ISSN 0964-5640
And Innovative Computer Applications
The Logarithmic Spiral Dr Art Quaife 5
Pi fight, round n+2 Dr M. Ecker 7
Unusual Map Files for Fractint John Sharp 8
Ford Circles Malcolm Lichtenstein 10
Sierpinski Odd-even Cellular Automaton Malcolm Lichtenstein 11
Musical Ants Malcolm Lichtenstein 12
Fuzzy Mandelbrot Intersections Malcolm Lichtenstein 14
New Twist to the Mandelbrot Malcolm Lichtenstein 15
Kolmogorov Pre-Fuzzies Malcolm Lichtenstein 16
Bilinear Irreversibility Malcolm Lichtenstein 17
Radial Polar Transforms of Mandelbrot Iterations Roger Bagula 18
Future Basic - a bear with a byte Roger Bagula 19
Programming Environments Roger Bagula 20
Julia Effects in Mandelbrot Iterations Roger Bagula 21
Hop - Fractals in Motion Michael Peters 23
Mathematica coding Yvan Bozzonetti 25
New Book Dr Clifford Pickover 25
Synopsis of Pulsor Model for Space-Time Events David C. Manchester 26
Towards Justice Brian W. Haines 28
Relativistic Thermodynamics From Intermediate Nanomachines. Yvan Bozzonetti 31
Fractal Report is published by Reeves Telecommunications Laboratories,
West Towan House, Porthtowan, Truro, Cornwall TR4 8AX, United Kingdom.
Internet: [email protected]
Volume 8 no 41 First published March 1996. ISSN 0964-5640.
Editorial
There seem to be few "innovative applications" coming forward for Fractal Report. However in Longevity Report we have published a paper by Dr Leonid A. Gavrilov (Moscow) on statistical research on ageing taking data from Russian records. As a result of his work he (and co-authors) have been able to make observations about genes and ageing. He does not go into the computational aspects of the work, so I will not reproduce it here. The interesting thing from our point of view is that this innovative work was done with small computers as are now freely available to individuals.
If any Fractal Report readers have done any scientific project using a PC or other small computer, details of the practical computational aspects may interest our readers, who want more from their computers than just playing games or using standard applications.
The self similarity of fractals is still taking its toll, and apart from the works of Lichtenstein & Bagula we seem to have little input in the form of programs. Thanks anyway to them and the other stalwarts who continue to provide Fractal Report with articles.
Announcements:
News of contemporaries.
Roger Bagula continues with TFTN, and indeed sent me some black and white "shortened demo copies" for "distribution to friends". If anyone (UK Europe only) wants one please write in. I won't bother asking you for stamps, but please if you don't hear realise they have all gone.
In the full issue there is the now usual striking colour cover and a selection of articles including Supercalculus: A Primer, being part of the first chapter of a book he has written or is writing, which makes various claims for his "Supercalculus" ideas.
[Fractal Translight Newsletter $20/yr ($50 overseas) from R.L. Bagula 11759, Waterhill Road Lakeside CA 92949 USA]
REC is still continuing on its course publishing mathematical puzzles etc. I included one of its articles here, with permission of course! [email makes getting permission so easy these days, at least provided the originator hasn't been conned into having a "legal department". Once someone has a legal department any sensible discussion about anything is impossible, so "legal department"="no" and that's it.]
[REC 909 Violet Terrace Clarks Summit PA 18411 USA, $36 pa worldwide, $28 Canada, $27 USA.]
From Mr John Sharp:
The chapter The Fractal Golden Curlicue is Cool in Clifford Pickover's Keys to Infinity is the same as the article published in the journal Visual Computer which was based on my Curlicue article in Fractal Report. In the book he has mentioned both my name and Fractal Report.
Can you put in a note about a weekend run by the Oxford University Department of Continuing Education on Mathematics and the Visual Arts which I am helping to organise. The program includes Martin Kemp talking on The Mathematics of Growth and form in art and nature, Roger Penrose on Escher and the Visual Representation of Mathematical Ideas, Ronnie Brown on Knots in Art and Mathematics with artists working using mathematics in their art. This is on the weekend of May 18-19 1996. Details from OUDCE, 1 Wellington Square, Oxford, OX1 2JA.
Readers might also be interested in a one day meeting on June 1 on the History of Recreational Mathematics, organised by Professor David Singmaster, at the South Bank University in London. The program is not fixed yet, but he has twisted my arm to give a talk, which will be on geometry and the only other person I know a subject for is Adrian Fisher talking on mazes. This is a meeting of the British Society for the History of Mathematics and details can be found on the Internet on http://www.dcs.warwick.ac.uk/dcs/bshm. I hope to be able to give a plug for Fractal Report, even if only when I talk, and will give you more information later.
I commented in an email on the fact that you are running out of material. When you search on the Internet, you only talk about Fractal areas. I hoped you might get more recreational maths in and perhaps you might search in this area. For example, Martin Gardner describes some problems of cutting polygons into similar and congruent parts on http://www.utoronto.ca/math/007-news.html#Gardner and there is some interesting stuff if you start at http://www.geom. umn.edu/ especially in the Geometry Junkyard.
Regards
Exchanges on the Internet:
John de Rivaz wrote in a reply to this
question about the Menger Sponge:
In article: <[email protected]> [email protected] (Nils > Lohner) writes:
>I saw a picture of the mendel [sic] sponge in the Chaos book by Gleick, but he does not mention the fractal dimension of it. Anyone know what it is, or how to calculate it? Please email results if possible...
Roger Castle Smith wrote an article about what he termed the "Menger" sponge in Fractal Report 32. He included a BASIC program to display it. He said it "has a dimension between 2 and 3".
As a REM in his program, in Transera's HTB (High Tech BASIC), he said: "Each iteration decreases the volume of the sponge by a factor of 20/27 whilst the surface area increases by a factor of 4/3. In the limit the surface area will be infinite and the volume will be zero.
Since the object has no volume it must have fewer dimensions than a sphere. But its infinite area suggests that it must have more dimensions than a circle."
Dr Gabriel Landini wrote:
The self-similarity dimension of the Menger sponge should be: log(number of pieces)/log(reduction factor),
in which case D=log(20)/log(3)=2.73
cheers, Gabriel
From Y. Radai
The discussion about whether images, PAR entries, etc. are copyrightable seems to me to have ignored a very important point.
Suppose you create an image (or PAR entry) and I come along and alter one tiny ingredient in it, e.g. I slightly change the colour corresponding to one particular iteration count or I pan to the right by one pixel. Suppose then that I publish this as my own. Have I violated your copyright?
If the answer is No, then it seems to me that the entire idea of copyright of such entities becomes a farce. If, OTOH, your answer is Yes, then what if I make a larger alteration? Suppose I change all the colours, zoom out by a factor of 10, pan in some direction, and/or zoom in by a factor of 1000. Now have I violated your copyright?
If the answer to this last question is still Yes, then take into account that every Mandelbrot image can be considered a modification of Mandelbrot's original image and therefore a violation of his copyright. (We might get off on the technicality that when Mandelbrot created his images the U.S. was not yet a signatory to the Berne convention, but such technicalities are irrelevant to my point. Also, the M-set was actually discovered 3 years before Mandelbrot by Brooks and Matelski, but that doesn't change the basic problem.)
If the answer to this question is No, then we are faced with the following perplexing question: Where is the LINE OF DEMARCATION between the two situations described above? Just when is a modification significant enough that we can say that we have created a different image and therefore have not violated any copyright?
It's all well and good to say that it's up to a judge or jury to decide. But suppose you are the judge or a member of such a jury. Suppose also that your decision is going to set a precedent, and you have to state your decision and reasoning in as general a form as possible. What should the general rule be for demarcating between the two situations?
I'm interested in this more as an ethical question than as a legal one, but it's very close. Although my web site will contain a number of images created by others, I think it's fair to refer to it as "my" collection, just as someone can speak of his art collection without having drawn a single one of the pictures.
Unfortunately(?), I have this problem: I like to improve images according to my particular aesthetic sense .... For example, I might say of an image created by someone else, "Wonderful shape, but what atrocious colours!" So I replace the original colour map by another one. (BTW, I think almost everyone, with the possible exception of the original author, would agree that the picture has indeed been improved.)
Or I might say that the image is very nice but there's a filament sticking out at the edge which completely spoils the composition. Let's pan a tiny bit so as to crop off this filament.
Or I might zoom in by a factor of 2 (or 20 or 200).
Or I might zoom out to get "the big picture" and then zoom in on a different region.
Now when I say that for me the question is more ethical than legal, I mean that even if a court were to decide that I have not violated any copyright, I would still not publish it without giving the original author credit if the modification is minor. But just when is a modification "minor"? The problem is essentially one of translating a multi-dimensional continuum of possibilities into a simple binary answer: Is my modification "far enough" from the original for it to be considered as a different image: Yes or No. If the answer is No, then I should (according to my ethics, at least) give credit to the original authors (which is easy if I know their names), and probably I should ask permission to use their work, which (even if I know their e-mail addresses, which is not always the case) is quite a bother when we're talking of 30 or more authors.
I don't know if such questions bother other people anywhere near as much as they bother me. I have seen many collections in which the impression is given that the images are original with the author, yet I know that in at least some cases they're not, since I have seen *the same* image in two or more collections.
Your opinions, please ....
Y. Radai
Hebrew Univ. of Jerusalem,
Israel [email protected]
The Brighter Side of Death
by Art Quaife, Ph.D.
Eadem mutata resurgo
(Though changed, I shall arise the same)
-- from the tombstone of Jakob Bernoulli
Mathematics offers a variety of symbols that could be used in an immortalist logo. The lazy eight, , is a well-known symbol for infinity. Mathematicians have developed a very elaborate theory of infinite sets, with symbols for the various sizes of infinity. The first infinite ordinal number is (omega). If everything works right, your lifespan will have years.
The logarithmic spiral has many features that make it a nice symbol for cryonics goals. For any point (x, y) in the plane, draw the line from the origin (0, 0) to the point. Let the length of that line be r, and its angle with the x-axis be . Then (r, ) are the polar coordinates of (x, y).
Basic Properties
The logarithmic spiral is defined by the equation
log(r) = a + b,
or equivalently
r = e(a + b).
The growth rate of the spiral is a. During each revolution, the spiral grows by the same factor e2a. If a is positive, we get a left-handed spiral. If a is negative, we get a right- handed spiral, which is the mirror image of a left-hand spiral. When a is zero, we get the circle as a special case of the spiral.
If we start at any point on the spiral and trace it inward, we will go through an infinite number of turns and never reach the origin (although we come arbitrarily close). Even though we circle the origin infinitely many times, the distance we cover is finite! If instead we spiral outward, we approach infinity, although we never actually get there. Thus, the logarithmic spiral has no beginning and no end.
The logarithmic spiral is the preferred growth pattern of many natural forms, such as shells, horns, tusks, sunflowers, and even spiral galaxies. It is closely connected to the "golden rectangle", the rectangle that allegedly has the most pleasing proportions. The artist M.C. Escher embodied two systems of logarithmic spirals in his print Whirlpools.
Invariance
Cryonicists expect to undergo substantial changes in their physical self, through freezing, reanimation, and extensive reconstruction, yet fervently hope that throughout these transformations their identity survives.
The collection of logarithmic spirals (for different a and b) is a hardy beast that embodies such hopes. It remains unchanged under many transformations. Let us look at a few simple ones. If we dilate a spiral, by multiplying each distance from the origin by the same positive factor c, we get another spiral, which could also be obtained by rotating the spiral though an angle -- log(c)/a. For another example, if we perform the inversion in the unit circle that replaces each point (r, ) by (1/r, ), a logarithmic spiral is mapped into its mirror image spiral, rotated by an angle 2b.
The evolute of a logarithmic spiral, the pedal curve of a logarithmic spiral, the caustic by reflection and the caustic by refraction of a logarithmic spiral, are all again logarithmic spirals. Some of these invariance properties follow directly from the invariance of the exponential function under differentiation: d(exp(x))/dx = exp(x).
Bernoulli's Fate
Jakob Bernoulli was the first to investigate the properties of the logarithmic spiral, in the late 1600s. He was moved to say "Since this marvellous spiral, by such a singular and wonderful peculiarity . . . always produces a spiral similar to itself, indeed precisely the same spiral, however it may be involved or evolved, or reflected or refracted . . . it may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self." He called it the spira mirabilis (the marvellous spiral), and had it engraved on his tombstone.
This story has a sad ending, as do all stories that end in the grave. The mason inscribed Bernoulli's tombstone with the Archimedean spiral, which has much different properties than his beloved logarithmic spiral. Rumour has it that this error set Jacob spiralling in his grave; we have no word that he was ever resurrected to his perfect self.
Reference
Maor, E. e: The story of a number. Princeton: Princeton University Press (1994).
Postscript
I have requested that my cryocapsule be labelled with
Cryono ergo ero
(I am frozen, therefore I will be)
by Yvan Bozzonetti
Fractal displays or any computer generated pictures are commonly coded using some form of the Basic language. There is two problems: a careful reading to look at necessary translation between different dialects and a practical limitation of the results because of the poor performance of Basic.
I think a far more evolved language is in order to get good results, I suggest using Mathematica coding. mathematica is the registered trademark of Wolfram Research and the mathematical software "Mathematica" is one of the most powerful on the market. Unfortunately, it cost some �1,000! But that is not really a problem for casual users as I sow shortly: in fact there is a way to use mathematica in its full latest version for nothing.
Wolfram Research publishes a small full colour journal about Mathematica, distributed freely to registered users; More interesting, any amateur mathematician can ask for a free subscription. Many pictures here are really outstanding and there is no reason to lost this opportunity.
Now when you want to use Mathematica at no cost, you can call the Wolfram Research web site and teleoperate the software on their computer. What if you have no web link? Well, the best thing to do is to get one! if not, I can run your file on my computer or the Wolfram's one and send you the result on diskette. When you'll have seen some colour pictures, you'll be convinced, at least I think!
You can subscribe to the Mathematica journal Mathuser at:
Past issues are not lost, they can be reached at:
http://www.wri.com/MathSource.html
To use Mathematica on the web, log on:
http://www.wri.com/demo/
New book on the use of computer graphics, fractals, and musical techniques
by Clifford Pickover
Pickover, C. (1995) VISUALIZING BIOLOGICAL INFORMATION.
World Scientific: Singapore, New Jersey, London, Hong Kong. ISBN 981-02-1427-8. (For information, 1-800-227-7562 in U.S. & Canada; Fax: 44-171-836-2020 in Europe; Fax: 65-382-5919 in other countries)
From the jacket blurb:
Biological data of all kinds are proliferating at an incredible rate. If humans attempt to read such data in the form of numbers and letters, they will take in the information at a snail's pace. If the information is rendered graphically, however, human analysts can assimilate it and gain insight at a much faster rate. The emphasis of this book is on the graphic representation of information-containing sequences such as DNA and amino acid sequences in order to help the human analyst find interesting and biologically relevant patterns.
Pickover's goal is to make this voyage through molecular biology, genetics, and computer graphics as accessible to a broad audience as possible, with the inclusion of glossaries at the end of most chapters, and program outlines where applicable. The book will be of most interest to biologists and computer scientists, and the various large reference lists should be of interest to beginners and advanced students of biology, graphic art, and computer science. Contributors find pattern and meaning in the cacophony of sequence data using both computer graphics, fractals, and musical techniques.
for Space-Time
Events
by David C. Manchester (C)Copyright 1996
The purpose of this paper is to briefly sketch an outline for a new quantum theory of Space-Time based on a modified version of Roger Penrose's Twistor and Spinor work.
CONTENTS: Preface
1. The Basic Idea
2. A Simple Model
3. Image and Message
4. A Reason for Chaos.
Preface
For the last 22 years I have thought long and hard about quantum physics, various gauge theories, chromodynamics, information and communication theories. My reason for bringing forth my thoughts here are twofold. First, I am aware of my limits as a mathematician; and second, I would share these ideas with those who may be better equipped to refine them, validate or disprove them.
Please forgive any errors on my part. In the words of Valentine Michael Smith, "I am only an Egg."
1. The Basic Idea
The Basic Idea I have is this: Space-Time is a quantised phenomena. Each quantum of Space-Time serves as a channel through which Mass-Energy may flow. Each moment inflates, transpires, and collapses in the phase space described here.
The phase space for these events is a mutually orthogonal, ortho-normalised Hilbert Space. All vectors are taken to be unit-vectors.
2. A Simple Model
The model for these events I have chosen is a somewhat modified version of a Penrose Twistor. The difference from a classical Twistor is that it Pulses. I shall call it a Pulsor.
A Pulsor has 3 nested portions. (as I currently think of it)
When all 3 portions are present, we have a Space-Time event, like a quark.
When only the inner 2 portions of the Pulsor are present, Space-Time is in one of two states of flux: Inbound collapse, or Outbound inflation.
When only the inmost portion of the Pulsor is present, Space-Time is absent, and only a "shadow" trace of the immediately adjacent fully inflated Pulsor is reflected there.
A complete cycle (starting with a fully inflated Pulsor) defines a quantum of Space-Time, alternately able to be considered as a "moment", or a "Stitch in Time" if You will.
Thus, in this model a quantum event could be described as
EVENT--->Inbound Collapse--->IMAGE--->Outbound Inflation--->EVENT
...and so on.
3. Image and Message
One particularly intriguing (to me) aspect of this model is that it reduces Space and Time to the roles of Media through which Mass and Energy may flow. And it permits the tools of Information Theory to be applied to the analysis of quantum phenomena.
In this model information is conserved. For the Physicist it may offer a way to balance the trend of Space-Time toward maximum entropy against an equivalent amount of negative entropy within the inmost portion of the Pulsor, which in this model is not subject to the one-way arrow of Time, since it exists (a misnomer) beneath the outermost portions of the Pulsor, a sort of "sub-Space-Time".
4. A Reason for Chaos
About 400 years ago Leonard Euler interrelated our five most important mathematical constants in his famous equation
e(i*pi) = -1.
Normally we take this a just a convenient way to interpret our number system.
But what if there is more to it than that? Why is this the way it is? I think that Euler's equation says something quite profound about number theory, and the limits of mathematical epistemology.
I contend that these 5 constants (e, pi, 1, 0, -1, and i) behave as strange attractors in defining any possible mathematical system.
There is one other hypothesis I would make in this regard. Some observers have looked at the Mandlebrot Set and come away with the impression "Who ordered that?"
I speculate that it may well be that there is a relationship between Euler's equation and the particular shape of the Mandlebrot set, even though I must leave it at the moment to others better versed in the mathematical techniques necessary to prove it. I must therefore put my speculation in the form of a final question:
Could it be that the Mandlebrot Set is a sort of "error-correction" mechanism that uniquely ties together our Universe across the Pulsor discontinuities?
Is the Mandlebrot Set the Strange Attractor for Our particular Space-Time? If so, does it also serve to unify all possible World-lines across the entire surface of existence?
As a scientist, as a human, I don't know. I have more questions than answers. Perhaps I always will.
Afterword
I wrote this and sent it out to get feedback and constructive criticism. Thank You in Advance for any insights You want to offer.
...David C. Manchester ([email protected])
...February 9, 1996
...Niantic, CT
by Brian W. Haines
editorial note - this subject deals with applications of computers and I hope it may interest some of you.
Whatever may be the future in store for medicine and extended life there will always be a need for some method of dispute resolution. In the English tradition which includes the American system the law has traditionally taken the form of an adversarial proceeding. This has produced injustice. It cannot be proved it has produced injustice, but sufficient people are dissatisfied with the results to assume the Courts are inequitable. The reasons are clear, the best lawyers win, and not the best cases. You can buy justice, those with few resources either are barred from legal proceedings by lack of money or they suffer from the inadequacy of representation. It is a fight and the finest wins.
The argument put forward by the lawyers and politicians is this may be a poor system but it is the best we have, no one has produced a better solution to the problem of dispute resolution. But there are those who believe somehow or other there must be a way of sorting out who should pay the penalty for misbehaviour, who should pay for an accident and who should pay for the faults in the various aspects of difficulties and frictions that arise from day to day living.
Enter as they say "Solomon". "Solomon" is a computer programme. From time to time various experts contend there is a solution to all our manifest difficulties in mathematics. The world, so the theory goes, runs upon mathematical principles. There is a rhythm that can be calculated. Life itself is the result of the combinations of numerous elementary particles and so forth. What could be more logical than to believe all answers can be found in the complex set of calculations that are now possible by reason of the latest technology.
[Editorial note - since this was written it was discovered that "Solomon" was some journalist's idea of a joke. However I see no reason why it shouldn't be a reality some time, so Mr Haines' arguments are still worth considering as it could happen one day!]
If people submit to the law of a computer then it is evident the computer can solve their problem. This is as logical as the light the day. Whether this is a feasible operation depends upon two initial questions; firstly are people going to accept the solution given by the computer and secondly; has the computer sufficient material to process an adequate solution. There is a very old and truthful computer adage that is very easily over looked when computers are called in aid for any operation.- Garbage in, garbage out -. Computers are as fallible as the people who have constructed the programme and the inputs.
[Editorial note: really the law is just a computer using humans as its computational elements. The rules of evidence are a sort of computer programme.]
It is very tempting to believe there is such a thing as abstract justice: the idea that somewhere out there a form of absolute right can be found. This is wrong. Unfortunately the designers of "Solomon" the computer programme have fallen into this error. Even their own publicity material falls short of logic. It goes almost without saying that all the famous cases when fed into their computer come up with a different answer to that found by the Courts. This in itself practically condemns the programme as being inaccurate. It is logically impossible for the computer to have been programmed with all the case histories from legal literature which by definition have to be accurate findings, and then to claim the computer can find later cases to be wrong. Obviously the later cases should also have been programmed in as part of the data base as to the way the Courts function.
It is a chicken and egg situation. You cannot have law without legal cases with solutions, for that is the raw data from which all law literature derives. The sources of law are to be found in the legal decisions, if the legal decisions are thought to be wrong, then the elemental data is wrong.
It was thought twenty or thirty years ago that all law could be reduced to mathematical equations. The problem this theory encountered was the same one the computer has to face. Which law, and what is the underlying philosophy. Roman law, Communist law, English law and Palm tree law all have very different approaches to problems. The concept of murder for instance can be very different viewed from other legal systems. The idea of murder can also be different depending upon circumstances within a system. It is not good enough to claim there is a basic definition and anything outside it cannot be adjudged to be murder. There are many types of action which can only be seen to be murder after the whole action has been examined so that it is the end product as it were that must be judged. It also works the other way round as well.
Much is claimed for artificial intelligence. The worst aspect of this new discipline is the name. It is essentially a misnomer. It is not intelligence nor anything like it. Intelligence is totally a human attribute, it does not exist in machines. One of the most famous misconceived tests is that of a person interacting with a keyboard and being asked to judge whether a machine or a person is operating the responses. People are very easy to fool. It does not follow that because a number of people are unable to say whether they are interacting with a machine or not does mean the machine has intelligence. It merely means some-one has been able to anticipate a range of problems. It is the same as some one watching a magic show, being unable to see how a conjurer does a card trick does not mean the conjurer is a real magician.
So we return to computerised law. Who decides whether the input has been sufficient? Who decides upon a mal-function in the computer? How is case law up dated in the data base?
The case law from which the computer draws in the future would be from itself. This could be suspect law by reason of various input anomalies. Perhaps later information would show earlier decisions to have been wrong. The question is would it automatically alter the constructs within the computer programme for decision making?
At present a wide number of Courts have a variety of Judges and a variety of juries. This means there can be an inbuilt method in the system whereby poor results are automatically over ridden by other cases dealing with similar facts. A computer system will only have one decision making centre it cannot correct itself.
In the brave new world of extended life, long memories will look at the old Court system with nostalgia. It is like today when you go to the Bank. There are endless mistakes because the Computer is always right, the individual is always wrong. Many are looking with longing at the places where Computers do not intrude, where personal service gave instant answers.
The Computer is not a universal panacea for the manifest failings of mankind. It is not a substitute for a bad political or legal system. There is no point at all in replacing a Court system which works some injustice with a mechanical system that will work even more injustice. The common sense approach is to get the present legal systems working properly and then replace the parts that can be computerised with a Computer; nothing else makes any sense at all, it is a case of the blind leading the blind otherwise.
Of course things are bad, but to computerise all these mistakes is absolute madness.
by Brian W. Haines
Odd isn't that France where no one cares a hang for healthy living should produce a woman of 121 years old.
In Devon there are people who are touching 111 none of whom have heard of oxidising agents. Yet the U.S.A., which worships health and youth and spends a fortune on every crank theory, can hardly raise a few active centenarians.
Relativistic Thermodynamics From
Intermediate Nanomachines.
by Yvan Bozzonetti
Nanotechnology can open, it seems, a new scientific domain to the realm of technological possibilities, namely the relativistic infinite dimensional phase space of relativistic thermodynamics. That may be very interesting and allow understanding of the origin of life.
Classical ideas about nanomachines fall into two separate domains: The first is a micron level analogue of macroscopic technology. It contains gears, springs, pipes and so on. Such systems are produced as an offspring of microelectronics and use the same surface etching technology on silicon. Another approach starts from simple molecules and try to build larger systems. This is more chemical or biochemical at the start. On physical grounds, the first top down technology is firmly classical, the second, on the other side is inevitably rooted in quantum mechanics.
In physics, the classical/quantum boundary has always been a problem. How to mix together a real number deterministic space with a complex probabilistic one? Because nanosystems encompass the mixing scale, there is the possibility to explore and use that domain with them.
Now it seems there is a theoretical understanding of the subject, thanks to Arlen Anderson from Imperial College London1. The information was summarized by John Maddox in the 9 February 1995 issue of Nature2. The idea goes as follows:
On the quantum side we start with a pair of conjugate variables, say the space coordinates q and impulsion p. We must have [p,q] = pq - qp = i where i is the square root of minus one.
On the classical edge, dynamics may be formulated in a number of ways: Force = mass x acceleration in Newtonian formalism, L = kinetic energy - potential energy in Lagrange, H = kinetic + potential energy for Hamilton ... The analogue of the quantum bracket is the Poisson's formalism: Starting with two variables x and y, two functions are produced: F(x,y)and G(x,y). Using the partial derivative along x and y: Dx and Dy the Poisson's bracket writes down : {F,G} = { Dx F Dy G - Dy F Dx G } = 1. Anderson's deceptively simple idea is to put the two formulas end to end:[ A,B ]* = [A,B] + i{A,B} = i.
Any system is so partly quantum, partly classical with no conflict at the boundary where one part take over to dominate the other.
Now, p and q form a six dimensional nonrelativistic phase space. That space is the one used to count the number of allowed states of a system in the probabilistic form of entropy. The entropy itself is the Log of that number, the zero be taken at the absolute zero Kelvin temperature for a crystallised object. This is the so called third thermodynamics law. Now there is the background to go to nanosystems.
If we have some objects, for example a set of molecules moving on a small surface, their displacements are not fully constrained to the surface, because this one is small. There is then a mixing of two spaces: One limited to the surface and endowed with two dimensions, D2 and another with three dimensions, D3. The corresponding thermodynamic phase spaces have respectively 4 and 6 dimensions.
If the surface is relatively large at molecular scale, the four dimensional phase space will dominate and the thermodynamics will be mainly on the surface. Nevertheless, it remains a small component in three dimensions giving two extra coordinate in the entropy phase space. This added room will accommodate far more states so that the D2 entropy (or disorder) will see an endless sink : The added dimension of D3 displaces the D2 thermodynamics equilibria towards more order. That order-making process could have worked as a scaffolding on small clay crystal surfaces at the start of biochemical evolution. Life could well be the end product of that process.
On the nanotechnology side, that opens the possibility of producing at no energy cost some out of equilibria synthesis. This thermodynamic catalysor could be tailored with three parameters: the surface's form and its x, y lengths.
Now enter the quantum - classical mixing: The preceding results can be duplicated in the two domains without problem. There is nevertheless a new possibility: If the quantum space is seen with a four dimensional phase space and the classical domain with a six one, entropy will flows from quantum to classical realm : Anderson formalism provides the bridge between the two worlds.
For a quantum observer, entropy seems now to escape at infinity and disappear completely, the classical domain acts then as an infinite dimensional quantum phase space. Infinite dimensional phase space is the hallmark of relativistic thermodynamics3. Without going to speed near the one of light or giant gravitational fields, there is a door towards relativistic process at low energy. Now, from Anderson, there is no system only classical or quantum, so the classical phase space must have a quantum analogue. In quantum mechanics, relativity is the signpost of the second quantification. So, the first quantum space is associated with a classical domain and that domain is associated with another quantum space, the second quantification one. The infinite entropy sink so created would build ever and ever more organized structures, this is life.
At some point in the evolution, life has lost its roots in the small surface domain, from here it has lost too its entropy sink and the order capital has thrived on a continuous input of energy. When the order get erroded in some place, the result is death. Death is, in this view, the far reaching effect of the infinite dimensional phase space lost. Can we fight ageing with the help of entropy sink nanosurfaces ? The question would deserve some thought.
The next step is : What if the nanosurface is designed so that the four dimensional phase space is mainly associated with classical (Euclidean) domain ? That time, the infinite sink is the first quantum space. There is no "second Euclidean" world, so that quantum space takes simply the part of relativistic Euclidean thermodynamics. What are the prospect of ordering Euclidean space ? What technologies could follow from it? Some answer must be left to further articles.
Fine grained clays such kaolin could turn from ceramics stuff to a new high tech domain. That material is a good potential support for nanotech thermodynamic systems.
References
1 Anderson Arlen, Phys. Rev. Lett. vol. 74 p. 621-625; 1995.
2 Maddox John, Nat. Vol. 373 p. 469 1995.
3 Tolman, Richard C., Relativity Thermodynamics and Cosmology, Dover New York (1987) Repr. from Oxford Univ. Press (1934).
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