FRACTAL REPORT 20

Editorial and Announcements 2

Review: Fractals in the Fundamental and Applied Sciences Dr Ian Entwistle & John de Rivaz 7

Fractal Power Yvan Bozzonetti 5

Producing Sharp Fractals John Sharp 7

Fractal Music: the 1/f algorithm Dr Gabriel Landini 14

The Mandelbrot Explained: A Beginner's Guide Mike Curnow 20

Fractal Report is published by Reeves Telecommunications Laboratories Ltd.,

West Towan House, Porthtowan, Truro, Cornwall TR4 8AX, United Kingdom.

Volume 4 no 20 First published April 1992. ISSN 0964-5640.

Editorial

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IBM Fractal Calendar

Dr Ian Entwistle writes: I have acquired the 1992 IBM Fractal Calendar. This is just such a beautiful compilation of fractals. It is A3 size and the resolution on the Mandelbrot Sets is phenomenal. Edited by Voss, it has a whole range of fractals including scenics and lovely DLA plots. Lots of information on how to produce the images is given on the date pages. I am not sure how anybody acquires one outside of having an IBM contact, but it is surely a must for the avid fractal enthusiast. It makes Fractal Cosmos calendars look very parochial and amateur. It also makes me wonder why I am bothering to try to produce quality output!!

Weapons of Chaos - Sci-Fi Trilogy

Dr Ian Entwistle writes: This Sci-Fi trilogy by R.E. Vardeman is a splendid read for chaos and attractor fans. It shows how a little imagination can be used to weave a good yarn around chaos and attractor theory. I have had my copy since purchased in 1989 (New English Library, paperback, 590 pages). It is all done without mentioning the Mandelbrot Set! I am only sorry I didn't read it earlier. I do have a collection of unread volumes to go still. [Don't we all, -ed]

More on Poem

Jean van Mourick kindly sent us a photocopy from Compute January 1992 which detailed advances made with this image compression system. Diligent readers of this column may recall that it is a fractal image compression system that can be used to reduce large pictures to a few kilobytes of ram for storage, using iterated function systems. Jones and Bartlett Publishers, 20 Park Plaza, Boston, Massachusetts, 02116, USA, have brought out a 1.44MB floppy disk that stores 100 pages of full screen 24 bit images and text. That's packing 77MB of data onto an ordinary 3�" HD floppy. P.OEM PC floppy books can contain two minutes of compressed moving video per disk with digitised sound and ASCII text. There is talk of them supplanting CD roms. When transferred to hard disk, the access speed is far faster with P.OEM, and the makers claim that CD roms will never match it. You don't need special hardware to play it either, although to record one you need $13,000 worth of equipment. (Which I should imagine is FAR cheaper than equipment to make CD rom disks.)

It looks as though P.OEM could well prove to be a new medium for DTP applications. This is especially so where the material is likely to be discomforting to the authorities and therefore the originator is unwilling to use outside printers.

Considering that we are already talking about 2 minutes of video on an HD disk, it could also be possible to store several complete films on a single CD using this technology. [Iterated Systems 5550A, Peachtree Parkway Suite 650 Norcross Georgia 30092 USA]

Fractal Artist has Photos for Sale

Jean van Mourick, of 4, Pantllym Llandybie Ammanford Dyfed SA18 3JT produces colour and monochrome fractal pictures and offers them for sale at �5 per ten including postage. They are post card size but are not printed on the back as is a normal post card. The samples sent were impressive, and the monochrome ones made effective use of the medium.

Save Mandelbrot Data

Mr Stephen Shaw wrote to say that he hadn't seen in Fractal Report the suggestion to save Mandelbrot escape numbers for each pixel in a file, or on disk, and then use this data to generate different images with different colours etc. Of course commercial or public domain programs such as Fractint use this or a similar method, but Mr Shaw may be correct in suggesting it to others who may wish to write their own programs.

Letter from Mr Denis McMahon, (New Zealand)

As I've been fishing blue cod from the drowned glacial valleys of the Marlborough Sound I missed sending you the renewal on the palindromic date 29/1/92. It may have been laying it on a bit after you so kindly referred to the Antipodean date in issue 16.

I'd like you to know how pleased we were with the Fractint version 16.0 disk from Tim Perkins. It was a revelation to all members of our computer club. I passed on a copy to a local college where there are many students who needed just this inspiration.

Editorial Comment: I felt it was again suitable to share Mr McMahon's poetic description of his home and life with our readers: I hope that you all appreciate the fractal nature of coastlines now! Perhaps Mr McMahon's computer club may come up with some interesting Fractint formulae to share with the rest of the readership of Fractal Report in the future.

Physics Academic Software

Named for 1911 CIP Awards

Computers in Physics has announced the winners of its 1991 software contest, and among the 14 winning software packages were fractal titles from Physics Academic Software (see Fractal Report 19, page 17).

Mapper, a program on chaos and fractal demonstrations, received top honours. James B. Harrold, a University of Maryland graduate student, developed the program which Physics Academic Software expects to publish later this year after peer review. The judges were impressed with the breadth of applications in mapping fractals. "The manual is a model of clarity and discusses 22 different maps and differential equations, all of which are included in the program", Computers in Physics reported.

Chaotic Dynamics Workbench, presently available from Physics Academic Software took an honourable mention. The software provides nine non-linear systems for study, including Duffing oscillators and driven pendulums. The program also recently won a 1991 Award of Merit from the North Carolina Chapter of the Society for Technical Communication.

This is an ongoing tradition for Physics Academic Software. in 1991 Chaos Demonstrations by Julien Sprott and George Rowlands received the accolade.

Physics Academic Software is a project of the American Institute of Physics in cooperation with the American Physical Society and the American Association of Physics Teachers.

A free catalogue of all Physics Academic Software titles is available from Physics Academic Software, Campus Box 8202, North Carolina State University, Raleigh, NC 27695-8202, U. S. A. The programs cost a shade under $70 for single users, or a site license is available at three times the cost of one program per ten users.

Computerised Lava Lamps in

Algorithm 24

Algorithm, Dr A.K. Dewdney's recreational programming magazine, carries an article in issue 24 on Lava Lamps simulations. Lava Lamps are glass cylinders filled with immiscible oils in which globules move up and down with convection currents when heated by a filament lamp in the base. They were regarded as products of villainy in Lynn Faulds Wood's Watchdog television programme because if children broke open the lamps and drank the coloured liquids they would be harmed.

However Dr Pickover's computer simulation contains nothing more harmful than the contents of the average computer, and Mrs Faulds Wood hasn't complained about that yet!

There are a number of short items in this magazine in letter form, but as its title implies they don't give BASIC listing - it is left up to the readers to implement the code. Fractal Report readers would find letters on subjects such as studying the early behaviour of mapping functions during their initial phases, and simulating the chaotic behaviour of a dripping tap by computer interesting, but would need programming knowledge to implement them.

There are reviews of computer programs - in this issue Autodesk's Chaos - The Software comes under the spotlight. As Autodesk's adverts usually give a phone number, telephobes (and poor people) may like the address, printed at the bottom of the article: 2320, Marinship Way, Sausalito, CA 94965, USA.

All back numbers of Algorithm are available. Most issues have something to do with fractals amongst the articles. A subscription to the current issue costs $19.95 plus $4 extra for people outside North America. Back numbers are $5 each, with the exception of no 1.1 which is $9. Address: Algorithm, P.O. Box 29237, Westmount Postal Outlet, 785, Wonderland Road, London, Ontarion, Canada N6K 1M6. Dollar checks on U.S. banks are accepted. (Canadian funds C$24.95 including taxation.)

Fractal Cosmos Calendar 1991

Dr Ian Entwistle has some copies available for �2 to cover post and packing. Please also include an SAE which will be used to return your cheque if sold out, otherwise it will be returned unused with your calendar. The prints are worth cutting out and framing.

Send to: Dr I.D. Entwistle, 44, Woodside Gardens, Sittingbourne, Kent ME10 1SG.

Waite For New Spectaculars

Following their success with Fractal Creations, the Waite group are to publish two new books, one on image manipulation for the PC (by Fractint's Tim Wegner), and the other on virtual reality, including DIY interface plans. Both books will include PC disks. Further details will be announced in Fractal Report. Publication of the books is scheduled for June 1992.

Review: Fractals in the Fundamental and Applied Sciences

(Ed.) O. Peitgen, J.M. Henriques, L.Penedo

Elsevier Science Publishers BV, PO Box 211, 1000 AE, Amsterdam, The Netherlands. $125.50, Dfl220.00

Ian Entwistle writes:

The publication of the proceedings of scientific conferences as hard bound quality books rather than as give away paper back abstracts is frequently used to widen the readership of scientific research papers. It also represents an alternative to review books on specific topics especially those whose titles contain "Advances in ....". The above title contains the full text of papers presented at the First IFIP Conference on Fractals in the Fundemental and Applied Sciences in Portugal during 1990. Among the papers were contributions from Saupe, Prusinkiewicz, Lindenmeyer and Ushiki. The books objective is clearly to bring the advances in fractal studies and usage to a wider audiece of mathematically literate scientists. It is a high quality bound book, hence it is rather expensive ($125.00 US). The length and content of the papers is very variable. Ranging from 2 to 26 pages and from a few non-mathematical to a majority containg a great deal of formal and theoretical mathematics. A number of contributions are worthy of attention by untrained mathematicians. Even visual inspection of the more acedemic papers may be inspirational. "Out of todays research may come a new algorithm for Fractal Report!" Of particular interest was an experimental study of microbial growth patterns. By painstaking measurement the authors established a growth map (rather like a Diffusion Limited Aggregation map) which showed that the fractal dimension changed with age. Such a model algorithm turned ino a computer program would produce interesting fractal patterns. Several of the papers contain splended colour illustrations in among the mathematics. The 3-D IFS generated fractals illustrated are certainly of the type the average fractal enthusiast would wish to know how to generate. This particular chapter would be worth a read by enthusiasts of IFS. Another paper entitled A Garden of Fractals contains some very readable material on L-Systems and their combination with IFS methods to produce some attractive fractal flowers. More dissappointing is a beautiful series of colour Newton type mappings in among a paper of formal mathematics without much other text! The generation of dragon curves by L-system approach in another paper is also readily followed by lay mathematicians. Other papers have superficial attraction because of their exellent quality graphics.

This book like most costly conference proceedings more rightly belongs on a library shelf or in the bookcase of a university researcher. If you have access to a local University or Technical College library it is worth perusing if any of the comments strike an interest chord. Books such as this are ultimately available through the Inter Library Loan scheme. If you are unaware as to just how fast the science of fractals has progressed from the Mandelbrot Set then looking at this type of book will be an eye opener. If you are interested in the more scientific aspects of fractals application then there are better books available.

John de Rivaz writes:

Maybe this will be the last book I am ever sent to review. I hope not, because I will aim to honestly evaluate its value for the sort of person that Fractal Report is aimed at: the individual who is interested in getting fractal images up and running on his own computer, whether it be for his own use or in the course of his works as an educator or whatever.

Many people ask "What is the point of studying Fractals?" From its title, one would expect to find out from this book. In fact, it is a collection of papers, supplied camera ready very much like Fractal Report. However they have been reduced down to a size of 225 x 150 mm, and perfect bound between hard covers. The papers originated from a conference held at Lisbon, Portugal, between 6 and 8 June 1990.

Some of the papers are highly mathematical - well beyond my ability to comment on, at least without spending an inappropriate amount of time studying them. Few of the papers had algorithms, although many had black and white images and some colour plates.

A small proportion of Immortalists also read Fractal Report, and so I will mention the existence of paper no 2: The Statistical Cluster Dynamics in the Dendroid Transport Systems. Although not about cryonic suspension, its contents will be of particular interest to cryonics technicians. Other papers covered items of interest to fluid mechanics and physical electronics people.

A Fractal Approach to Music Composition is the title of one paper. However the information it gave left me none the wiser as to how to do it. This may be more of a criticism of me than the article, but nevertheless I was disappointed. Early on the article refereed to a Levy Flight. Although by context it is possible to get a rough idea of what this is, not knowing exactly must hinder comprehension.

I would also draw Fractal Report readers' attention to Herr Ernst G. Giessmann's article on L-systems. There are ideas to try using a L-system program, and this will draw one into reading the text properly and getting a greater insight into the subject. I would make similar comments about Daryl Hepting & Przemyslaw Prusinkiewicz's long paper on Iterated Function Systems. Many of these papers obviously had very interesting information buried within them. The problem for many Fractal Report readers will be in getting it out in simple terms. The great truths of science are every often simple, but they can easily be obscured in "erudite" material. In Techniques for Solving Inverse Problems on page 255 there is undoubtedly some useful information on how to find the attractors for images or for data, such as stock market quotations. But the paper is in no way a simple "how to do it" article. There is an amusing typo on the first page where the word "appalling" is used instead of "appealing". Lest I assume it was a typo. Possibly the author felt that the consequences of using fractals to break codes or find the most probable predictions of the future of chaotic systems would be appalling! The last article returned to this subject, but similar comment applies.

There are many "application of fractals" articles in this book, and anyone even glancing at it will never think that the study of fractals as an esoteric waste of time again. Uses are shown in texture analysis, fluid-field interactions, microbial growth, earthquake analysis, and many more.

Peter Petek described a Julia set arising from the Toda Lattice. I don't know whether I have found the right one, but it inspired the picture from Fractint reproduced to the right.

A Garden of Fractals is an article that could interest Fractal Report readers, as it gives some ideas for writing a program that produces elaborate pictures of plants in a garden setting. Colour plates and a how-to-do-it section are included, although rather more skill will be needed to implement the ideas than is assumed with Fractal Report articles.

Fractal power by Y. Bozzonetti.

Abstract

If fractals follow the same course as non-Euclidean geometries, then there may be a new powerful physical theory using them in store. I suggest a possibility here in the domain of nuclear interaction force. Here are many potential experiments to verify it. The most interesting possible application is astounding: squeezed states of the field may allow to us control chance (see Longevity Report 32, April 1992).

Readers have two possibilities: To undertake a long period of study in higher mathematics to understand the theory or skip at the conclusion and use the idea for selling fractal displays to interested people without studies. (Give them a copy of the paper, they will be happy to see they can buy something they cannot understand, money is more powerful than mind!).

In the first part of the 19th century, some doubts about the generality of Euclidean geometry brought to light what we know as curved space, or non Euclidean geometry in two dimensions.

Discovery of that domain was not a single event, at least three men worked out the problem independently: Lobachevski, J. Bolay and C. G. Gauss. Half a century after that, B. Riemann generalized the theory at an arbitrary number of dimensions. Forty years later, A. Einstein exploited Riemann's results in the General Theory of Relativity.

Fractals may be on a similar course. They have some great ancestries: Peano, G. Julia, Hausdorff... and a generalizer, B. Mandelbrot. Who will be the Einstein of that new domain? What will be its theory?

Here, I hope to give a possible hint: The General Nuclear Force Theory. Well, I am not the new field's "Einstein" and my subject is only a possibility, not a full theory. Nevertheless it may gives a taste of what is in store in fractal physics and an answer at the question: Why study fractals?

Today, Quantum Electro-Dynamics (QED), the Feynman's modern theory of electromagnetism, stands as the most accurate scientific theory. It sees all electromagnetics phenomena as a phase shift (an arrow rotation in a plane) of the wave function associated with a four dimension electromagnetic potential. Because a simple plane rotation asks only for a single axis, the symmetry of the interaction can be described by the unitary, unidimensional group, or for short: U(1).

Nuclear force at short range works in a similar way. Simply, the single axis plane rotation gives the way to three axis, four dimensional, rotation with symmetry group SU(3). In U(1) QED, there is only one charge, the electric one. In nuclear SU(3) we get three charges, nicknamed "colour". So we inherit the name: Quantum ChromoDynamics (QCD) for the theory. QED produces one massless intermediate particle: the photon, on the other hand, QCD colour charges need eight similar objects, termed gluons. Inside nuclear particles, the 8 kinds of gluon bounce between three electron-like particles: the quarks. Gluons exchange continuously colour charge between quarks so that the sum adds up always to zero or white, colourless particle.

In SU(3), a product of two elements A and B is not equal to the product of B by A:

A.B is different from B.A

By a long but well established (at least experimentally!) reasoning, it implies a finite range for the SU(3) field. That is why we cannot see it outside nuclear particles. All of that forms the modern, well established and verified theoretical frame of particle physics.

Now, enter an abstract branch of mathematics: Topology. Topology teaches us a strange fact: There we find 17 kinds of spaces with local euclidean properties. How to select from them the really euclidean space? For QED U(1), we see no problem: With the infinite range of electromagnetic interaction, the field "can see at a distance" if it is in the good space. On the other hand, QCD SU(3) with its limited range cannot do that. For it, all spaces look equally good. There is no escape: nuclear field must expand in 17-space. We need then to include in it a symmetry transforming any space into another, that is a 17-axis rotation described by group SU(17).

Near the centre of ordinary particle, all spaces are the same and SU(17) remains irrelevant. Near the border, things look not so pretty: the world "sees" 17 spaces, sadly, gluons with colour charge (photons have no electric charge) are always interacting and cannot see far away. It is not a surprise if SU(3) theory breaks down at nuclear level, where particle boundaries are interacting. Here we predict a SU(3)xSU(17) symmetry. SU(3) stands at the limit of present day computational capabilities; SU(17) looks hopelessly intractable on that ground.

In a physicist's eye, the preceding idea looks bad: We want to describe a single fundamental object, namely a particle with a non-fundamental symmetry: The group exploited is a compounded one built from two simple groups: SU(3) and SU(17). This is not a simple idea and only simple ideas produce great physics! Can we find a simple symmetry group with a number of elements (its order) giving prime factors 3 and 17? The fist possibility is a group with 3x17=51 elements. Sadly, no simple group have that dimension.

Here are some better possibilities: for example the SU(16,3) group. I call it the "Pearl sphere": It rotates along 16 axes, a sphere in 17 dimensions, one per space. Now, each "point" on the sphere surface stands as a tiny pearl with its own rotational axis. In that case we have three axis on a four dimensional pearl. Each pearl is a SU(3) theory, the result looks not simple to unravel!

Another potential solution seems to come with the super algebras D(2,1,). We can think of it as a stack of six groups unknown at the moment. That object seems to be the natural one in 17 dimensions, unfortunately, stands for a free parameter taking any value. When we shift it, we jump to another algebra. This cannot be the final answer.

S(16,3) may be broken into simpler systems because 16 is not a prime number (16 = 2 X 2 X 2 X 2), D(2,1,) displays an infinite number of possibilities... They are at best intermediates solutions at the basic question. So we need to come back to the idea of finding a good simple group.

Mathematicians know how to build an infinity of simple groups, all of them organised in a small number of sets. If one such group fits our requirements, many more will do the same in the set. What is the good one? A theory with many possibilities seems bad: If a symmetry group predicts some property, you can do an experiment to verify it. If the experimental system "says nay", you can jump at the next group and never accept that basic fact: the theory is false.

To overcome that problem, we need a symmetry group isolated in some way from others: A simple, finite, sporadic group. Outside euclidean space, there are twenty-six of them. how many have the factor 17 in them? The answer is: ONE!

That looks very interesting: we have a theory with a single possibility. Any experiment may disprove it. In most case, when a theory meets that requirement, it stands near the reality.

Now comes the explanation: Why I am not the new field Einstein. The group is the Fischer-Griess FG group, dubbed "The Monster" by J. H. Conway. It contains near 8 x 10⁵³ elements! Can we do something with that? We have touched the worst thing in the Universe, so we cannot hope to go very far with it.

I list here some personal remarks, following a hint by J.C. Conway: First, 19 other simple, sporadic groups are implied in FG building. The whole set of 20 groups fits well as the 20 essential coordinates in a four dimensional space. More: this bounded space allows a double boundary in three dimensions. Curvature in 3 d. forms six essential coordinates: exactly the number of remaining sporadic groups; (and what we need to build a super algebra as D(2,1,)!. Let me choose one of them, the Janko-1 or J1 group with 175560 elements "only".

Fractals, at least!

Now, how to pack J1 for example, on the three elements group SU(3) in the centre of a particle? The answer: A fractal dimension of content larger than ordinary space. From Mandelbrot we know its number of dimensions is:

D(J1) = log(175560)/log(3) = 10.9918...

For FG group we get in the same way:

D(FG) = log(8.10^53)/log(3) = 112.9756...

We can do the same with the remaining 24 groups.

Now, we understand what is the nuclear force: a sum of fractal spaces built on the "simple" central SU(3) field of particles. There is more than one nuclear force, a fact well known experimentally: Each fractal space gives one of them. We have six boundary groups and so, six nuclear forces at first. Now if we "displace one of the twenty groups in four dimensions, we get a new set of six force-fields. At FG level there are 20 x 6 = 120 nuclear forces!

It seems simple to play with the new system: Assume each element of FG must be included at each instant in a particle, what will be its scale? To find it, we need only to divide a proton diameter by the cube root of the number of elements in FG. The answer, near 10^-31 cm falls very near the great unification scale predicted by experiments and some theories. On the other side, current theories ask only for U(1)xSU(2)xSU(3) symmetries. Only six elements, not 8x10⁵³! There may be some surprises in store in particle theory.

One of them may come to light soon: If we ask, what is the scale of elements in the boundary group J1, we need to take the square root of 175560 = nearly 419. (We use sq. root for J1, not the cube root of the FG case, because we work here with boundary in N-1 dimensions.)

To see something 400 times smaller than a proton, we need an energy 400 times greater than the proton mass. With the proton mass near one Gev (Giga electron-volt) we need 420 Gev to see at a single J1 element. This is well in the range of capabilities of the largest particles accelerators.

What is the physical meaning of a single group element? On computer, nuclear force simulations exploit a meshy space, here we discover a physical significance for that kind of lattice: A group element defines a mesh in the lattice. To see a single mesh allow to broke it. How gluons propagate in the broken-mesh fractal space? We cannot find smooth wave functions here, all displacements become chaotic, another favourite of fractal users.

A wave in chaotic space produces a kind of pattern called caustics, seven of them may be seem in three dimensional space. The smooth J1 nuclear force gives the way to seven, or more precisely six plus one caustic forces. What single out the 7th caustic, named hyperbolic ombilic is its capacity to concentrate power indefinitely. No bag model of QED seems able to holds against it. The practical outcome is a leaky nuclear force spilling outside particle limits.

With a dimension of content near eleven, the fractal nuclear field may extends at great distance if there is plenty of energy. We need 11 time one Gev by 10^-13 cm, -a proton diameter- unit of length. ( 10000 watt-second per cm).

There is a powerful theory in particles physics domain: the super symmetric super strings model. It works well, at least on the paper, in 11 and 26 dimensions. Here, we have 26 groups each one of them attributed at a coordinate. The J1 dimension of content space stands very near 11, (D(J1)=10.99..), I cannot think of that as mere chance.

A leaky fractal J1 nuclear force will be super symmetric. Who will do the experiments? Who will develop the theory?

We can extract from it some simple order of magnitude, for example the level of nuclear power. Assume we want to know what it is in the J1 linear case: First, we are in fractal space. Now, FG looks so large at SU(3) or J1 scale than we can neglect them, so what matter is the 113 dim. space. To put it in another way, I assume nuclear fusion to take place in 1-dim fractal space. As a last simplifying assumption, I assume all the particle mass is stored in the SU(3) field, so nuclear fusion with FG will let no mass behind. Now, the mass dissipated into energy is simply given by the fraction written with the two fractal spaces. It read : 1/113 in unit of particle mass. For near 1 Gev proton rest mass this translates into 7.5 Mev, the real value for most stable nuclear structures is 8.4 Mev. This stands fairly good for such a crude estimate.

What about the same calculation for 7th J1 caustics? Here, we are not in a fractal space and we have burned up all the possibilities of a group. Twenty five remain untouched, so we have burned one part in 26 of the nuclear mass: near 36 Mev per proton. We do not know that form of nuclear fusion on Earth today. Because 26 dim. super symmetry works only with whole spin particles, if the process is indeed super symmetric invariant, it must play on paired half spin objects.

If all of that is true, we can certainly see it at work in the natural world. Astrophysics is the high energy laboratory of the poor, so we have to look at the sky.

A supernova explosion seems the right place for breaking a nuclear force, or more than one, may be up to some effects at FG level. With that group, we cannot limit our scope to ordinary euclidean space, all topological solutions are possible. Imagine the decay of nuclear force in non Euclidean space: It will radiates neutrinos and electromagnetic radiations on all frequencies and spaces at hand.

When a radio wave starts in some topological space, it rests in it, there is no way for electromagnetic radiation to slip from a space to another. Locally, all goes well, the space is Euclidean. At some distance all goes wrong, the space is no more flat. Even in empty space that produces a strong nonlinear effect. The wave "sees" on each side, a strongly charged conducting plate. In fact, curved space reflects the wave's own electromagnetic field.

In that setting, quantum fluctuations of empty space are reduced ... and the velocity of light in empty space becomes slightly greater. We have a supraluminal effect! The radiation is nonlinear and even squeezed, its distribution of quantum "heat" is distorted. The same holds true for neutrino radiations. Now I turn to facts:

In 1987 a powerful supernova lighted up in the Great Magellanic Cloud. Many neutrino detectors registered the event as a pulse of particles. The most precise machines gave the same arrival time. Unfortunately they was not working some hours before. Near four hour before the main event, some less perfected detectors were shaken by something. A spurious event or a pulse of neutrinos travelling faster than light in non Euclidean space?

At the same time, gravitational wave detectors reacted to something. Certainly not any kind of neutrinos and neither a gravitational wave: The received energy was more than one million times too great. It was surely some kind of very penetrating electromagnetic radiation ... Again at least four hours before the supernova lit up. Two simultaneous detections by two kinds of detectors without common point is too much for a chance event. May be we have here the first example of a squeezed, topological, supraluminal radio wave. Who want to build a dedicated radiotelescope with an antenna well isolated from the outside?

Light comes in packs of photons. This is a consequence of the Bose-Einstein statistics of particles with whole number of spin, such the photon. One of the simplest effect of squeezed state is to induce an "unbunching" effect on photons propagation. This translates in a photon counter to a kind of regular modulation of the photon flux. If the emitting source is demonstrated unable to produce the modulation process, then we are looking at a non linear light. In empty space, this is possible at long distance only in non euclidean space. I think I can do the experiment, so I build a telescope with photon counter to do the work. A young astronomer wrote to me, some time ago, he thinks he has seen something like this, but cannot be sure.

Finally, I shed a light on the matter side: If a nucleus, in an atom contains at least one particle leaking nuclear field, then the nuclear force will extend farther than in common matter. Normally radioactive materials may become less radioactive or even fully stable. Larger nuclear systems can be built, giving new elements.

At large distance, that nuclear field may even overcome electric force: Positive charges in the nuclear matter are screened by electron, on the contrary, nothing can screen a nuclear force spilling out. Its only limit must be another nuclear system. This introduces the strange concept of nuclear molecules: Many nuclei bounded by nuclear force and surrounded by a cloud of common electrons. Near the centre of the system, the electric field may becomes very powerful. Its potential energy may even grow beyond an electron-positron mass. In that case, a pair of particles is created from nothing in empty space. The electron remains bounded in the nuclear electric field with a negative mass (its mass is deduced from the mass of energy contained in the electric field, the total mass remains positive). On the contrary, the positron cannot stay in a positive electric field, it is expelled from the atom. At the periphery, a loosely bounded electron starts a free life, the positive charge holding it before is now cancelled by the inner, negative mass electron. Soon or later, free electrons and positrons will collide releasing high energy gamma rays. Many sources of annihilation electron-positron are known, are they powered by that system?

The strange neutron star SS433 radiates two opposed beams of matter with well defined energy giving them a velocity near 27% of light velocity. The corresponding energy, in unit of mass, stands at 3.86% of the rest mass. This is all we can hope to extract from nuclear power. And 1/0.0386 = 25.9 : The beams have an energy near or equal to 1/26 of the rest mass of matter (We have seen before that kind of nuclear force). In my opinion, FG nuclear force may extract nearly all of the rest mass of a particle in the form of free energy. Each of the 26 fields may do 1/26 of the job. What we call nuclear power is only the tip of the iceberg: the first J1 nuclear field. If the ground on the neutron star SS433 is fully converted to the second nuclear field, matter falling on it from the nearby companion star must radiate away all its first nuclear field. That J1 ombilic field may then be picked up by free atoms in the nearby space. The energy gained propels them; on the other side it carry no impulsion, so that two opposed beams are formed to let the total quantity of movement equal to zero.

Well, that is not all the story, this is true only if we can separate right and left boundaries in three dimensions. If we do not, they will cancel each other, leaving unbounded four dimensional space, a thing not possible in a quantum mechanical system. We need then to build "broken boundaries": one with 1+2 dimensions and the other with 2+1 dim. If the first uses a symmetry group, say SU(16,3) and the second a super algebra, say D(2,1,), then they cannot cancel each other. Poincarre (flat) and de Sitter (spheric) spaces may be associate with 1 and 2 d. space respectively. This is the kind of system required to separate right and left full three dimensional boundaries without mutual destruction.

Conclusion.

All of that looks very complicated, may be only because our educating system is so primitive and far away from physical realities of the world. On the other hand, may be we have here some part of a great theory. In the 18th century there was scientific books for ladies giving them a small idea of current science. The objective was only to fuel maintain conversation.

In the same way, at the start of the century, many artists were searching science for new ideas. It was the case for Pablo Picasso. After reading some vulgarizing papers on relativity and its foundation, Riemann's geometry, it started a new career: It stopped painting what it was seeing and started to produce what, in its mind, the world must look like in the new geometry. That was not for the common people, only for the superior mind. Money induces a fairly good feeling of superiority, so there was no problem to suggest in potential buyers the idea than distorted pictures on its walls was a mark of intellectual superiority.

I think the preceding theory may play the same purpose for fractal art. Here are some suggestions: The pearl sphere may be a kind of picture to develop, unfortunately, as like all computer products it may be duplicated. Its place is more as a new function in a FRACTINT XX release.

Specific "painting" must be impossible to reproduce. Fractals in caustics space open a possibility: Looking through a disturbed surface of water reproduces the caustics effect. The recipe now looks simple: put a large fractal drawing on the bottom of a bath, induces some waves and takes a picture. The power of a theory starts with its simplest applications! If it makes money, do not bother about its mathematical consistency.Shigehiro Ushiki, known for his Phoenix, is the author of a paper on a series of Julia sets, again superbly illustrated with beautiful colour plates.

I have commented on this book from the viewpoint of the type of reader that Fractal Report is aimed at: someone who wants to get images on his own computer, either for his own amusement, to help with educational projects, or artistic reasons, or for that matter just for the hell of it! I hope to also include a review from the academic viewpoint, as we do have a number of academic readers. I would have reservations in suggesting that every reader buys this book. However if you want to do a project in "popularising" some erudite exposition, then this is a good place to look for them.

REM ****************

REM * CONCHA *

REM * *

REM ****************

REM José E. Murciano

REM Apdo. 192

REM 44080 Teruel

REM Spain.

SCREEN 9:KEY OFF:CLS

PI=4*ATN(1)

FOR C1=0 TO 15:READ C2:PALETTE C1,C2:NEXT C1

FOR I=0 TO 12*PI STEP PI/120

RC=EXP(I/7.5)

XC=320+RC*COS(I):YC=175+RC*.7*SIN(I)

FOR N=I TO PI+I STEP PI/120

Q%=ABS(16*SIN(N))

R=RC/2.75

X=R*COS(N):Y=R*SIN(N)

PSET(X+XC,.7*Y+YC),Q%-1

IF INSTAT THEN END

NEXT N

NEXT I

W$=INPUT$(1)

DATA 0,17,17,3,3,35,35,51,51,23,55,55,55,62,62,63

REM data 0,18,18,38,38,44,44,45,45,5,5,17,17,25,3,7

REM data 0,33,33,,21,21,37,37,53,53,39,39,47,47,55,55,63

REM data 0,61,8,33,41,43,51,30,54,46,60,36,37,45,5,13

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