Archive for September, 2013


September 28th, 2013 Comments off


I’ve written several times, most recently last year, about the Pythagorean Three-Body Problem, which has just marked its first century in the literature (See Burrau, 1913).

Assume that Newtonian Gravity is correct. Place three point bodies of masses 3, 4, and 5 at the vertices of a 3-4-5 right triangle, with each body at rest opposite the side of its respective length. What happens?

The solution trajectory is extraordinary in its intricate nonlinearity, and lends itself to an anthropomorphic narrative of attraction, entanglement and rejection, with bodies four and five exiting to an existential eternity of No Exit, and body three consigned to an endless asymptotic slide toward constant velocity.

This past academic year, I worked with Ted Warburton, Karlton Hester, and Drew Detweiler to stage an interpretive performance of the problem, along with several of its variations. The piece was performed by UCSC undergraduates and was part of the larger Blueprints year-end festival. Here is a video of the entire 17 minute program.

The first of the four segments is an enactment of the standard version of the problem (As set above), and was done with a ballet interpretation to underscore that this is the “classical” solution. Prior to joining the faculty at UCSC, Ted was a principal dancer at the American Ballet Theater, and so the cohoreography was in an idiom where he has a great deal of experience.

The score for the performance was performed live, and is based wholly on percussion parts for each of the three bodies. The interesting portion of the dynamics is mapped to 137.5 measures, which satisfyingly, last for three minutes and forty five seconds.


The nonlinearity of the Pythagorean Problem gives it a sensitive dependence to initial conditions. It is subject to Lorenz’s Butterfly Effect. For the second segment of the performance, we chose a version of the problem in which body three is given a tiny change in its initial position. Over time, the motion of the bodies departs radically from the classical solution, and the resolution has body three leaving with body five, while body four is ejected. A more free-flowing choreography was drawn on to trace this alternate version.


A fascinating aspect of the problem is that while the solution as posed is “elliptic-hyperbolic”, there exist nearby sets of initial conditions in which the motion is perfectly periodic, in the sense that the bodies return precisely to their initial positions, and the sequence repeats forever. In the now-familiar solution to the classical version of the problem, the bodies manage to almost accomplish this return to the 3-4-5 configuration at a moment about half-way through the piece. This can be seen just after measure 65, at which time body 4 (yellow), body 5 (green), and body 3 (blue) are nearly, but are not exactly, at their starting positions, and are all three moving quite slowly:

If the bodies all manage to come to rest, then the motion must reverse and retrace the trajectories like a film run backward. With this realization, one can plot the summed kinetic energy of the bodies, which is a running measure of the amount of total motion. Note the logarithmic y-axis:

The bodies return close to their initial positions at Time = 31, at which time there is a local minimum in the total kinetic energy.

Next, look at the effect of making a small change in the initial position of one of the bodies. To do this, I arbitrarily perturbed the initial x position of body 3 by a distance 0.01 (a less than one percent change), and re-computed the trajectories. The kinetic energy measurements of this modified calculation are plotted as gray. During the first half of interactions the motion is extremely similar, but that the second half is very different. Interestingly, the gray curve reaches a slightly deeper trough at Time = 31. The small change has thus created a solution that is slightly closer to the pure periodic ideal.

I next used a variational approach to adjust the initial positions in order to obtain solutions that have progressively smaller Kinetic energy at time 31. In this way, it’s easy to get arbitrarily close to periodicity. The motion in a case that is quite close to (but not quite exactly at) the periodic solution is shown just below. After measure 65, the bodies arrive very nearly exactly at their initial positions, and, for the measures shown in the plot below, they have started a second, almost identical run through the trajectories.


The perfectly periodic solution occurs when bodies 4 and 5 experience a perfect head-on collision at time ~15 (around measure 33). If this happens, bodies 4 and 5 effectively rebound back along their trajectory of approach, and the motion retraces, therefore repeating endlessly. Here’s the action which shows the collision:

Ted suggested that Tango and Rhumba could be the inspiration for the choreography of the perfectly periodic solution. I was skeptical at first, but it was immediately evident that this was a brilliant idea. The precision of the dancing is exceptional, and the emotion, while exhibiting passion, is somehow also controlled and slightly aloof. No jealousy is telegraphed by motion, allowing the sequence to repeat endlessly in some abstract plane of the minds eye.


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September 21st, 2013 Comments off


Fall quarter at UCSC has arrived, and with it, the latest iteration of my astrophysical fluid dynamics course.

This class covers the workings of bodies that are composed of gas, ranging from molecular clouds and accretion disks to stars and giant planets. These objects are complicated enough so that numerical calculations can often help to generate insight, so I’ve traditionally distributed some simple numerical routines for use in the class problem sets.

My first exposure to computers was in the mid-1970s, when several PLATO IV terminals were set up in my grade school in Urbana. My mid-1980s programming class was taught in standard Fortran 77. Somehow, these formative exposures, combined with an ever-present miasma of intellectual laziness, have ensured that Fortran has stubbornly remained the language I use whenever nobody is watching.

Old-style Fortran is now well into its sixth decade. It’s fine for things like one-dimensional fluid dynamics. Formula translation, the procedural barking of orders at the processor, has an archaic yet visceral appeal.


Student evaluations, however, tend to suggest otherwise, so this year, everything will be presented in python. In the course of making the sincere attempt to switch to the new language, I’ve been spending a lot of time looking at threads on stackoverflow, and in the process, somehow landed on the Wikipedia page for Malbolge.

Malbolge is a public domain esoteric programming language invented by Ben Olmstead in 1998, named after the eighth circle of hell in Dante’s Inferno, the Malebolge.

The peculiarity of Malbolge is that it was specifically designed to be impossible to write useful programs in. However, weaknesses in this design have been found that make it possible (though still very difficult) to write Malbolge programs in an organized fashion.

Malbolge was so difficult to understand when it arrived that it took two years for the first Malbolge program to appear. The first Malbolge program was not written by a human being, it was generated by a beam search algorithm designed by Andrew Cooke and implemented in Lisp.

That 134 character first program — which outputs “Hello World” — makes q/kdb+ look like QuickBasic:

(‘&%:9]!~}|z2Vxwv-,POqponl$Hjig%eB@@>}=m:9wv6wsu2t |nm-,jcL(I&%$#”`CB]V?Txuvtt `Rpo3NlF.Jh++FdbCBA@?]!~|4XzyTT43Qsqq(Lnmkj”Fhg${z@\>

At first glance, it’s easy to dismiss Malbolge, as well as other esoteric programming languages, as a mere in-joke, or more precisely, a waste of time. Yet at times, invariably when I’m supposed to be working on something else, I find my thoughts drifting to a hunch that there’s something deeper, more profound, something tied, perhaps, to the still apparently complete lack of success of the SETI enterprise.

I’ve always had an odd stylistic quibble the Arecibo Message, which was sent to M13 in 1974:


It might have to do with the Bigfoot-like caricature about 1/3rd of the way from the bottom of the message.


Is this how we present to the Galaxy what we’re all about? “You’ll never get a date if you go out looking like that.”

Fortunately, I discovered this afternoon that there is a way to rectify the situation. The Lone Signal organization is a crowdfunded active SETI project designed to send messages from Earth to an extraterrestrial civilization. According to their website, they are currently transmitting messages in the direction of Gliese 526, and by signing up as a user, you get one free 144-character cosmic tweet. I took advantage of the offer to broadcast “Hello World!” in Malbolge to the stars.


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