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Chaos, Fractal Nightclubs, and the Glamour of Math

Question: What does the word “chaos” mean to mathematicians?

Benoit Mandelbrot: The theory of chaos and theory of fractals are separate, but have very strong intersections.  That is one part of chaos theory is geometrically expressed by fractal shapes.  Another part of chaos theory is not expressed by fractal shapes.  And other part of fractals does not belong to chaos theory so that two theories which overlap very strongly and do not coincide.  One of them, chaos theory, is based on behavior of systems defined by equations.  Equations of motion, for example, and classical mathematics, and around 1900, Poincaré and ****, two great mathematicians at the time, have realized that sometimes the solution of very simple looking equations can be extremely complicated.  But in 1900, it was too early to develop that idea.  It was very well expressed and very much discussed, but did not – could not grow. 

Much later, of course, with computers this idea came to life again and became the very important part of science.  So both chaos theory and fractal have had contacts in the past when they are both impossible to develop and in a certain sense not ready to be developed.  And again, they intersect very strongly but they are very distinct.

Question: Do mathematical descriptions of chaos define some order within chaos?

Benoit Mandelbrot: Well a very strong distinction was made between chaos and fractals.  For example, the rules which generate most of natural fractals, models of mountains, of clouds, and many other phenomena involve change.  And therefore they are not at all chaotic in the ordinary sense of the word, in an ordinary, current, modern sense of the word.  Not chaotic in the old sense of the word, which doesn’t have any specific meaning. But I don’t like to discuss the question of terms.  The term chaos came, but you know something which was very confused, it helped it jell, but the use of a biblical name in a certain sense forces us to find the implications which were not important in mathematics.  That’s why when the time came to give a name to my work, I chose the word fractal which was new.  Before that, there was no need of a word at all because again there were only a few undeveloped ideas in the very many great minds.  But when a word became necessary, I preferred not to use an old word, but to create a new one.

Question: How did you come up with the word “fractal”?

Benoit Mandelbrot:  Well, it was a very, very interesting story.  At one point, a friend of mine, an older person, told me that he saw a paper of mine on a new topic.  And he said, “Look Benoit, I tell you, you must stop writing all of these papers in that field, that field, that field, that field.  Nobody knows where you are, what you are doing.  You just sit down and write a book.  A short book, a clear book, a book of things which you have done.”  So, I sat down and wrote the book.  Now, the book had to title, why?  Because the topics I had been studying had not been the object of any theory whatsoever.  And there are many words which mean nothing, but many fields which have no name because they don’t exist.  So, the publisher didn’t like this very ponderous title, said, “Look.”  And a friend of mine, another friend, told me, “Look, you create a new field, you are entitled to give it a name.”  So, I had Latin in high school and it turned out that one of my son’s was taking Latin in the United States, and so there was a Latin dictionary in our house, which was an exception. 

I went in there and tried to look for a word which fitted what I had been working on.  And when I was playing with the word fraction, and looked in the dictionary for a word where fraction came from.  It came from a Latin word which meant, how to say disconnect – rough and disconnected, it was a very general – the idea of roughness originally in Latin.  So, I started playing with fractus, which I named it that and coined the word fractal.  First of all, I put it in this book, Objets Fractals, in French as it turned out, and then the English translation of the book, and then the word took off.  First of all, people applied it in ways in which I didn’t find sensible, but there was nothing I could say about it.  So, then the dictionary started defining it, each a little bit differently.  And in a certain sense the word became alive and independent of me.  I could scream and say, I don’t like it, but it made no difference. 

I had once a curiosity of looking on the Web in different countries having different languages, what is a fractal, and found that in one country, I will not mention, it’s a word that has become applied to some nightclubs.  A fractal nightclub is a kind of nightclub.  I don’t know which, because I haven’t been there, but, and I don’t know the language, but I guess, from what I could guess, what it was.  It’s a word which has its own life.  I gave it a definition, but that definition became too narrow because some objects I want to go fractal did not fit the old definition. 

So some people asked me would I still believe the definition of whatever – 40 years ago.  I don’t.  But I have no control.  It’s something which works by itself.  The fact that very many adults I know never heard of it, but the children have, is what gives me particular pleasure because high school students, even the bright ones, are very resistant to, how to say, imposed terms.  And the combination of pictures and of deep theory, you can look at the picture and find something, some idea about this picture is sensible, and then be told that very great scientists either can’t prove it, or has taken 40 years to prove it, or had to be several of them together to prove it because it was so difficult.  And it can be seen by a child, understood by a child.  That aspect is one which very many people find particularly attractive in the field. 

In mathematics and science definition are simple, but bare-bones. Until you get to a problem which you understand it takes hundreds and hundreds of pages and years and years of learning.  In this case, you have this formula, you track in a computer and from a simple formula, in a very short time amazingly beautiful things come out, which sometimes people can prove instantly and sometimes great scientists take forever to prove.  Or don’t even succeed in proving it.

Recorded on February 17, 2010
Interviewed by Austin \r\nAllen

\r\n

The word "fractals," which Benoit Mandelbrot invented, has caught on with everyone from kids to club owners.

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A "very massive star" in the Kinman Dwarf galaxy caught the attention of astronomers in the early years of the 2000s: It seemed to be reaching a late-ish chapter in its life story and offered a rare chance to observe the death of a large star in a region low in metallicity. However, by the time scientists had the chance to turn the European Southern Observatory's (ESO) Very Large Telescope (VLT) in Paranal, Chile back around to it in 2019 — it's not a slow-turner, just an in-demand device — it was utterly gone without a trace. But how?

The two leading theories about what happened are that either it's still there, still erupting its way through its death throes, with less luminosity and perhaps obscured by dust, or it just up and collapsed into a black hole without going through a supernova stage. "If true, this would be the first direct detection of such a monster star ending its life in this manner," says Andrew Allan of Trinity College Dublin, Ireland, leader of the observation team whose study is published in Monthly Notices of the Royal Astronomical Society.

So, em...

Between astronomers' last look in 2011 and 2019 is a large enough interval of time for something to happen. Not that 2001 (when it was first observed) or 2019 have much meaning, since we're always watching the past out there and the Kinman Dwarf Galaxy is 75 million light years away. We often think of cosmic events as slow-moving phenomena because so often their follow-on effects are massive and unfold to us over time. But things happen just as fast big as small. The number of things that happened in the first 10 millionth of a trillionth of a trillionth of a trillionth of a second after the Big Bang, for example, is insane.

In any event, the Kinsman Dwarf Galaxy, or PHL 293B, is far way, too far for astronomers to directly observe its stars. Their presence can be inferred from spectroscopic signatures — specifically, PHL 293B between 2001 and 2011 consistently featured strong signatures of hydrogen that indicated the presence of a massive "luminous blue variable" (LBV) star about 2.5 times more brilliant than our Sun. Astronomers suspect that some very large stars may spend their final years as LBVs.

Though LBVs are known to experience radical shifts in spectra and brightness, they reliably leave specific traces that help confirm their ongoing presence. In 2019 the hydrogen signatures, and such traces, were gone. Allan says, "It would be highly unusual for such a massive star to disappear without producing a bright supernova explosion."

The Kinsman Dwarf Galaxy, or PHL 293B, is one of the most metal-poor galaxies known. Explosive, massive, Wolf-Rayet stars are seldom seen in such environments — NASA refers to such stars as those that "live fast, die hard." Red supergiants are also rare to low Z environments. The now-missing star was looked to as a rare opportunity to observe a massive star's late stages in such an environment.

Celestial sleuthing

In August 2019, the team pointed the four eight-meter telescopes of ESO's ESPRESSO array simultaneously toward the LBV's former location: nothing. They also gave the VLT's X-shooter instrument a shot a few months later: also nothing.

Still pursuing the missing star, the scientists acquired access to older data for comparison to what they already felt they knew. "The ESO Science Archive Facility enabled us to find and use data of the same object obtained in 2002 and 2009," says Andrea Mehner, an ESO staff member who worked on the study. "The comparison of the 2002 high-resolution UVES spectra with our observations obtained in 2019 with ESO's newest high-resolution spectrograph ESPRESSO was especially revealing, from both an astronomical and an instrumentation point of view."

Examination of this data suggested that the LBV may have indeed been winding up to a grand final sometime after 2011.

Team member Jose Groh, also of Trinity College, says "We may have detected one of the most massive stars of the local Universe going gently into the night. Our discovery would not have been made without using the powerful ESO 8-meter telescopes, their unique instrumentation, and the prompt access to those capabilities following the recent agreement of Ireland to join ESO."

Combining the 2019 data with contemporaneous Hubble Space Telescope (HST) imagery leaves the authors of the reports with the sense that "the LBV was in an eruptive state at least between 2001 and 2011, which then ended, and may have been followed by a collapse into a massive BH without the production of an SN. This scenario is consistent with the available HST and ground-based photometry."

Or...

A star collapsing into a black hole without a supernova would be a rare event, and that argues against the idea. The paper also notes that we may simply have missed the star's supernova during the eight-year observation gap.

LBVs are known to be highly unstable, so the star dropping to a state of less luminosity or producing a dust cover would be much more in the realm of expected behavior.

Says the paper: "A combination of a slightly reduced luminosity and a thick dusty shell could result in the star being obscured. While the lack of variability between the 2009 and 2019 near-infrared continuum from our X-shooter spectra eliminates the possibility of formation of hot dust (⪆1500 K), mid-infrared observations are necessary to rule out a slowly expanding cooler dust shell."

The authors of the report are pretty confident the star experienced a dramatic eruption after 2011. Beyond that, though:

"Based on our observations and models, we suggest that PHL 293B hosted an LBV with an eruption that ended sometime after 2011. This could have been followed by
(1) a surviving star or
(2) a collapse of the LBV to a BH [black hole] without the production of a bright SN, but possibly with a weak transient."

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