Six Degrees and the Erdös Number

Recently there has been a lot of talk about small networks and how people are connected to one another. One of the more popular manifestations of this in the world of movies is the Kevin Bacon number. An actor's Kevin Bacon number is 1 if he or she was in a movie with Kevin Bacon. The number is 2 if the actor was in a film with someone who has a KB number of 1, and 3 if in a film with a KB number 2 and so on. There are a number of web sites devoted to the Kevin Bacon Game, click here to see one of them. For a popular discussion of Small World Networks you can take a look at Duncan J. Watt's book, "Six Degrees."

The equivalent of the KB game for scientists, or at least mathematicians, is the Erdös number. Paul Erdös, who recently died, was a prolific mathematician with many collaborators, and any association with him is a statement of ones connectedness with the top echelons of the community. The interesting discovery about small world networks, as discussed in Watt's book, is that groups contain both specialists and generalists. While the specialists may be considered the most important members of the group, it is the generalists that connect the group to the wider world. Without such people the group's activities remain isolated. With them some surprising connections can follow. Therefore even non-mathematicians may have a relatively low Erdös number. So I wasn't too surprised when a friend of mine, Joe Marasco, sent me an email reference to his web site where he discusses how he found his Erdös number to be 5. Joe, the recent author of a book on software development (referenced on this site) was also an experimental high energy physicist. His work is far removed from the type of "pure" math that Erdös was involved in, so he seemed quite pleased to find he had such a low Erdös number. Being a native New Yorker, (Joe is also one) I couldn't help taking up the challenge. So after using the Google Scholar web site and the Erdös number project data page I found the following set of references:

The acoustic cavity containing small scatterers as a singular perturbation problem
LESSER, M B; LEWIS, J A
Journal of Sound and Vibration. Vol. 33, pp. 13-27. 8 Mar. 1974

Charge Singularity at the Corner of a Flat Plate, J. A. Morrison and J. A. Lewis, SIAM J. Appl. Math., 31:2 (September 1976), pp. 233-250.

Asymptotic Analysis of a Random Walk on a Hypercube with Many Dimensions, P. Diaconis, R. L. Graham and J. A. Morrison, Random Structures and Algorithms, 1:1 (1990), pp. 51-72.

On a Linear Diophantine Problem of Frobenius, R.L. Graham (with P. Erdös), Acta Arithmetica, 21, (1972), pp. 399–408.

To my delight my Erdös number of 4 is lower than Joe's 5. The key to this was that R.L.Graham, with Erdös number 1, also worked at Bell Labs (I worked there from 1966 to 1971). This of course was the connection from the world of pure to the world of applied math.

If you want to compute your own Erdös number you should take a look at the Erdös number project web site (linked to above). I would be interested to hear from any friends as to what their Erdös number might be. At least one person in my academic department has an Erdös number of 3 so all of his collaborators are also at least E4s.

Cheers,
Marty

Counting the Stars

One of the hurdles I had to face in the 1960's, as a graduate student at Cornell, was the PhD qualifier exam. After a year or two of study you had to prove that you were worthy of continuing by facing an oral exam by the members of your graduate committee made up of three or four professors from your areas of major and minor study. An interesting aid to this ordeal was a book where students that had undergone this rite of passage recorded their experiences. Naturally the entries of most interest were those in which the inquisitors matched the members of your own committee. Thus I was particularly concerned with one case where one of the professors was Phillip Morrison , a man known for his keen mind and interesting history. Morrison had the reputation of asking questions that called for imagination and that even reflected his views of how one should think about science. In this case the question asked was; What is the number of visible stars? The student was supposed to arrive at an answer to this question using whatever data he had at hand, and it had to be reasoned out, he couldn't just give a number he recalled from reading a book on astronomy. As I recall the student didn't answer the question to Morrison's satisfaction and Morrison's own answer met with an argument from another member of the committee, the mathematician Mark Kac (Kac is pronounced kahts). Kac was as much a "character" as Morrison and according to the student, they became so involved in the argument that they forgot all about him.

The argument involved Morrison's use of probability. The hint he gave the student was that there is a North Pole Star but no South Pole Star. Now consider how close two objects can get and still be distinguished as individuals by the human eye. In optics the distinguishablity of objects is calculated by taking account of diffraction, a property due to the wave nature of light. The English physicist Rayleigh suggested a measure of this, now called the Rayleigh Criterion. The angular separation in radians that is separable by a lens of diameter D is, according to this, given approximately by L/D, where L is the wavelength of the light. This corresponds to a discernible area on a unit sphere of π(L/D)² . Now the area of a unit sphere is 4π , so dividing the total area by the discernible area gives the number of objects that can be distinguished in the entire sky by a particular lens for a specific light wavelength. The wavelength of visible light is roughly 400 to 500 nanometers and the size of the eyes lens is about one centimeter. But the human visual system is not quite up to the Rayleigh Criteria and we can adjust for this by decreasing the effective size of the lens to account for this (same as increasing the distance for an object to be observable. Doing this by something like 400 gives about 20,000 visible stars that could be seen. But the "experimental result" that there is only a North Pole star says that we only observe half of them (in one out of the two experiments a star is observed). If light pollution is minimized this is about what is obtained by other methods. The other method most often suggested for counting visible stars is very similar to Morrison's procedure. Take a cardboard tube (say from a paper towel roll) and point it at a number of places in the sky as randomly as possible. Count the stars seen in each case and average the result. Divide the area of a sphere with the radius of the tube by the area of the opening and multiply by the average to get your result. There are a number of web sites that discuss this method.

Given all this you can see why an argument developed. The small number of two observations makes the statistics questionable, and this is what drew Kac's attention. On the other hand getting the right result also depended on making an estimate of the eye's acuity. Maybe the experiment is more of a way of measuring this than the number of visible stars? Still I think the problem does show a way of thinking when one has limited data. The acuity of the eye could presumably be determined by another experiment. Frankly I have not looked into this.

By the way, Morrison gave me another problem at my exam. It was easier in a way. He asked me to guess an answer to the three body problem of computing the positions of the Earth, Moon and Sun system. But I'll leave that for another day.

The Trouble With Ringworlds

Hard science fiction writers play by the rule that for the most part their stories should not violate the known laws of physics. This still gives plenty of latitude. Thus cosmic scale engineering, though not feasible at our level of technology, is allowed. One of the more gigantic cosmic constructs was proposed by the physicist Freeman Dyson. In the search for extraterrestrial intelligence, Dyson suggests we should look for large infra red sources. The rational is that an advanced civilization would make use of the entire energy output of its star. Hence it would surround the star with a shell that would intercept all the emitted radiation. Dyson's idea has been misinterpreted by most science fiction writers, including the writers of the Star Trek scripts. They literally interpret the Dyson sphere as a solid shell, with the inhabitants living on its inside. Dyson's idea is impressive enough, his thinking being that the star would eventually be surround by so many space habitats that essentially none of its radiation would be able to escape. In either case the only indication of the star's presence would be the infra red re-radiation of its energy output.


The SF writer, Larry Niven, proposed an impressive variant of the Dyson sphere, suggesting that a rotating ring be constructed around the star. In his book, "Ring World", the ring spins about the star fast enough to provide enough centrifugal force to simulate the gravity of an earth like planet. Thousand mile high walls on the edges of the ring are used to retain the atmosphere. The resulting living area is vast and provides a nice background for his characters to roam around.

But hard core SF writers have to live with their fans, and Niven's quickly pointed out that his ring world was unstable, if slightly perturbed it would move off center and the edge would collide with the central star. It takes a fair amount of mechanics and math to show this, but this was no obstacle to many of his readers.

Of course being a creative author, Niven turned his error into a plot element for the next book, "The Ringworld Engineers".

What I would like to show here is that an elementary knowledge of Newtonian principles can show why the Ringworld is unstable. All we need know is that gravitational force falls off as the inverse square of distance and that the velocity of body such as a planet in orbit about its star becomes smaller as the distance from the star increases. Thus the orbital velocity of the Earth is much higher than the orbital velocity of Mars.
The first figure shows the basic configuration of the Ringworld rotating about its star. The spin of the Ringworld also maintains the orientation by the principal of conservation of angular momentum.

In the next figure the Ringworld is displaced with the star no longer in the center. The figure actually shows both the original position and the displaced position. The arrows indicate the orbital velocity of a body at the given distance from the star. The "a" arrows show the velocity needed to maintain orbit. This is the classic situation that the acceleration of gravity is countered by the tangential velocity of the orbiting object, which is the Newtonian explanation of why the moon doesn't fall down! The upper "b" arrow is the orbital velocity needed at the displaced position, which is closer to the star. Similarly the lower "c" arrow is the orbital velocity needed at the greater distance from the star. In the upper case this means that the tangential spin velocity will be too small for maintaining orbit, that the gravitational force will move this part of the Ringworld toward the star. In the case of the lower position, now further from the star, the velocity will be too great to maintain orbit, hence this part of the Ringworld will move further away from the star. The net result is that what starts out as a small displacement will tend to increase, which is the essence of instability.

The last two figures show what happens if there is no rotation and we just consider the force of gravity. The arrows just indicate the force of gravity on the various portions of the Ringworld, taking into account the inverse square law. Clearly this is an unstable situation.
Off course this arguments are only approximate as the Ringworld is postulated to be a continuous object. If we go back to Dyson's original idea, the Ringworld would be a discontinuous cloud of space habitats and there would be no stability problem. The Dyson Spheres of Science Fiction are another matter.

Even so, as a setting for a science fiction story, Niven's Ringworld is a lot more interesting. If you want to find out how he deals with the stability problem you'll have to read the book!

Cheers,
Marty

Beyond the Second Millenium

It’s the early part of the 21st century and a group of refugees have stolen Earth’s second interstellar vessel in order to escape from persecution at the hands of the government. The ship is designed to carry a large crew on a lifetime long voyage to a nearby star, but happily one of the refugees is the famous “slip-stick Libby”, a natural genius in physics and mathematics. Libby has cobbled together a faster-than-light drive from some spare parts and the ship is now diving into the sun. The reason for this is that the drive needs energy from the sun’s gravitational field. Naturally the calculations are delicate, but Libby is a genius with mental arithmetic. the name “slipstick” refers to Libby’s trusty slide rule, which he artfully uses to make the last minute calculations!

Now jump ahead 10,000 years to the planet Dune. The Atreides Family has just taken over the planet from which the mysterious “spice” is harvested. The spice provides a drug that permits interstellar voyaging by means of some sort of mental effect. The logistics of the takeover are under the control of a so-called “Mentat”, a human being who uses specially trained mental abilities to carry out complex calculations. Computers have long since been banned as being too threatening to the supremacy of humanity.

Both these scenarios are taken from still popular science-fiction novels written in the middle of the 20th century. The first describes a scene from Robert A. Heinlein’s “Methuselah’s Children”, the second is from the writer Frank Herbert’s series that started with the book “Dune.” They are both books that I read as a student and that helped shape my expectations for the future. My slide rule, purchased in my first year of engineering school, still sits in my desk drawer, but I only dimly recall how to use all of its features. Somewhat in the style of today’s computer software the manufacturers of these instruments competed on the bases of added features, most of which were seldom if ever used! An important design point was the matching of materials in order to prevent stick-slip friction as one moved the inner rule. Hence I could imagine the special features of Libby’s slide rule, which were achieved with the advanced star ship building technology of the 21st century. The story actually appeared in the magazine “Astounding Science Fiction” as part of Heinlein’s future history series. Today the magazine has morphed into the publication known as “Analog.” The book version, published in the 1950’s was fleshed out a bit, but the slide rules were left in. Both Heinlein and Herbert were professional writers of what is called hard core science fiction, or science fiction that promotes conceivable if not currently possible science and technology. Despite this they really didn’t come close to any conception of how we would be using computers fifty years later. Software like Matlab or Maple or the current Internet had no place in their fictional futures. The science fiction author Vernor Vinge, in a story called “True Names”, written in the late seventies did have something with the flavor of the modern internet, but as he is a professor of computer science in San Diego this is no big surprise.

What does this lack of insight into the future tell us about our present conceptions of how engineers and scientists will be using computers and carrying out their professional activities in the years ahead? In 2040 will Matlab Version 27 be anything like Matlab version 8? The view of Herbert in the Dune stories is that at some point the computers become dangerous and are outlawed. His future society requires computing power, but this is obtained by genetic manipulation and training of human beings. Since we are only at the beginning of developments such as cloning and genetic manipulation we may be surprised. The physicist Roger Penrose believes that human consciousness is a product of unknown physics at the level of quantum gravity. IN one of his books on this subject, “the Emperor’s New Mind”, he “proves” this by showing how our brains can in a sense arrive at results that cannot be calculated. While I find Penrose’s ideas unlikely, if they do turn out to be true we may find that genetically produced human computers, like Herbert talks about are quite able to outperform silicon-based machines. Just think how we are able to “know” things like how to catch a ball or even walk around without any apparent computation, and the difficulties of computing such dynamic problems in real time.

As an educator my job was to train students so they could both create the technology of the future and adapt to change. The conventional response to this problem is to emphasize basic science. This has a lot of merit, but it isn’t the whole story. I spent a number of years integrating the tool of computer algebra into my mechanics course. Eventually this led to a text book. I included with the text a number of software tools for using computer algebra to solve typical mechanics problems. It was a difficult choice as it implied that as computer algebra systems evolved my text would become obsolete. Also it would not be of much use to students who didn’t have the particular computer algebra system I used. In the end I really had no choice, even though I knew that my system would be eventually be as silly as Libby’s slide rule on an interstellar space craft. Todays engineering and science students will be living in an unpredictable future and our only choice is to provide the mindset of an ever changing world. Part of this is to show them how bad the predictions of fifty years ago have proven to be.

Cheers,
Marty

Note. This piece is a slightly altered version of a short article I wrote for the February 1998 issue of the in house magazine of the Stockholm based software firm Comsol, ComsolNews. Maple and Matlab are trademarks of Maple Software and The Math Works respectively.

The Future May be Under Our Feet

On The New York Times Editorial Page: 3 July 2005 you can read a piece with the title: Fusion Power, Elusive and Alluring. The first line states, "A standing joke among scientists is that fusion power - the holy grail of those seeking a boundless supply of energy to supplant fossil fuels - is always decades away." Actually I heard the joke in a slightly more optimistic form, where the last phrase stated “...is thirty years away at any given time.” For over fifty years now some of the best minds in science have worked on the problem. Every once in a while we even hear about “cold fusion” or“table top fusion”, the latter from sono-luminescent collapsing bubbles. But maybe we are putting our bets on the wrong horse.

An obvious source of fusion power is the Sun. Almost all our fuel sources can even be traced to that great fusion reactor in the sky. Space enthusiasts have proposed putting large solar electric arrays in orbit and beaming down the gathered power. As attractive as this may seem, there are enormous practical obstacles to the idea. Still in the long run it may be feasible.

Off course as we run out of hydrocarbon based fuels we are going to get pretty desperate. I’m thinking of the plight of the Easter Islanders as discussed in Jared Diamond’s book “Collapse.” If one believes in the Hubbert Peak argument this is going to happen soon. So are there any other possibilities that might give us the fuel we need before we are at each other’s throats?

One possibility, that seems very under-exploited to me, is geothermal energy. Heat pumps to supplement other forms of energy extraction in heating our homes and work places are gaining in popularity as hydrocarbon fuels become more expensive, but there’s not enough to keep up our industrial civilization going. Then there is Iceland, a place where the whole economy depends on the readily available geothermal energy supply. Great for the Icelanders but what about the rest of us? Actually even in the US and Europe there is a significant use of geothermal energy. This is discussed in the links given below.

An interesting aspect of oil and coal is that these fuel sources also have a geothermal origin of sorts. Rotting vegetation sank into the earth and was heated by our planets molten core to form our treasure trove of hydrocarbon fuel. Notice that the core supplies heat. We now know that radioactive sources maintain the core. Fantastic, we have a giant heat engine right under our feet, so why don’t we explore it more?

It may be right under our feet, but it’s not easy to get at, not at all. At first glance you might say that all we need do is drill into the core. Unfortunately we don’t have the technology to do this. The most that has been accomplished is one to two kilometers into the surface under the seabed. The lithosphere or shell covering the hot interior is up to one hundred kilometers deep. A long drill string is just that, in the sense that it’s like trying to make a hole in a layer of jello with a piece of string. Corrosion is also a major problem in the heat of the depths. Locating and utilizing deep wells has reached depths of up to 4 kilometers. The problem is finding them or having them where you want.

The only way geothermal energy is going to really pay off is by reaching lithosphere scale depths, and that calls for something new. Maybe we should put our brainpower and resources into doing this. It’s not as scientifically interesting as developing fusion power, and with luck we might solve the problem in thirty years! Why are we sinking huge sums of money into fusion research and practically nothing into utilizing this fuel source beneath our feet? Is the problem really more difficult than building a practical fusion reactor? Maybe it’s just not as interesting to our more creative scientists and engineers. But engineers do tend to follow the money, so a few billion spent on research here might be better than supporting a fusion reactor project in France. What are your thoughts on this?

To help your thinking here are a few links to look at. The last has a great summary of where we stand now with geothermal energy. It's illustrations show just how difficult it is to reach the really hot interior.


What is the Hubbert Peak?

On Deep Sea Drilling

Geothermal Energy Association

A nice brochure about Geothermal Energy

Explaining the Obvious

Question: If you fill a drinking glass with water and then turn the glass upside down, what happens? What a dumb question that is! The water falls out of the glass, what else would you expect? Yet this is not as obvious as it seems. Here is an experiment that puts the obvious into question.

As before fill the glass of water, but now make sure you fill it to the brim, absolutely full. Now take a thin piece of cardboard and place it over the top of the glass. Holding the cardboard in place, turn the glass upside down. Now remove your hand from the cardboard. What do you think will happen?

If your too timid to try this you'll just have to take my word for the result, which is that the cardboard stays in place, and the water does not fall out of the glass. Note that the cardboard is not holding the water back as it isn't glued or fixed to the glass rim! So why does the water and the cardboard both stay put?

The answer is air pressure, which exerts enough force to hold the cardboard and the water in place. Lots of kids know this trick, after turning the glass + cardboard upside down they place it on a table and slide the cardboard out. When mom or dad pick up the glass they get a surprise.

But why does air pressure suffice to hold the water in this case but not the first? The answer to this is quite subtle and involves the notion of stability. You can in principle balance a pencil on its point, but if the center of gravity of the pencil deviates in the slightest from being directly above the pivot point, the pencil will fall. The same is true of the water surface. If the surface is flat it has minimum area, however if it even slightly deviates from the flat position the surface area increases. As the water is denser then the air, this sets the forces acting on the water to further increase the area, and the whole surface falls apart. The cardboard serves to keep the area in a stable or flat form. Another way to look at this, if you remember some basic physics, is that the total potential energy of the water-air interface is at a maximum when the surface is flat. Just think of a little bit of water getting lower while a equal volume of air gets higher. Again, as the air is less dense, the total potential energy will decrease. It's just like a ball rolling off the top of a hill.

I'll end this with a question? Why does the cardboard fall off if you place it on the top of an empty glass and turn the glass upside down?

If all this gives you a headache try the experiment with whiskey, then when you are finished you can at least have a good drink.

Cheers,
Marty

Faster than Sound and Backwards in Time: A Science Surprise

If you're a sci-fi fan you probably have heard about Tachyons, particles that move faster than light. No experiment has demonstrated the things exist, but the possibility isn't denied by the theory of Relativity. Not only doesn't the theory deny their existence, you can even use it to examine their properties. That's led to a number of sci-fi plots; because the theory says you can use Tachyons to send signals into the past. Maybe some of those super rich guys have used Tachyons to tell their younger selves to make the right investments. One of the best stories using Tachyons is by Greg Benford. In his novel "Timescape", written in the 1970's, physicists from the 1990's send warnings back about environmental disasters. Since we're still around I guess it worked.

But why should particles moving faster than light travel back in time? Well, we can get some idea of how this might occur by looking at what happens when an airplane is traveling supersonically, that is, faster than the speed of sound. To see how events in time can be seen to be reversed, we need to forget about communicating with radio and other electromagnetic waves. Instead for our situation only sound waves may be used for communication.

Following physicist Brian Greene's use of the Simpsons in his brilliant book, "The Fabric of the Cosmos: Space, Time, and the Texture of Reality", I'll borrow Bart and Lisa Simpson for this demonstration. Bart is racing faster than the speed of sound, say twice as fast or at Mach 2, toward Lisa on his jet skate board. Looking at his digital watch he counts the number of seconds it will take him to reach Lisa. In his loudest and highest pitched voice he says, one, two, three ... ten. But he's traveling faster than sound! This means that as soon as he shouts "one", he leaves the sound behind. Now he shouts two, but again he speeds past his own shout. Finally he reaches Lisa and stops, quickly asking her if she has heard him. As soon as he says this they both hear the word ten (almost, I'll get to that in a moment) followed by nine, eight ... one, that is they hear the message in reverse order. So Lisa perceives Bart's message in reverse time order. It's worse than that, just give it a little thought to see why.

Not only will the order of numbers be backwards, the sound of each number will also be reversed. Lisa will have to do a bit of work to decode the message as it will not sound much like English. So if you're expecting to get news from your older self on how to beat the stock market, you had better get to work building a Tachyon decoder!

Another Blog?

Yes, just another blog. Well as it says on the mug my wife gave me a few years ago, "Everyone has a right to my opinions." Most likely it will all be lost in the blogosphere anyway, so enjoy or ignore as you will. A short bio probably is in order to get started. I'm living in Stockholm and am a retired professor. For the last 18 years I worked at the Mechanics Department of the Royal Institute of Technology. Before that I was at such diverse places as the University of Pennsylvania and Lausanne Switzerland. My interests vary, but are generally centered on the world of technique and science.

By the way "Mechanics" is not repairing cars or the like. It is a fairly theoretical subject, mostly involving the application and development of Newton's Laws of Motion. My neighbor, who is a real mechanic, knows this. Every time he sees me working around the house or near my car he rushes out to help me. Generally I'm not too pleased at this, but what the heck, it does give him pleasure. When I first told him I was a professor in mechanics, he was impressed. Now he has learned better.

I digress, but after all this is a blog. Students love it when the prof digresses, It's a good chance to get some sleep and stop taking notes.

So what is this blog going to be about?

Now that I've stopped teaching I need a platform to vent my thoughts. My captive audience is gone, so if I want anyone to listen now I've got to make it interesting. Also a lot of things happening in today's world bother me, so this is a chance to see if others agree or not with me. Finally I can't help wanting to educate, which should really turn off potential readers.

By the way, now that I'm retired, I'm trying my hand as a novelist. So maybe I can use this to drum up some interest in my writing. I've just finished the second draft of an action adventure novel with some slight sci-fi overtones. The working title is Arctic Heat.

So if any of this interests you, or your a relative or friend, or you owe me money, let me know, or subscribe to this ego trip.


Cheers,
Marty

PS Like many scientists my spelling is somewhat creative, so if this bothers you as much as it did my 8th grade English teacher, well, too bad.