the bikes. I had initially planned to do only four of the gaps, but after climbing the first one and not feeling dead, I elected to ride all six. This was by far the longest ride I had ever done; I was riding in miles what my previous longest ride was in kilometers. Because I was pushing distance and pushing height to the extreme, I didn't want to also push speed, so stayed back from the main group of crazy people with Ethan and Roger. After each gap we stopped at a general store to refill our water bottles and get refuel, usually arriving when the front group was departing.

The ride was very clearly characterized by its constituent gaps, and I will describe each of those. The flats in between each gap were nice but not all too memorable, lots of farmland with pretty mountains in the distance. The roads were in good condition with very few potholes, the drivers in general were not jerks and gave us space while passing. There were more encouraging thumbs-up and waves from cars than there were aggressive honks.

My Strava is in metric, deal with it. |

Gap #1: Brandon Gap

We parked in between Rochester Gap and Brandon Gap, in a location carefully chosen by Brian to be at the bottom of a hill. We set out at about 7:30 and quickly came to Brandon Gap. We agreed to do this one no-drop so we could get a sense of which riding buddies would be appropriate. This climb was totally unpaved, which I was not really prepared for. It was gravel with a few tire-track lines of less-gravel, that each of us followed up. In terms of grade and altitude this was probably the easiest climb (although I'm perhaps biased by the fact that it was first) but the gravel made it a tad dicey. Fortunately the downhill was paved, and this was one of the fastest descents I had ever done: it was 4.5 miles and took 9 minutes, and I was breaking to avoid hitting turns too fast.

After this the groups naturally formed, with groups of three different speeds doing all six gaps, and one group doing four gaps.

The Pow-Wow |

Gap #2: Middlebury Gap

This was perhaps the longest and highest gap, but also the least steep, so it was manageable. I don't have a bike computer, so every time we got to a local maximum I'd ask my ride-mates if we were at the top, and every time they'd laugh and say no. I was starting to realize what I was getting myself into, that these gaps were more than an extended Eastern Avenue. The ride up Middlebury was pretty with a few small towns along the way.

Gap #3: Lincoln Gap

This was, objectively, the worst gap. It started out as a manageable climb with some occasional unpaved parts. Then, instead of pointing forward, the road pointed up. For 2.5 miles the road had an average grade of 14%, at times reaching 24%. I don't even know that's possible because I feel like the asphalt would just slide down the hill before it could set. 2.5 miles doesn't seem very far but it is when you can't exceed 4 mph. Each of us eventually stopped biking and started walking; the combination of strength, endurance, and gear ratios had failed. First I started doing the paper-boy and swerving back and forth to lessen the gradient. Then I started standing up and using my weight to force each pedal stroke down, until this failed as well. Loading up the cars at MIT I noticed that Daniel, who normally rides a carbon Cervelo, had brought his clunkier commuting bike because of its more favorable gearing. On Lincoln gap, I understood why. The road up Lincoln was narrower than the others, and had a lot of shade from forest lining both sides. Eventually the road levelled out enough for me to get back on my bike, and I made it to the top where there were lot people who had driven up to hike. The way down was unpaved, which was pretty sketchy given how steep it was, so I basically squoze the breaks the entire time.

Gap #4: Appalachian Gap

I was going to say that this was one of the more challenging ones, but who am I kidding, they were all extremely challenging. It had a fairly comfortable "are we there yet"-inducing foothill through some farmland, then a big descent before the climb proper started. This one was a fairly winding road up the mountain, such that you couldn't really see what coming around the bend, whether it was a mild lessening of slope or whether you were about the face a wall. The average grade of the climb-proper was 9%, with a mile straight at over 10%. At this point I had exceeded my previous distance record but was feeling pretty good, and decided to power up the top. Eventually I rounded a corner and could see the next two curves leading to the gap at the summit, which was an encouraging sight that gave me a boost to the top. This one was probably my favorite. I don't remember much about the descent, so it was probably uneventful.

Gap #5: Roxbury Gap

We had put in like seven hours of riding by the time we got to this one, and I was starting to feel fatigued. It was fairly similar to the Appalachian gap, turny and steep, except perhaps a little bit steeper. If I had done this earlier in the day I would have had a better time at it, but as it stood it was a bit too much to handle so I took a break and walked it for a bit before getting back on the bike and summiting. Towards the top and about 94 miles into the ride, my Garmin watch died, so I switched to phone stravving for the rest of the ride. The descent was unpaved and also had some extreme washboarding going on, so it was a bumpy ride despite my vigorous braking. Right at the bottom there were two kids with a lemonade stand, which we all gladly patronized.

Gap #6: Rochester Gap

The last gap! We got here around 6 PM, and 11 hours after setting out I was ready for this to be over. This was probably the easiest paved climb in terms of length and steepness, but I was not in the same condition I was in in the morning. I down-shifted and set to peddling, but noticed I was falling behind my ride buddies. I realized this was not because of fatigue per se, but because my ass was so sore that it hurt to stay in the saddle, so I did a lot of this climb standing up and pumping, finding a reserve of strength as I paper-boyed my way to the top. There was a sign marking the gap, which was a welcome sight. Right when I summited it started raining, and we rolled down the last descent into the Rochester school parking lot, and it was over.

I think overall I performed as well as I could have given how arduous this ride was; my bike worked well the whole time and neither my body nor my mind failed. I had a Clif bar at the top each peak and kept as well-hydrated and electrolyted as I could, which kept me going for the twelve hour journey. I only decided to do the trip a fortnight prior and didn't put in any specific training, but there are a few things I might do differently. I would try to get more appropriate climbing gears on my bike (I forget what my ratio is but it's not good), I would practice more on steep hills like School St in Arlington, and I would try to lose some weight so I'd have less to haul up. This was a really fun but extremely difficult ride; I don't regret doing all six instead of a truncated four. I recommend doing this once, but only once.

## Saturday, 16 September 2017

### Extreme High-Speed Dreidel Physics

(Warning: if you're viewing this post on a phone with limited data...you should probably leave before all the gifs load)

Every November or December, Jews around the world celebrate Hanukkah, a holiday that commemorates defeating and driving out the Hellenistic Seleucids from Israel and establishing the short-lived Hasmonean kingdom* around the year 160 BC. The celebration of Hanukkah involves lighting candles, eating greasy food, exchanging gifts, re-telling the story, and playing games with a spinning dreidel.

Legend has it that when the study of Judaism was banned by the Seleucids, scholars took to the hills to study in secret, and when authorities came to investigate, they pretended they were just doing some good wholesome gambling. That legend, however, is very likely made up and doesn't appear in print before 1890. I read an article arguing that the dreidel originated from a popular gambling device called a teetotum that was popular in Northern Europe in the medieval period, that eventually made its way into Yiddish culture. The letters on the four sides of the dreidel (equivalent to N, G, H Sh) are interpreted to stand for the Hebrew "Nes Gadol Haya Sham" (a great miracle happened there), but originally stood for the Yiddish rules for gambling with a dreidel: nit (nothing), gants (everything) halb (half), and shtel ein (put one in)**.

From a physics perspective, a dreidel is an example of a spinning top, a source of extremely difficult homework problems in undergraduate classical mechanics related to torque and angular momentum and rigid body motion and whatnot. I was chatting with a theorist I know who mentioned that it would be fun to calculate some of these spinning-top phenomena for the dreidel's specific geometry (essentially a square prism with a hyperboloid or paraboloid base), and I suggested trying to compare it to high-speed footage. A quick literature review revealed that most of the dreidel analysis has to do with whether the gambling game is fair and how long the games last. Annoyingly, the search was obfuscated by the fact that there's a publisher called DReidel.

My lab has a high-speed camera that is used to film gel particles and droplets as they deform. It is normally connected to a microscope, but with the help of a student in my lab, we removed it and connected it to a macroscopic lens we had lying around in ye-olde-drawer-of-optics-crappe. A representative of MIT Hillel graciously provided me with a few dreidels, and I some time spinning the dreidels in front of the high-speed camera and recording them at 1000 frames per second.

Before I get into the more quantitative analysis, let's just take a look at what a dreidel looks like in slow motion, because as far as I can tell from a the google, I am the first person to attempt this.

As I initially spin the dreidel, it spins in the air a few times, lands with an axial tilt, and gradually rights itself as its angle of precession comes closer to the vertical. After that, you can see it spinning rapidly and precessing a little bit, but not doing anything too crazy.

The self-righting behaviour is a lot more extreme when I do the advanced-level upside-down flip.

On those first few bounces, it really looks like it's going to fly out of control and crash, but still it gradually rights itself into a stable rotation. While this self-righting tendency is strong...it is not unstoppable.

It's also pretty fun to watch what happens when they eventually lose stability and fall over.

This self-righting is too complicated-looking for me to understand right now, so let's look at something simpler: the steady-state (a) rotation and (b) precession.

To perform quantitative analysis of the dreidel's motion, I would want to be able the measure the phase of its rotation over time. Because the dreidel is made of reflective plastic, as it spins it reflects light into the camera, moreso when its face is parallel to the plane of the camera. Thus by measuring the total intensity, we should have a proxy for the phase of the dreidel, each intensity peak being a quarter-turn, and can investigate how that evolves over time. I wrote a MATLAB script that summed the total intensity of each frame and plot it over time.

There was initially a problem with this method of analysis, however. You can figure it out if look at the wall behind the dreidel in the above gifs (especially the wide crashing one), and notice that it's flickering. This is because the room is illuminated with AC electric light with a 60 Hz frequency. The light intensity is proportional to the square of the current, so it has a maximum twice per cycle, and the light flickers at 120 Hz. That is exactly the frequency at which the intensity fluctuates; the flickering was swamping the contribution from the dreidel. However, the quarter-turn frequency isn't that far off, so I was getting some neat beat frequency dynamics as well***.

This caught me off guard and it was skewing all my videos, so I took another few recordings using my cell phone flashlight with all the AC lights turned off. The videos don't look nearly as good, but the time-series are cleaner.

We can the intensity fluctuating periodically every ~8 ms, corresponding to a rotation period of 32 ms (nicely close to the square root of 1000, so it's also around 32 rotations per second), and a slower mode of about 200 ms or 5 precessions per second. 32 rotations per seconds is 128 quarter-rotations per second, so you can figure out why it took me a while to figure out that I had to disentangle it from the 120 Hz light flickering.

The fourier transform shows two peaks, one corresponding to rotation and one to precession, which is of stronger amplitude (I believe this is due to my analysis method and not to actual physics). The reason the peaks are smeared and not as at a sharp frequency is because the angular velocity gradually decreases as the dreidel loses energy to friction, so the peaks get smeared to the right.

With the flickering out of the way, I can also calculate how the rotation period evolves over time, using a peak-finding function in MATLAB. It gradually gets slower, as expected, which 6-7 ms between peaks at the beginning, and 14 ms between the peaks before it crashes. If this is caused by dry friction at the base, we would expect the frequency to decrease linearly with time. If it's caused by viscous drag, we would expect an exponential decrease. What do we see? The fact that it's discretized by the frame rate makes it harder to tell, but applying a rolling average on the frequency decrease suggests that it is linear and thus caused by dry friction.

That was mostly a discussion of the rotation, although precession presented itself as well. Let's take a closer look the precession. I wanted to measure the angle the dreidel was at with respect to the vertical, and how that evolved over time. This is not as easy to measure as the total image intensity; I had to use Principal Components Analysis. I found an algorithm on this blog post, and it worked as follows:

Two things are apparent from looking at this graph: both the amplitude and frequency of precession are increasing over time. The fourier spectrum of the precession angle contains only the precession peak, without the rotation peak at higher frequency. What's happening is that gravity is exerting a torque on the dreidel at an angle relative to its principal angular momentum vector, which induces a precession in the direction determined by the cross product of spin and down. The angular frequency of precession is inversely proportional to that of rotation, so that as the dreidel slows due to friction, its precession speeds up, which is what we see. The spinning is essentially preventing the gravitational torque from pulling the dreidel down, and as it loses angular velocity, the precession angle gradually increases.

This whole project started as a discussion with a colleague about how the term "Jewish physics" should be repurposed from a label the Nazis used to discredit Einstein, and dreidels seemed like a natural thing to focus on. After fiddling around with a high-speed camera for a bit I got some cool videos, and thought I'd share them. I didn't really cover anything in this post that isn't explained in great detail in dozens of analytical mechanics textbooks, but it's perhaps the first time anyone has used a high-speed camera to record a dreidel. I thought it was neat.

In addition to being a fun little diversion, it also spurned an improvement in my DNA image analysis code. It was taking so long to open the dreidel movies in MATLAB that I looked into a more efficient way of doing so, which improved loading time by like a factor of 20 (from 55 seconds down to 3 seconds), which I now use to open DNA movies as well.

*If you grew up hearing the Hannukah story every year, you probably will not recognize the words Hellenistic, Seleucid, or Hasmonean.

**For those unfamiliar with Jewish linguistics, Hebrew is a Semitic language related to Arabic and Ethiopian (Amharic), whereas Yiddish is a Germanic language that uses Hebrew letters, so some of the words are similar to the also-Germanic English, e.g. halb and half.

***This is what the FAKE NEWS wrong analysis looks like:

Every November or December, Jews around the world celebrate Hanukkah, a holiday that commemorates defeating and driving out the Hellenistic Seleucids from Israel and establishing the short-lived Hasmonean kingdom* around the year 160 BC. The celebration of Hanukkah involves lighting candles, eating greasy food, exchanging gifts, re-telling the story, and playing games with a spinning dreidel.

Legend has it that when the study of Judaism was banned by the Seleucids, scholars took to the hills to study in secret, and when authorities came to investigate, they pretended they were just doing some good wholesome gambling. That legend, however, is very likely made up and doesn't appear in print before 1890. I read an article arguing that the dreidel originated from a popular gambling device called a teetotum that was popular in Northern Europe in the medieval period, that eventually made its way into Yiddish culture. The letters on the four sides of the dreidel (equivalent to N, G, H Sh) are interpreted to stand for the Hebrew "Nes Gadol Haya Sham" (a great miracle happened there), but originally stood for the Yiddish rules for gambling with a dreidel: nit (nothing), gants (everything) halb (half), and shtel ein (put one in)**.

From a physics perspective, a dreidel is an example of a spinning top, a source of extremely difficult homework problems in undergraduate classical mechanics related to torque and angular momentum and rigid body motion and whatnot. I was chatting with a theorist I know who mentioned that it would be fun to calculate some of these spinning-top phenomena for the dreidel's specific geometry (essentially a square prism with a hyperboloid or paraboloid base), and I suggested trying to compare it to high-speed footage. A quick literature review revealed that most of the dreidel analysis has to do with whether the gambling game is fair and how long the games last. Annoyingly, the search was obfuscated by the fact that there's a publisher called DReidel.

My lab has a high-speed camera that is used to film gel particles and droplets as they deform. It is normally connected to a microscope, but with the help of a student in my lab, we removed it and connected it to a macroscopic lens we had lying around in ye-olde-drawer-of-optics-crappe. A representative of MIT Hillel graciously provided me with a few dreidels, and I some time spinning the dreidels in front of the high-speed camera and recording them at 1000 frames per second.

Before I get into the more quantitative analysis, let's just take a look at what a dreidel looks like in slow motion, because as far as I can tell from a the google, I am the first person to attempt this.

As I initially spin the dreidel, it spins in the air a few times, lands with an axial tilt, and gradually rights itself as its angle of precession comes closer to the vertical. After that, you can see it spinning rapidly and precessing a little bit, but not doing anything too crazy.

The self-righting behaviour is a lot more extreme when I do the advanced-level upside-down flip.

On those first few bounces, it really looks like it's going to fly out of control and crash, but still it gradually rights itself into a stable rotation. While this self-righting tendency is strong...it is not unstoppable.

It's also pretty fun to watch what happens when they eventually lose stability and fall over.

This self-righting is too complicated-looking for me to understand right now, so let's look at something simpler: the steady-state (a) rotation and (b) precession.

It rotates several times while precessing from left to right. |

There was initially a problem with this method of analysis, however. You can figure it out if look at the wall behind the dreidel in the above gifs (especially the wide crashing one), and notice that it's flickering. This is because the room is illuminated with AC electric light with a 60 Hz frequency. The light intensity is proportional to the square of the current, so it has a maximum twice per cycle, and the light flickers at 120 Hz. That is exactly the frequency at which the intensity fluctuates; the flickering was swamping the contribution from the dreidel. However, the quarter-turn frequency isn't that far off, so I was getting some neat beat frequency dynamics as well***.

This caught me off guard and it was skewing all my videos, so I took another few recordings using my cell phone flashlight with all the AC lights turned off. The videos don't look nearly as good, but the time-series are cleaner.

The fourier transform shows two peaks, one corresponding to rotation and one to precession, which is of stronger amplitude (I believe this is due to my analysis method and not to actual physics). The reason the peaks are smeared and not as at a sharp frequency is because the angular velocity gradually decreases as the dreidel loses energy to friction, so the peaks get smeared to the right.

The measurement gets a bit hairy towards the end. |

That was mostly a discussion of the rotation, although precession presented itself as well. Let's take a closer look the precession. I wanted to measure the angle the dreidel was at with respect to the vertical, and how that evolved over time. This is not as easy to measure as the total image intensity; I had to use Principal Components Analysis. I found an algorithm on this blog post, and it worked as follows:

- Define a threshold intensity such that everything in the image above the threshold is dreidel, and everything below it is background. Set the dreidel equal to 1 and the background equal to 0.
- Define two arrays, one containing all the x-coordinates of all the 1's, and the other containing all the y-coordinates of all the 1's (such that each 1-pixel is represented in the array).
- Calculate the 2x2 covariance matrix of the location array.
- Find the eigenvectors and eigenvalues of the covariance matrix.
- Find the angle associated with the eigenvector with the larger eigenvalue (e.g. the arctangent of the ratio of its components)

Two things are apparent from looking at this graph: both the amplitude and frequency of precession are increasing over time. The fourier spectrum of the precession angle contains only the precession peak, without the rotation peak at higher frequency. What's happening is that gravity is exerting a torque on the dreidel at an angle relative to its principal angular momentum vector, which induces a precession in the direction determined by the cross product of spin and down. The angular frequency of precession is inversely proportional to that of rotation, so that as the dreidel slows due to friction, its precession speeds up, which is what we see. The spinning is essentially preventing the gravitational torque from pulling the dreidel down, and as it loses angular velocity, the precession angle gradually increases.

This whole project started as a discussion with a colleague about how the term "Jewish physics" should be repurposed from a label the Nazis used to discredit Einstein, and dreidels seemed like a natural thing to focus on. After fiddling around with a high-speed camera for a bit I got some cool videos, and thought I'd share them. I didn't really cover anything in this post that isn't explained in great detail in dozens of analytical mechanics textbooks, but it's perhaps the first time anyone has used a high-speed camera to record a dreidel. I thought it was neat.

In addition to being a fun little diversion, it also spurned an improvement in my DNA image analysis code. It was taking so long to open the dreidel movies in MATLAB that I looked into a more efficient way of doing so, which improved loading time by like a factor of 20 (from 55 seconds down to 3 seconds), which I now use to open DNA movies as well.

*If you grew up hearing the Hannukah story every year, you probably will not recognize the words Hellenistic, Seleucid, or Hasmonean.

**For those unfamiliar with Jewish linguistics, Hebrew is a Semitic language related to Arabic and Ethiopian (Amharic), whereas Yiddish is a Germanic language that uses Hebrew letters, so some of the words are similar to the also-Germanic English, e.g. halb and half.

## Thursday, 23 February 2017

### Comfortable Relativistic Space Travel: New paper on arXiv

I have written a new paper which has just come out on arxiv.org. It has the somewhat long title of "Relativistic minimization with constraints: A smooth trip to Alpha Centauri" and can be found here. My co-author on this paper was Riccardo Antonelli, who also writes a physics blog, called Hologrammata. It discusses and derives the most comfortable way to plan a relativistic journey, which may become relevant in several thousand years when we can actually go fast enough for this to matter. In this post, I will explain what this paper is about and what inspired me to write it.

Every month, the American Journal of Physics puts out a new edition with about twenty articles. These are usually a delight to read and can be read by anyone with a basic physics background. Some of the articles are about physics education, but most of the research articles involve applications of established physics to problems nobody had quite thought of before. (This is where my tunnel-through-the-Earth paper is published).

In the October 2016 edition, I read an article called "The Least Uncomfortable Journey from A to B" (possible free version here) by three Swedish and possibly French researchers, Anderson, Desaix, and Nyqvist. In it, they ask the following question: on a linear trip taking time T from point A to point B separated by distance D, what is the trajectory (instantaneous velocity as a function of time or distance) that minimizes the time-integrated squared acceleration? If you can't mentally convert that sentence into LaTeX, the "discomfort" here is the total acceleration that the passenger feels throughout the trip, pushing them into their seat on the first half of the trip, and thrusting them out of their seat on the second half.

The solution, which they find through Lagrangian-style variational calculus, turns out to be a trajectory with constant jerk, where the acceleration decreases linearly over the entire trip, such that the velocity is a quadratic function of time. A lot of the paper talks about various analytical approximation methods, and also computes an alternate "least uncomfortable" trajectory where the squared jerk is minimized instead of the squared acceleration.

The interesting thing about the solution is that there is only one. The relative velocity (v/vmax) as a function of the relative position (x/D) or time (t/T) does not actually depend on the distance or time. No matter how far or fast you want to go, there is only one set least-uncomfortable way to get there. However, there is a problem with this. If you want to calculate how to plan a train ride that covers 500 miles in an hour, the solution is fine. But if you want to plan a trip that goes from Earth to Jupiter in an hour, then it starts running into some problems. Although it wasn't stated in the paper, the solution only holds when D/T is much less than the speed of light. I was curious about finding a relativistic solution to this problem, and if I get curious enough about a problem that is actually possible to solve, I can be quite persistent.

In the relativistic version of this problem, when you consider the integral over time of the squared acceleration, it matters in which reference frame the time and acceleration are being measured. Because we care about the comfort of travellers and not of stationary observers, we consider the proper acceleration and not the coordinate acceleration. This problem can be formulated either in terms of "lab" time or proper time on the ship, there are benefits to both. My first attempt at this essentially involved following Anderson et al.'s derivation except with a lot of extra Lorentz factors, which is generally not a good way to go about things, and I stalled pretty quickly.

Then one day, I was talking to my internet friend Riccardo, who had just finished his master's thesis in string theory and holography. I figured a lot of the relativistic analytical mechanics formalisms might still be fresh in his mind, so I mentioned the problem to him. He too found the problem interesting, and came to the realization that if the problem was formulated in terms of the rapidity (the hyperbolic tangent of velocity relative to c) since its derivative is proper acceleration, it could be expressed as a much neater problem than my "add a bunch of Lorentz factors" approach.

The way to derive the solution, Riccardo discovered, is to treat it as a Lagrangian function of the rapidity rather than the position (such that you write down L(r, $\dot{r}$, ) instead of L(x, $\dot{x}$) and apply the Euler-Lagrange to higher-order derivatives of position than normal). Even though we were unable to derive a closed-form solution, it turns out, the rapidity of the least uncomfortable solution evolves like a particle in a hyperbolic sine potential, and I was able to generate solutions using the Runge-Kutta algorithm, which has been serving me well since I learned how to use it in 2008.

As I said, there is only one classical solution, it is universal for all distances and times. However, when relativity it taken into account, the solution depends on how close to the speed of light it gets: there is now a single free parameter that characterizes the solution, and we can express it in terms of its maximum velocity, rapidity, or Lorentz factor. Our solution recovers the classical solution in the low-velocity limit (which is good, otherwise we'd be obviously wrong), but as the maximum speed increases, a greater fraction of the time is spent close to the maximum (which makes sense, as the path is some form of "accelerate close to light speed, decelerate down from close to light speed" and as you approach light speed, your velocity changes less and less)

Now, we had solved the problem that I was inspired to solve, the relativistic version of Anderson et al.'s least uncomfortable journey. However, this whole thing may be moot for space travel: you don't necessarily want to keep acceleration to a minimum, instead you might want to contintually accelerate with Earth-like gravity for the first half of the trip, then reverse thrust and decelerate with Earth-like gravity for the second half, so that the entire ride (besides the switcheroo in the middle) feels like Earth. We calculated a trip like this to Alpha Centauri, which would take six years relative to Earth and 3.6 proper years on-board due to time dilation, reaching 95% light speed. With our solution, covering that distance in the same proper time would only reach 90% the speed of light, and might be more appropriate for sending mechanically sensitive electronic probes or self-replicating machines than it would be for sending people.

Anyway, we wrote up the paper and now it's online for the world's reading pleasure. It was fun seeing this paper go from a persistent idea to a finished product in a few weeks time, and dusting off some of the long-forgotten special relativity tools. It was a nice little collaboration with Riccardo, where I came up with the problem and he came up with the solution. This is the first random physics idea I've had in while that has actually come to fruition, and I hope there are many more to come.

Artist's impression. |

Every month, the American Journal of Physics puts out a new edition with about twenty articles. These are usually a delight to read and can be read by anyone with a basic physics background. Some of the articles are about physics education, but most of the research articles involve applications of established physics to problems nobody had quite thought of before. (This is where my tunnel-through-the-Earth paper is published).

In the October 2016 edition, I read an article called "The Least Uncomfortable Journey from A to B" (possible free version here) by three Swedish and possibly French researchers, Anderson, Desaix, and Nyqvist. In it, they ask the following question: on a linear trip taking time T from point A to point B separated by distance D, what is the trajectory (instantaneous velocity as a function of time or distance) that minimizes the time-integrated squared acceleration? If you can't mentally convert that sentence into LaTeX, the "discomfort" here is the total acceleration that the passenger feels throughout the trip, pushing them into their seat on the first half of the trip, and thrusting them out of their seat on the second half.

The solution, which they find through Lagrangian-style variational calculus, turns out to be a trajectory with constant jerk, where the acceleration decreases linearly over the entire trip, such that the velocity is a quadratic function of time. A lot of the paper talks about various analytical approximation methods, and also computes an alternate "least uncomfortable" trajectory where the squared jerk is minimized instead of the squared acceleration.

The interesting thing about the solution is that there is only one. The relative velocity (v/vmax) as a function of the relative position (x/D) or time (t/T) does not actually depend on the distance or time. No matter how far or fast you want to go, there is only one set least-uncomfortable way to get there. However, there is a problem with this. If you want to calculate how to plan a train ride that covers 500 miles in an hour, the solution is fine. But if you want to plan a trip that goes from Earth to Jupiter in an hour, then it starts running into some problems. Although it wasn't stated in the paper, the solution only holds when D/T is much less than the speed of light. I was curious about finding a relativistic solution to this problem, and if I get curious enough about a problem that is actually possible to solve, I can be quite persistent.

In the relativistic version of this problem, when you consider the integral over time of the squared acceleration, it matters in which reference frame the time and acceleration are being measured. Because we care about the comfort of travellers and not of stationary observers, we consider the proper acceleration and not the coordinate acceleration. This problem can be formulated either in terms of "lab" time or proper time on the ship, there are benefits to both. My first attempt at this essentially involved following Anderson et al.'s derivation except with a lot of extra Lorentz factors, which is generally not a good way to go about things, and I stalled pretty quickly.

Then one day, I was talking to my internet friend Riccardo, who had just finished his master's thesis in string theory and holography. I figured a lot of the relativistic analytical mechanics formalisms might still be fresh in his mind, so I mentioned the problem to him. He too found the problem interesting, and came to the realization that if the problem was formulated in terms of the rapidity (the hyperbolic tangent of velocity relative to c) since its derivative is proper acceleration, it could be expressed as a much neater problem than my "add a bunch of Lorentz factors" approach.

The way to derive the solution, Riccardo discovered, is to treat it as a Lagrangian function of the rapidity rather than the position (such that you write down L(r, $\dot{r}$, ) instead of L(x, $\dot{x}$) and apply the Euler-Lagrange to higher-order derivatives of position than normal). Even though we were unable to derive a closed-form solution, it turns out, the rapidity of the least uncomfortable solution evolves like a particle in a hyperbolic sine potential, and I was able to generate solutions using the Runge-Kutta algorithm, which has been serving me well since I learned how to use it in 2008.

As I said, there is only one classical solution, it is universal for all distances and times. However, when relativity it taken into account, the solution depends on how close to the speed of light it gets: there is now a single free parameter that characterizes the solution, and we can express it in terms of its maximum velocity, rapidity, or Lorentz factor. Our solution recovers the classical solution in the low-velocity limit (which is good, otherwise we'd be obviously wrong), but as the maximum speed increases, a greater fraction of the time is spent close to the maximum (which makes sense, as the path is some form of "accelerate close to light speed, decelerate down from close to light speed" and as you approach light speed, your velocity changes less and less)

Now, we had solved the problem that I was inspired to solve, the relativistic version of Anderson et al.'s least uncomfortable journey. However, this whole thing may be moot for space travel: you don't necessarily want to keep acceleration to a minimum, instead you might want to contintually accelerate with Earth-like gravity for the first half of the trip, then reverse thrust and decelerate with Earth-like gravity for the second half, so that the entire ride (besides the switcheroo in the middle) feels like Earth. We calculated a trip like this to Alpha Centauri, which would take six years relative to Earth and 3.6 proper years on-board due to time dilation, reaching 95% light speed. With our solution, covering that distance in the same proper time would only reach 90% the speed of light, and might be more appropriate for sending mechanically sensitive electronic probes or self-replicating machines than it would be for sending people.

Anyway, we wrote up the paper and now it's online for the world's reading pleasure. It was fun seeing this paper go from a persistent idea to a finished product in a few weeks time, and dusting off some of the long-forgotten special relativity tools. It was a nice little collaboration with Riccardo, where I came up with the problem and he came up with the solution. This is the first random physics idea I've had in while that has actually come to fruition, and I hope there are many more to come.

## Friday, 3 February 2017

### New article on the early history of science

I wrote an article on the first science experiments, over at PhysicsForums. Check it out!

## Saturday, 14 January 2017

### Empirical Searches for Nominative Determinism

Nominative determinism is the idea that a person's name influences their life or career. The Seven Dwarfs are an example, and humorous lists have been written about people with names that suit their vocation. The world's fastest man is named Bolt, there was a Russian hurdler named Stepanova, a meteorologist named Julie Freeze, and eugenics pioneer Eugen Fischer. Considerable attention is given to a urology paper by Weedon and Splatt.

To my knowledge, the idea is not taken too seriously. However, I was recently curious about the extent to which people have looked for evidence of nominative determinism. I went on Google Scholar and found some results which I'm sharing with you now.

Most analyses of nominative determinism focus on medical professions, looking at whether specialists tend to have names associated with their speciality.

Although this study was not entirely serious, it had a number of methodological flaws as it failed to consider the prominence of each surname in the medical fields relative to its prominence in the population at large.

One recent study was by four members of a family surnamed Limb, all doctors (two of whom named C. Limb are avid climbers). They surveyed a directory of over 300,000 medical doctors grouped by speciality, and went through the list and independently noted which were suited to medicine in general and to their speciality. The lists were then merged by consensus. They found that 1 in 149 doctors had medically appropriate surnames, and 1 in 486 had names relevant to their field. They noted:

In the analysis of particle physics experiments there is something called the "look elsewhere effect" where any significant effect has to be considered in light of all the other effects that are being looked for (this xkcd gives an example of the folly), which reduces the overall significance of the interesting part. I suspect that if they had looked at what other surnames were over-represented in bradycardia diagnosis, the significance of Brady would be diminished. I think this proviso should be applied to any search for nominative determinism: look at the boring names as well as the silly ones.

There are a few other studies out there, mostly tongue-in-cheek. These do go a step beyond compiling humourous lists, but not too much further.

To my knowledge, the idea is not taken too seriously. However, I was recently curious about the extent to which people have looked for evidence of nominative determinism. I went on Google Scholar and found some results which I'm sharing with you now.

Most analyses of nominative determinism focus on medical professions, looking at whether specialists tend to have names associated with their speciality.

Examples of nominative determinism in medical research, with statistics of N=1. Ref. |

One recent study was by four members of a family surnamed Limb, all doctors (two of whom named C. Limb are avid climbers). They surveyed a directory of over 300,000 medical doctors grouped by speciality, and went through the list and independently noted which were suited to medicine in general and to their speciality. The lists were then merged by consensus. They found that 1 in 149 doctors had medically appropriate surnames, and 1 in 486 had names relevant to their field. They noted:

"Specialties that had the largest proportion of names specifically relevant to that specialty were those in which the English language has provided a wide range of alternative terms for the same anatomical parts (or functions thereof). Specifically, these were genitourinary medicine (eg Hardwick, Kinghorn, Woodcock, Bell) and urology (eg Burns, Cox, Dick, Koch, Cox, Balluch, Ball, Waterfall). Some terms for bodily functions have been included in the frequency counts but are not mentioned because they are colloquial terms that may lower the tone of this publication."Another study looked at nominative determinism in patients rather than doctors, trying to see if surname influences health. Based in Ireland, the authors looked through the Dublin phone book to estimate the percentage of people surnamed Brady (0.36%), and then tried to ascertain what fraction of them had been diagnosed with bradychardia (slow heart rate). The only data they had available was for pacemaker implants, and found that 8 out of 999 pacemakers had gone to people named Brady. Despite the small numbers, this was statistically significant (p=0.03) relative to Brady's prominence in the Dublin telephone directory. The authors don't speculate on the causes of this, but neglected to discuss the idea that doctors may be subtly biased towards diagnosing Bradys with bradycardia.

"This increased bradyphenomenon in Bradys could be attributable to increased levels of bradykinin"

In the analysis of particle physics experiments there is something called the "look elsewhere effect" where any significant effect has to be considered in light of all the other effects that are being looked for (this xkcd gives an example of the folly), which reduces the overall significance of the interesting part. I suspect that if they had looked at what other surnames were over-represented in bradycardia diagnosis, the significance of Brady would be diminished. I think this proviso should be applied to any search for nominative determinism: look at the boring names as well as the silly ones.

There are a few other studies out there, mostly tongue-in-cheek. These do go a step beyond compiling humourous lists, but not too much further.

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