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Poetry, Space-time, and Entropy


Plotkin's Entropy by Donna Bellas. See text for info. With the amazing success of Isaacson's bio of Einstein, and the translation and release in the US of Neffe's uniquely-German bio, Einstein is certainly in the air. Not as much as 100 years ago, when he was perhaps the most famous person in the world, but nevertheless many are reading, and writing poetic about Albert.

From Bernhardt Blumenthal, a colleague at La Salle who is a Rilke scholar and poet (and in his spare time chair of the Dept. of Foreign Languages & Literatures and the Master’s Program in Central & Eastern European Studies) describes Einstein’s influence on his latest work:

…my latest poem, Schwere Seelen, (Heavy Souls), which just appeared last week in a German literary periodical plays on Einstein’s General Relativity. The souls, heavy with love and suffering, warp reality and pass over the space-time curve through hyper-space into another universe–one without entropy. The female lover, incidentally, transits space escaping on a light beam exiting a black hole (thus faster than our conventional absolute speed of light) and rescues the beloved from his entropic situation.

Bernie has long been fascinated with the 2nd Law of Thermodynamics, and his work and teaching often embrace the "entropic" topic in a fine blending of art, science, and literature. In an interview shortly after receiving the Robert L. Kahn Lyric Prize (given annually by the Society for Contemporary American Literature in German (SCALAG), Bernie describes his unique approach and genre: “I like to apply physics to literature to form a type of science fiction lyric poetry."

I am a firm believer that creative, informed marriage of science and literature can help students of both to understand the concepts and worldview of the other.Schwere Seelen is such a creation. I am thankful that Bernie has given me permission to reproduce his latest poem here, along with the English "reconceptualization."

Schwere Seelen Sternkönigin, mit Leid und Liebe überschwer bist du nach lichtloser Fahrt in meine Welt gekommen. Die Lichtgeschwindigkeit überspringend, kommst du siegend, mir den Weg in das neue Universum zu weisen. Meine bleierne Seele führst du rettend über die Raumzeitkurve in das blendende Licht hinein. Wir gleiten unsichtbar nun aus berstenden Sternen entfliegend in unvorstellbare Welten hinein. Hinter uns die Strahlen ausgehender Sonnen und vor uns die Unendlichkeit.

The English "reconceptionalization"…

Heavy Souls Astral queen, laden with love and suffering, after blackened flight, you have come into my world. Soaring past the speed of light, you come victorious lighting the path to the new universe. My leaden soul you transport with saving grace across the space-time curve into the blinding light. We glide invisible now escaping imploding stars into unimaginable worlds. Behind us the smoldering rays of dying suns and before us rising eternity.

Schwere Seelen appeared in Trans-Lit2, eds. Irmgard Hunt and Jolyon T. Hughes (Society for Contemporary American Literature in German), Vol. 13, No. 1, Spring 2007, p.69.

Heavy Souls has been accepted for publication in an anthology to be published by the International Library of Poetry.

About the image in this post: Plotkin’s Entropy is available at DonnaBellas Zazzle and DonnaBellas CafePress. In another beautiful marriage of art and science, Bellas writes that the image "captures the chaotic energy of a new protein as it seeks it final folded state. This painting is named after Dr. Steven Samuel Plotkin , a physicist who researches the energy pathways of protein folding." See Bellas’ site for many more science-inspired images.

Categories Art Literature & Poetry Science

A Myth of Gaussian Proportions


When I was a freshman in high school, my home room teacher gave us a very nasty assignment during an after-school detention session - to calculate 35 to the 35th power!

This assignment was particularly cruel and unusual punishment because there were no such things as calculators back in 1967.

What I really needed was something I didn’t know about until college: a closed-form solution.

Finding closed form expressions for partial sums is a standard calculus exercise. The ur-example of this type of problem is the sum of the first n integers, which is easily shown to be n(n+1)/2.

This closed form expression collapses (n-1) operations into three. Because it yields an exact answer, it is not really a predictor, but, in a sense, it is a model of a process.

When this example is done in a calculus class, a typical accompanying story is how young Gauss solved this problem in record time, totally showing up the teacher who had given out the onerous task of adding the first 100 integers. (The version I always heard was that this was a punishment because the students had been particularly noisy that day. The sadistic mathematical punishments of my high school teacher certainly lends credence to this tale.)

Brian Hayes, in his American Scientist Online article Gauss’s Day of Reckoning, questions whether the story is true or myth. Along the way he does prodigious research into Gauss’ writings and those of his biographers, and he reaches some very interesting conclusions that are pertinent to all students and teachers of mathematics. Some excerpts:

...soon I was wondering about the provenance and authenticity of the whole story. Where did it come from, and how was it handed down to us? Do scholars take this anecdote seriously as an event in the life of the mathematician? Or does it belong to the same genre as those stories about Newton and the apple or Archimedes in the bathtub, where literal truth is not the main issue? If we treat the episode as a myth or fable, then what is the moral of the story? After reading all those variations on the story, I still can't answer the fundamental factual question, "Did it really happen that way?" I have nothing new to add to our knowledge of Gauss. But I think I have learned something about the evolution and transmission of such stories, and about their place in the culture of science and mathematics. Finally, I also have some thoughts about how the rest of the kids in the class might have approached their task. This is a subject that's not much discussed in the literature, but for those of us whose talents fall short of Gaussian genius, it may be the most pertinent issue.

Hayes does a great service by pointing out the "fable" may convince some students that they are not capable of doing mathematics. He ends with a "moral of the tale"

The story of Gauss and his conquest of the arithmetic series has a natural appeal to young people. After all, the hero is a child—a child who outwits a "virile brute." For many students, that is surely an inspiration. But I worry a little that the constant repetition of stories like this one may leave the impression that mathematics is a game suited only to those who go through life continually throwing off sparks of brilliance. On first hearing this fable, most students surely want to imagine themselves in the role of Gauss. Sooner or later, however, most of us discover we are one of the less-distinguished classmates; if we eventually get the right answer, it's by hard work rather than native genius. I would hope that the story could be told in a way that encourages those students to keep going. And perhaps it can be balanced by other stories showing there's a place in mathematics for more than one kind of mind.

As I think back on my hichschool detention, I try to picture myself solving the problem instantly, showing it to my teacher (I wish I knew phrases like "virile brute" back then), and jauntily walking out of that classroom hours before anyone else. Of course, I had no idea of what a logarithm was (and how it could be used to do the problem quickly) and had no slide rule, nor could I use one if I did - a corollary of not knowing anything about logarithms. So my solving this problem quickly was never going to happen.

Now, with my trusty TI-84 calculator, I can easily compute 35 to the 35th power (and see the first 10 or so significant digits of the answer.) Ironically, it is easier to do this with a calculator than to find the sum of the first 100 integers.

I want to believe that, even if everyone else in his class had a calculator back in the 18th century, Gauss would still beat the other students (and professor) to the answer. And, while there is a certainly a place in mathematics for "more than one kind of mind" - there is no myth about Gauss’ genius.

Categories Education Mathematics

Fractal Solar Wind


Massive solar flare starting and ending on earth’s surface. Click to enlarge.Sunspot cycles are hot right now - literally and figuratively. After just posting about Cycle 24 - the about-to-begin 11-year cycle of sunspot activity, I now see a lot of references to scientists at Warwick University reporting on observed fractal nature of the solar wind, and the ramifications for prediction and understanding of sunspot cycles

The articles announcing the findings have been very exuberant about this latest finding, but so far they are short on some crucial facts. They are also woefully inadequate when it comes to reporting previous work.

Most web sites are just reproducing the press release from the University of Warwick:

The researchers, led by Professor Sandra Chapman, have also been able to directly tie these fractal patterns to the Sun's 'storm season'. The Sun goes through a solar cycle roughly 11 years long. The researchers found the fractal patterns in the solar wind occur when the Sun was at the peak of this cycle when the solar corona was at its most active, stormy and complex - sunspot activity, solar flares etc. When the corona was quieter no fractal patterns were found in the solar wind only general turbulence.

From this description it is not clear what the fractal pattern is. The New Scientist site provides more details, including a possible reason for the fractal pattern:

Sandra Chapman and colleagues at the University of Warwick, UK, used sensors on the ACE, Wind and Ulysses spacecraft to measure the sun's magnetic field strength in the solar wind as it fluctuates over time. When plotted on a graph, the team found that this pattern becomes fractal when the sun is in the most stormy phase of its 11-year cycle, they report in an upcoming issue of Physical Review Letters. "The field describes a wiggly line, just like a crinkly coastline," Chapman says. She says the fractal pattern could be created by energetic "ropes" of swirling magnetic flux, which cross over in an ordered and organised way during the stormy season. The observation could provide an insight into the driving force that accelerates the solar wind to such high speeds away from the sun, something that is poorly understood, Chapman says.

This is exciting news, but I wonder what is the major difference between what the Warwick group has observed and what was noted way back in 1993 when Russian physicists published a paper titled "Fractal and Multifractal Structures in Solar Wind. This work, by L.M. Zeleny and A.V. Milovanov, was published in Geomagnetizm i Aehronomiya. From the abstract:

Results of a study of interplanetary magnetic field turbulence at heliocentric distances of 1-30 AU are analyzed. It is shown that the fractal distribution of magnetic force tubes over the sun surface, discovered at distance scales of 400-40,000 km, leads to the formation of magnetic clouds at heliocentric distances of about 10 solar radii, with the spatial distribution of the magnetic clouds in solar wind characterized by a fractal dimension close to 3/2. The value of the spatial fractal dimension that determines the fine structure of the magnetic cloud as a fractal cluster of magnetic force tubes is obtained. The relationship between the spatial fractal dimension and the turbulence spectrum of interplanetary plasma is examined.

On the surface, this sounds remarkably like the current findings. I have not yet seen the article by the Warwick group. Titled "Self- similar signature of the active solar corona within the inertial range of solar wind turbulence", the work was published on May 18th 2007 in Phys. Rev. Lett. 2., by K.Kiyani, S. C. Chapman, B. Hnat, and R. M. Nico. I assume that they refer to the Russian work. If anyone has access to this particular issue of Phys Rev. Lett., please post a comment describing whether this is true.

Categories Fractals Science

Solar Cycle 24 Predictions


Cycle 23-24 sun-spot predictions. Click to enlargeIn a post last year titled Solar Activity Modeling: Great Predictions, Lousy Understanding? I described the current state of solar cycle modeling. Solar storm cycles are approximately 11 years in length.

We are now entering Cycle 24, which will start in March 2008, and reach a peak in 2015. The prediction from the NOAA (National Oceanic And Atmospheric Administration) is that "the Earth will soon experience a period of intense solar storms and the exact number of solar storms expected will become clearer in time."

Read the NOAA release of these findings here. This is an excellent article that describes how sunspots form, the nature of the 11-year cycles, and how the predictions are made. Sun spot activity can be extremely deleterious to transportation and communication infrastructure, and therefore predictions have to be accurate. The care with which the predictions are made is evident in the statement by solar-physicist Doug Biesecker:

“…there are approximately six techniques used to predict the intensity of a solar cycle,” said Biesecker. “The first three are based on statistics and provide a sound historical baseline upon which to forecast future cycles. The other three are based on physics and the sun’s dynamo conveyer belt theory.”

An overview of these techniques can be found in last year’s post.

Categories Physics Understanding & Prediction

Visualizing Web Pages - HTML Graphs


FractaLog web graph. Click to enlarge. As an interesting follow up to my recent post on KartOO and Google Browser, check out the HTML Graphlet applet created by Sala (no last name), and posted on the Aharef blog. To use the applet, just enter the URL of the page to be graphed.

The applet constructs multi-colored graph of nodes and edges, with each color representing a different HTML tag. As the graph is produced, it grows outward, with branches sprouting - all in a very kinetic/organic way.

The color of the nodes refer to specific HTML tags: blue: for links (the A tag) red: for tables (TABLE, TR and TD tags) green: for the DIV tag violet: for images (the IMG tag) yellow: for forms (FORM, INPUT, TEXTAREA, SELECT and OPTION tags) orange: for linebreaks and blockquotes (BR, P, and BLOCKQUOTE tags) black: the HTML tag, the root node gray: all other tags

The image at the top of this post is a map of FractaLog as of the date of this posting.

Sala asks that those with flickr accounts post a screenshot of their site tree, using websitesasgraphs for a tag. Click on this link to see a wide variety of web page graphs.

Some of these graphs are more fractal-like than the others. I can only hope that someday the FractaLog graph will look suitably fractal to deserve its name. Update on Thursday, May 31, 2007 by Registered Commenter

R.A. DiDio
Note: Just as I published this post, the link to the Graphlet is not working - nor is the link to Aharef. I am getting a "WEb Site Suspended" message.

If it is working for anyone, please let me know with a comment here.

Categories Fractals Maps Visualization

Web Search Visualization & Fractal Maps


KartOO fractal search. Click to enlarge.There is a growing trend in using connectivity-visualization mapping when web searching. Search engines such as KartOO and TouchGraph Google Browswer, attempt to show the linkage among web sites/pages by web-like 2-D graphics, where colors and icon-sizes indicate the strength of the site linkages. The effect can be described best as a "concept map", with nodes representing sites, and connections between nodes representing links. Of the two engines, KartOO is much more visually interesting. (I have included screen shots produced by each engine in a search for "fractal".)

Whether this enhances the search experience is debatable - at present. While KartOO’s pr claims that the non-linear nature of the way information is distributed on the internet should be matched in the non-linear nature of the visualization, so far I haven’t seen anything that makes me want to switch from Clusty, which does a great job of organizing themes in web searches, and provides the links/themes in a nice old-fashioned list.

I find the same lack of compelling features for the Touch Graph Google Browser. I should note that TouchGraph is a company that designs custom-tailored visual search features for other companies and projects, with a focus on "creating tools that enable decision makers to display, navigate, and analyze complex data simply and intuitively…Individuals and organizations have more vital information at their fingertips than ever before. Traditional search engines provide a way to sift through this data. However, the greatest insights can be achieved not by sifting, but by looking at the big picture to see how items are connected."


TouchGraph fractal search. Click to enlarge.I’m still underwhelmed by what the visual engines do that’s better than Clusty. They do produce some fascinating, kinetic displays, because nodes and connections expand when clicking on different parts of the graphic display, and various pop-ups appear with further detail about the sites. Because clicking often produces new sets of nodes, there is a fractal-like quality to the search map. I can see the efficacy of such a search for different types of data, or business functions, but I doubt that they will become the main search option for an ordinary web search.

Categories Fractals Visualization

When Monty Met Erwin

It may be that only Monty Python’s exquisite blend of slapstick/farce/erudition/absurdity can take on the infamous cat of Erwin Schrödinger…


Schrödinger, one of the giants of Quantum Mechanics, truly believed that his cat presented an absurdity that undermined the probabilistic interpretation of Quantum Mechanics. QM was not undermined, and still continues to beguile and bedevil all who try to grasp its implications about the nature of the world.

Cecil Adams’ "epic" poem The Story of Schrödinger’s Cat, is a great riff on the Krazy Kat. A choice excerpt:

Now, you'd say the cat either lives or it don't But quantum mechanics is stubborn and won't. Statistically speaking, the cat (goes the joke), Is half a cat breathing and half a cat croaked. To some this may seem a ridiculous split, But quantum mechanics must answer, "Tough @#&!

Of the many thousands of S-Cat web sites, one of my favorites is The Well-Intended, but not quite interactive Schrödinger’s cat: A Rather Silly Experiment in Quantum Mechanics.Click here to try your luck at predicting whether you can indeed let the cat out of the bag… Cartoon info: By Paul Dlugokencky, concept by Zachary H. Levine, for the American Physical Society

Categories Philosophy Physics

There's Danger in Them Thar Equations


A very interesting piece by Howard Wainer in the latest American Scientist (May-June 2007) concerns dangerous equations, which he describes as falling into two classes:

  • equations that are dangerous because we know them - they "may pose danger because the secrets within its bounds open doors behind which lies terrible peril," with E=mc2 the most obvious candidate
  • equations that are dangerous because we don’t know them - mot because there is no theory that has yet yielded these equations, but rather because they are not known by those who need to know them. This is especially true for policy makers that base their decision on mathematical models, and specifically statistical models.

Wainer’s top choice for most dangerous statistical equation is due to Abraham de Moivre, who showed in 1730 that the standard error of the mean of a sample is the standard error of the mean of the population divided by the square root of the sample size. A significant prediction of this equation is that small sample sizes lead to large fluctuations in sample means. It is this simple statement:

small samples → large fluctuations in sample means,

that provides the biggest danger when not used, or not understood, by both policy makers and the average citizen.

Most with the most rudimentary knowledge of statistics know that reported poll figures come with some degree of uncertainty because of sample size (e.g. a typical political poll will yield a statement such as "Candidate A lead the race with 48%, plus or minus 3 percentage points"). The point that Wainer raises is that, when faced with seeing a large number of sample means, we all tend to forget the main point we should see quite a variation of these means if the sample sizes are small.

Wainer goes on to list five examples in which a large number of reported sample means measured in a large number of small-size samples leads to unwarranted conclusions:

  • maps of disease rates by county
  • the relation of student performance to class size
  • the relation between safe cities and city size
  • gender difference in academic performance

In each of these cases Wainer convincingly illustrates how De Moive’rs equations is ignored by policy makers, with disastrous results - often resulting in the mis-allocation of scarce resources. (e.g. are small class sizes the best way of using school tax funds to increase student learning?)

There’s even an illustrative story of determining standards for gold coins in 12th century England - a case where a quite a few cases of extreme punishment for certain minters. Of course, de Moive was 600 years in the future. It was only then that those punished could be considered "exonerated."

Wainer’s article is a thorough, fascinating, and extremely well-written look at the dangers of basing decisions on mathematical models without knowing what the models are really saying. Given its readability, it can easily be used in a Quantitative Literacy course as well as a traditional Stat class.

Categories Modeling Politics Understanding & Prediction

The Two Einsteins


A lot has been written about the latest Isaacson bio of Einstein, and it’s now on all US best-seller lists. Remarkably, there is another Einstein bio just released by Jüurgen Neffe (actually released 2 years ago in Germany and then most of Europe where it has been a best-seller equivalent to Isaacson) .

I was lucky enough to get the Inquirer assignment for both of them. See the full review. (Because the Inquirer piece was restricted to 850 words, I will have much more to say about both of these books shortly.)

They’re both amazing books, with Neffe’s maybe the more interesting… If you have the time, read them both!

Categories Physics Politics Science

Football Earth and the Degree Confluence Project


Now this may be the wildest web-based community project to date: The Degree Confluence project. The project goal is amazingly audacious - to "to visit each of the latitude and longitude integer degree intersections in the world, and to take pictures at each location. The pictures, and stories about the visits, will then be posted …"

A basic calculation shows that there are 64,442 confluences . (To see this, forget the poles for a second - they are points of only one latitude (90° or - 90°), and 360 longitude-values. The remaining confluences then are 179 latitude lines * 360 longitude lines = 64,440 confluences. Add the poles to get to 64,442)

The project was started in 1996 by Alex Jarrett because he "liked the idea of visiting a location represented by a round number such as 43°00’00"N 72°00’00"W. What would be there? Would other people have recognized this as a unique spot? "

He also writes that he had recently purchased a GPS and was looking to"come up with something to do with it."

After posting about the confluences he was "claiming" to his web site, readers "claimed" some of their own and posted about them there, and the project " just snowballed from there."

Boy, did it. Currently there have been over 5,000 successful confluence claims!

I can’t possibly describe all of the incredible detail that has gone into the project, e.g. are the confluences on land, water, or ice? You have to check out the web site for an amazing set of tools that will help you find confluences not yet "marked", where confluences are in a given country, plus much more. (There’s even a listing of letters in a variety of languages that you can use to describe the project to the owner of the land on which you’re trespassing, and hopefully avoid being shot.)

There are 2 reasons I am posting about the DCP here:

Because the earth is not perfectly spherical, but flattens out at the poles, and the degree-lines of longitude get closer and closer, the confluence points start getting so close to each other to not make sense to have so many claims from such a small area. (Not to mention that they are in an arctic region). The DCP has come up with a classification system that denotes confluences as primary or secondary depending on distance from the poles. The calculations of confluence locations are based on modeling the earth as an oblate spheroid (i.e. an ellipsoid of revolution).

OK - so there’s a modeling connection. But a better connection is provided by a quote from a confluence hunter:

Confluences are interesting to me because they represent randomness that emerges from strict order. It goes far beyond a silly quest for invisible man-made boundaries. The confluence latticework is an open defiance of the order our culture imposes on us, which frowns on tourists who abandon the traveled roads, the sanitized vistas, and the stops designed to conjure up dollars for empty memories.

Here the randomness described is the wonderfully diverse events that can happen as you travel to a confluence - the vagaries of life and culture on this degree-by-degree grid.

So go to your nearest confluence (the web site has a tool to help you find this point), photograph it, and claim it on the site - there are still over 11,000 available.

And be sure to bring your letter about the project translated into the correct language!

Categories Randomness