ISSN 0964-5640
And Innovative Computer Applications
Letters 3
Are Infinite Carotid-Kundalini Functions Fractal? Dr Clifford Pickover 7
Powerful Fractals Dr Jules Verschueren 11
Sunflower Spirals John Sharp 17
Fractals, Logic Functions and Bits Mikel Paternain Machin 19
Fractal Report is published by Reeves Telecommunications Laboratories,
West Towan House, Porthtowan, Truro, Cornwall TR4 8AX, United Kingdom. CompuServe 100431,3127
Volume 6 no 36 First published December 1994. ISSN 0964-5640.
Call for Papers
The next issue will be the first of our larger, quarterly issues, designed to get the best out of international airmail rates, and it will appear early in March. Overseas subscribers will receive it by airmail again. Surface mail was too erratic. Sometimes it took three weeks, other times three months. The service appears virtually useless, I suggest the Post Office make it more consistent or close it. They probably will when they become a Public Limited Company!
Although changing from bi-monthly to quarterly looks like a regressive step, at least there will be more pages per volume, therefore a greater demand for articles.
The article pile is nearly empty, so readers are requested to get their thinking caps on again. Remember that we now cover other unusual computer applications, not just fractals. (But no games, business applications etc.) Articles on disk or via e-mail tend to get in ahead of articles on paper, although camera ready paper articles are accepted.
Editorial
Image Compression
The price of the Images Incorporated compression disk has fallen to $140 at Media Magic, so I bought one. However I then needed a new hard disk as it needs masses of disk space. Having bought one I then found that my BIOS wouldn't support it, and so I then spent the next fortnight writing a software driver. The driver is actually only a few lines: the time was spent in finding out how to do it! In the end, it was abandonned as software was found on CompuServe - see next item.
The Images Incorporated program's user interface seems a bit primitive compared to modern Windows based software, but that is just a niggle. What it does is amazing!
It makes me wonder whether something similar couldn't be done with text files. If the result can be run through a grammar and spelling checker after decompression any mistakes could be sorted out. If a workable system could be developed, it would be very valuable to people into electronic mail and the information superhighway. Anyone any ideas?
Hard Drive Problems solved by
simple program:
A short assembly language shareware program is available on CompuServe's IBM Hardware+ Forum by searching the libraries for BIOS and dowloading ANYDRIVE. When run, it alters the boot files of hard disks so that if your BIOS won't accept the parameters of a new drive you have bought (up to 1024 cylinders by 56 tracks), you can override it. The program also has a remove option if you want to put things back as they were. Registration is $5, a good alternative to paying £60 for a new drive controller card or BIOS upgrade set. Incidentally if you download a program by email and then register by sending money abroad, you save VAT!!! Address: Christoph Lameter 244 N. Oakland Avenue Apt 6 Pasadena CA 91101 USA. If you're not on CompuServe and you want him to send you a disk, send $20 to include postage and registration ($15 USA), state disk size.
Communications
Your editor is now on CompuServe, no 100431,3127. The box is usually emptied at about 10pm, and replies posted the following day at the same time. Sometimes there is another access, resulting in a quicker response, but this should not be relied upon. Text files only, please, if communicating from outside CompuServe. Other files have to be encoded with UUENCODE (download it from the IBM Communications Forum in Library 2 under UU521.ZIP)
Announcements
News of Contemporaries
Amygdala is back! Issue 31-32 is dated August 1994, and owes quite a bit to e-mail. They have doubled the size of their issues and halved the frequency - undoubtedly beset by the monopolist mailing prices. The slide supplement has been withdrawn, with compensation to be announced for those who have paid in advance for subscriptions to it.
The lead article is about a method of speed up for iteration fractals known as synchronous orbits. It refers to an assembly language program for the Amiga (680XX) but no actual code or algorithms are given.
There is a large section of a printout of a FAQ (Frequently Asked Questions) file from an Internet Usenet newsgroup alt.fractals. It also mentions a fractal mailing list on BITNET. To subscribe, send a message to <listserv@givtm1.bitnet> with the following as text:
<SUBSCRIBE FRAC-L Your Name>
To unsubscribe, put UN in front of the above message (with no space). If that doesn't work send a message to the sysop.
Further printouts of e-mail included discussion on how to find the fractal equation given the image, and how to produce solid objects from fractal iterations. No practical information was given on either, probably because it doesn't exist - yet. However the discussions of the people on e-mail may well produce such practical information on the future.
More excerpts were about producing abstract images of ceramic object from the noise they make when they are tapped to make a ringing sound. Again no practical information as yet. I had often wondered whether to lash out £50 or so on an a/d adaptor with oscilloscope and spectrum analyzer software, to see if it could be used along these lines. In the Amygdala discussion, mention was made of fundamental ratios used in the design of some ceramic artifacts, and whether this shows up in any results from these sounding experiments.
Music to Image Conversion
FSL, of 49, Nethergreen Road, Sheffield, S11 7EH offer a music (as audio signals) to image conversion package for £52.88 delivered. A 486DX33 is recommended. I have no idea of how good it is - perhaps someone will buy one and send us an article about it.
Rollo Silver's Slide Set
Amygdala is offering a slide set of 20 slides by Rollo Silver for $25 post paid anywhere. Payment must be in US funds or via credit cards.
Amygdala Box 219 San Christobal NM 87564 USA. (Subscriptions $39 pa overseas, $25 US.)
Nanotechnology Used
in New Disk Drive
According to New Scientist of 29 October 1994, Matsushita is to introduce a new computer read only disk drive that can hold one terabit (1012 bytes) of data. It will use an atomic force microscope, a device similar in principle to the scanning tunnelling microscope, to read the data. Not true nanotechnology in terms of replicating assemblers, certainly, but nevertheless it may well be the first application of technology on this scale to a mass market product. However it may be some years before it actually appears on the market, say the manufacturers.
Matsushita have, however, introduced a drive that will read existing CD ROMS and also record on special disks that will sell for £7. This drive will sell for £700, and runs at four times the speed of a conventional CD ROM drive. The new drive will be called the PD ("Par Disk") and is said to be available in November 1994. It does, of course, join an already bewilderingly large and diverse market of high capacity removable media storage devices for the PC.
Chose the wrong one, and in a few years you will be unable to buy media and you will find you have to buy another drive (ie the one that wins the race) to read everyone else's.
Reciprocal Advertisement:
International Planning Concepts, a quarterly ideasletter about tax, estate and asset planning, economics, politics and society. Subscribe for 1995 by sending name and address (block letters please) with £16 (UK draft/cheque) to PO Box 107, Douglas, Isle of Man IM99 1JF, British Isles.
In fact I would have been pleased to mention this quarterly newsletter even if they hadn't offered to advertise Fractal Report free in exchange. Their advice will be invaluable for anyone wanting to plan their affairs to keep out of the clutches of probate lawyers and death tax enforcers - leaving even more money to spend on computer hardware. I have seen the first two issues and can highly recommend it.
United States readers will be particularly interested in their ideas, which give a different angle to many similar newsletters offered in the USA. They also show the way into European financial institutions that are well outside US jurisdiction. Any US reader not wanting to spend the $10 costs of getting a UK bank draft can note that I will do the conversion for approx $4: send me a check (Payable on a US bank to "J. de Rivaz") for $30 and I will forward subscription request together with a UK cheque. (Or send me $30 cash, but at your own risk!) Please make it clear what you are subscribing to.
REC 63-64
In the past I have had some readers write and say that they are bored with REC or are cancelling their subscriptions. I think maybe they ought to look at it again now as the latest 30 page issue has plenty in it to interest people who like programs to type in. To name a few: Since cosine curlicues, snowflake curve, Primitive sound and colour bar graphics, fractal fern, "exotic" graphic arts. There is still a lot about number theory and special numbers. Maybe someone enterprising can devise a program for generating images from special numbers?
Also included was a flyer for "Mathemagical Farrago XV", a compilation of many programs on a set of 1.44M disks, some shareware, some disk versions of programs already published in REC. [REC subscribers $33.95, non-subscribers $43.95, includes trial subscription.]
[REC 909 Violet Terrace Clarks Summit PA 18411 USA, $36 pa worldwide, $28 Canada, $27 USA.]
From Mr Yvan Bozzonetti:
FROM:Yvan Bozzonetti, 100413,2614
TO: John de Rivaz, 100431,3127
DATE:9/29/94 7:11 PM
Dear John,
I send you separately a letter from Dr. Mark Stuckey and a copy of the reply. You can always publish anything I write, but that may not be really the stuff Fractal Report readers look for.
Are you interested by the geometrical shapes produced by three dimensional plot of functions? That looks as sculptures, not fractal yet at the moment; I use for that MAPLEV-2 to do these pictures.
I have received Longevity Report and Fractal Report, thank you for sending them. In Fractal Report P.7, the grid after "R" and "a" would read: squared, or power two or ^2, the first upper key at left on my keyboard, under Esc give me directly 2 in exponent, it is not the case on your system I assume or on CompuServe. In the same way the dot array would read : "angular degrees" or small 0 or o in exponent. Greek letters in the Christoffel symbol have done too some unexpected tricks.
One remark on chains as money maker: I think it would be better to use it with a legal entity such as a corporation rather than a physical person.
From: stuckeym@vax.etown.edu
Organization: Elizabethtown College
Dear Yvan,
Thanks for taking the time to explain your ideas. Keep in mind that I am not a quantum field theorist, as I respond to your letter.
You write, "If a space has more than one gauge symmetry, it contains more than one parallel transport." According to my understanding of affine geometry, one may introduce any number of covariant derivatives on a manifold (not just coordinate transformations of a covariant derivative), each with its own set of geodesics. Can you briefly explain what the gauge symmetries tell us about reality, so that I may align the gauge modelling of these properties with different families of curves on a manifold? Since each covariant derivative/geodesic family has its own distinct metric, it would seem that gauge symmetry might give rise to different causal relations on a spacetime manifold. Is this true?
You asked me what I meant by T^2. By T^2 I mean the two-dimensional torus. It is geometrically equivalent to R^2, but topologically distinct. Your statement "local properties studied by differential geometry do not preclude me from drawing conclusions on a larger scale" is correct. However, differential geometry does not also preclude someone else from drawing a topologically different conclusion. See for example "Nonuniqueness of Topology" in Misner, Thorne, and Wheeler, Gravitation, page 725.
Thanks for the information on chaos and quantum mechanics. I only have in mind to model reality as a discrete set with metric, equivalence relation, and mapping. Since QM can be described with group theory, my set theoretical formulation would achieve correspondence via the addition of an appropriate binary operation.
Best regards,
Mark
Re: Copie de : Your letter September 26.
From : yvan Bozzonetti, 100413,2614
Dear Dr. Mark Stuckey,
Yes, you may be not a quantum field theorist, but I am only an amateur scientist, so I cannot explain to you gauge theories. What I can do is tell you how I understand them, keep in mind I may be wrong and if not, I can be somewhat limited in my understanding. Nevertheless, there is how I see it:
I like to think to space as a curved surface (two dimensions are simpler than four to build mental pictures). At every point on that surface there are "hairs", and surfaces (cones, spheres...) sticking to the basic surface. A particular hair kind is formed of an unit diameter loop. If I discard all other objects on the surface, there is one unit loop attached at each point. I think of that as the U(1) fibre bundle. On each loop there could be a mark to single out the 0 angle. When we go from one place, say A, to another say B on the surface there are two effects: One linked to the curved space and another to a phase shift between loops at A and B. Because a single loop can be brought in any phase position by a rotation along a single axis, I think this symmetry is ruled by the U(1) group. On a physical ground, I would say a phase shift in a loop between A and B would describe an electromagnetic interaction (absorbtion or emission of a real or virtual photon).
Similarly, if I think about sphere attached at each point of the surface, then going from A to B the shere phase shift may ask for a rotation along two perpendicular axis, so I label that symmetry with the SU(2) group. This is the symmetry of spin in three dimensions and may be too that of electro-weak interactions. A cone shaped surface would describe a space where the perimeter of a circle is less than 2.Pi times its radius, something found it seems in superstrings. Nothing of that has any effect on the geometrical properties of the basic surface and so do not produce different causal relations. Now, I am somewhat uneasy with this picture of a single basic space with many fibre bundles built on it. I would find myself far more comfortable with many "basic" spaces, each with their own causal relationship. To get a step further, I would say each point could be included not in one, but many spaces, each with its own causality rules.
If there is a place where one particular space structure dominate at a given level of precision, then, there, the Universe would look deterministic. On the contrary, if there are many space with equal footing in a region, then there the Universe would seems utterly probabilistic. To summarize I see the Universe as fully deterministic, but that determinism would apply separately in each space. So I see quantum probabilistic law as an effect of another space. If we could get to this space (to get at a place where it is dominating) it would seems perfectly deterministic, with only a small perturbative effect from the probability induced by Euclidean space. Well, I have no proof of that nor any reference, it is simply my idea, I would be happy if you could point out its flaws so I could get a better view of the reality.
I have read many years ago the MTW's Gravitation book, at least track 1 and more slightly track 2, I do not recall the page 725. I have not the book at hand now, I'll look at it next week so I'll could give a more informed comment.
You say:"... my set theoretical formulation would achieve correspondence via the addition of an appropriate binary operation." I am sorry to be so limited but I do not understand. Could you explain more about it, keeping in mind I am only an amateur with limited background. If you have written some papers on that subject, may I get them? I know supersymetry implies a binary operator called sometime Z, but I do not know if that has anything to do with our idea.
Comment:
I have only printed the first part of these letters. Are readers interested in this exchange? Please indicate when you renew.
From Mr Malcolm Lichenstein:
I enjoyed Fractal Report 34, except that I couldn't really translate all those Fractint formulas by Yvan Bozzonetti into Amiga Basic.
A short tutorial into the meaning of elements in those statements might be a great idea for a future Fractal Report.
Reply:
With regards to .frm files, they are actually quite simple. They are built up from three elements: My fractal {Initial Conditions: Iteration, Condition}. The full colon separates the initial conditions from the iterations. The comma separates the iterations for them conditionals. Multiple statements should be separated by colons. Separate sections can be placed on different lines, but the punctuation must be included. All variables are complex. This makes for some unusual operators:
Imag() replaces the real component with the imaginary component, setting the imaginary component to zero. Imag(3+4j) is therefore 4+0j, ie 4 Real() Real(3+4j) is 3 + 0j ie 3 Conj() complex conjugate abs() makes both real and imaginary positive Predefined variables:
z used for periodicity checking p1 command-line parameters 1 and 2. p2 command-line parameters 3 and 4 If you set parameters = 1,2,3,4 then p1=1+2j, and p2=3+4j pixel screen coordinates being calculated defining variables: x=(2,3) makes variable x=2+3j
Unfortunately as your Amiga (as far as I know) has no high level languages that handles complex numbers, you have quite a bit of complex algebra to convert the formulae into Cartesian form for it. de Moivre's theorem makes the algebra simpler, but then you have made your computer run slower by calling exponential and trig functions instead of ordinary multiplications etc.
As Fractint is free and written for speed, there is a strong case for buying an old a cheap PC just to run it. As the price of new Pentium machines falls, there should be a large number of 386 and 286 machines on the second hand market. See your local paper advertisements!
From Dr Clifford Pickover
For those of you who can access the Usenet newsgroups and who enjoy various recreational computing issues such as computer art, fractals, and related, you might get a kick out of the following newsgroup created by Thomas Rage at the University of Oslo in Norway. Here is a brief description from the group's charter:
"The purpose of alt.fan.cliff-pickover is to provide a place on Usenet for the mathematical and computer science recreations based on the work of Cliff Pickover. A theme running through many of Cliff's works is that computer art can be discovered as much as it can be designed, by moving though parameter spaces and collecting objects of beauty. In general, Cliff is a general proponent of the creative use of the computer and poses many thought-provoking puzzles and problems. The group discusses new developments in the area of science, art, and mathematics, including interesting organizations and articles."
From Mr Bev Mason:
I have the following item for sale, price £40, and would be pleased to hear from anyone interested:
A complete set, including issue 0, of Fractal Report.
Mr Bev Mason Highlands Bromsash Ross-on-Wye Herefordshire HR9 7PR
Tel 0989 750353
From Mr John A. Colls
I find Roger Bagula's contributions quite interesting. However in the program Rabbits, Foxes and Opossums I was unable to see the connection with population simulation. None of the variables is identified with any of the creatures mentioned, and the pattern emerging does not seem to be in any way related to their survival or distribution. I would have expected any such program to contain provision for studying changes over time or in environment.
Comment:
I have e-mailed Roger Bagula's nephew who should pass the message on. Maybe there will be a reply ... maybe not! ...
Yes there was.
But unfortunately it was all written by hand with lots of equations, determinants, matrices etc and I am not going to copy all that out and generate endless correspondence about mistakes etc.
It does, appear, though that Mr Bagula did introduce a third dimension into the original equations to represent the third species. I photocopied the letter and sent it to Mr Colls.
From Dr Hugh Daglish
I like to find things in magazines that I can try out fairly quickly. So I've had a go at a couple of programs in issue 35 of Fractal Report.
First, Fishtank by C.J. Freeman. This is a simple program, but by using the particular features of QBASIC, it is effective. However there is a slight mistake in the script. The parameter in line 13 should be v not r. as it stands, the fish are all one colour! You queried line 999. This is actually correct: QBASIC uses the B parameter to draw an empty box and BF to draw a filled box. As the colours are numbers 15 and 9, this fills a white fishtank with blue water. TurboBasic's INSTAT is replaced with IF INKEY$ ="" THEN STOP.
A bit of guesswork was needed to get Repulsion by Paul Gailiunas going. I assume force()=0 means fill the array with zeros, and POINT is PSET. I omitted CLG as I don't know what it meant.
Reply
(CLG = CLS - PG)
Clear Graphics instead of Clear Screen? -ed
From Mr C.J. Freeman
I wish to point out some printing errors in the program Fishtank as it appeared in the latest edition of Fractal Report (No. 35, page 16).
1. The line
r = 2*u*v + cy/(cx*cx+cy*cy)
should read
v = 2*u*v + cy/(cx*cx+cy*cy)
2. A line reading
y = y-p
should be inserted between the lines
FOR p = 0 to 80 STEP 40
and
GOSUB puta.
I note that you have queried the line reading
LINE (20,0)-(620,300) 8, BF.
This line is quite correct; the statements 8, BF have the effect of filling the "tank" with light grey "water", a background for the "fishes" to be seen swimming in.
From Mr John Sharp
Inspired by recent Fractal Reports, here are a couple of article and since I have not written for a time, a few comments. The first article on sunflower spirals was inspired by Patterns of Repulsion in Fractal Report 35, which I could not be sure I had running correctly. When no illustrations are provided it is not easy to be sure what to expect.
The magic squares items recently have been different in two ways. Firstly, you gave a challenge/query with no solution. Secondly, someone did provide help. My contribution (to be published next issue )is in the line of a "help - how do I do this". Maybe we should be doing more in this line to give the essence of a group, rather than just having a means to show what we have done.
I wrote to Harold McWhinnie in July and didn't get a reply. Is he a subscriber or is it something you picked up on CompuServe?
I wrote to St Martin's Press about the pre-publication offer sending my credit card as did with his book on the Mazes for the Mind expecting a reply as before. So far I have seen nothing, but it may be coming surface mail and taking its time.
I am unsure about Peter James' book Host. It is going to take a lot to make me change from conventional books. Reading computer text (like help files) is fine if you want to search and use hypertext, but lacks a lot otherwise. I remember where I am spatially and can relate the current page to something I saw recently, especially when I skim. Also, the different shapes and designs of books helps to visually make me remember them. With computers there aren't such clues yet. We have had 500 years to get books honed, so a decade or so to have got where we are is good going. It needs explorers like Peter to make the evolution happen, so I hope he does well.
I hope someone comes up with a response to the request of Stephen Leech to produce a clear, definitive "How to program Fractint formulas". When I tried earlier, with a request to input a particular equation, I couldn't understand the method proposed.
Fractals have reached the art world with respectability. I went to the Stedlijk in Amsterdam in the summer. They have always been in the forefront of collecting contemporary art which has turned out to be significant later on. There is a work called "Mandelbrot? Fragen Sie mal meine mutter" created in 1993 by George Herold who was born in Jena in 1947. It is woollen, and best described as woven on a kind of spider's web with the Mandelbrot set in the centre with an interpretation using the colour bands in which are included other fractal like objects. I have a slide, and I will see if I can get a copy on paper.
In a not too dissimilar line, The Third Dimension Society Annual Competition of stereoscopy slides was won by Martin Willsher with a new method of producing 3D image pairs using Fractint. He did not used any of the methods available in Fractint for this purpose. I hope to write something up on this and other methods in future Fractal Reports. I am hoping to start a group within The Stereoscopic Society for computer generation of 3D images. (Membership of the society is £10 a year and I can provide details for anyone who is interested). They have been doing since 1893, and have had similar problems to yours, John, in the waxing and waning of interest. There are about 750 members currently.
Keep up the good work.
Comment
I agree with these points. Contributors are asked to provide illustrations wherever possible. Also it is a good idea when submitting work to any publication to put your name and possibly address on the back of every sheet, and also to put 1 of n, 2 of n ... n of n (where there are n sheets) on the back of each as well.
From Dolores García García
I have just received "Professor Compressor" Michael Barnsley's demo disk of fractal compression. I was surprised to see that images with high compression rate (ie the ones that had lost more information) looked very much like paintings. Probably our brains store images using some kind of biological fractal compression system and that is why paintings are the way they are. Probably engineers have been wrong all the time when they tried to teach robots to identify objects using data from geometrical figures like spheres, cubes, cones, and the like, and they should start trying to make robots see scenes as a combination of fractal objects instead. After all if evolution has chosen this method it must be a good one!
I also made some experiments with the compressor files, changing data here and there. In Fractal Report 25 you mentioned that you tried feeding the digits of pi to the decompression program but it only resulted in several error messages. Well that wasn't very subtle was it? I tried changing bits one by one, and discovered a few interesting things. For example changing the first bits always gave an error message. I think that probably Iterated Systems introduced an identification code at thee beginning of the file, to check that the file is truly a compressed fractal file, and it is the size it should be. Other bits at the beginning seemed to be the colour codes: changing them changed the colours of the image. And from there on small changes didn't seem to affect the resulting images. I tried changing more and more bits at the same time, but still couldn't see any difference. Why? At first sight a highly compressed file should be more sensitive to errors than an uncompressed one. Changing ten bits in a 20K file should be 100 times more visible than changing ten bytes in a 100M file, After experimenting a bit with "Fractal Graphics", a program that draws IFS fractals, I noticed why it wasn't so. Changing slightly the parameters of an IFS fractal will change only slightly the fractal itself. So it's perfectly natural that I couldn't notice any difference.
This reminded me of the best compressor in the world: DNA. Storing all the data of a human body into the nucleus of a single cell is the highest compression rate ever achieved! And it is also quite resistant to small errors. The explanation that geneticists usually give is that there is a lot of redundant information: chromosomes go in pairs. But geneticists often think of DNA as something like a computer program, and they know that a small mistake n a computer program usually leads to a disaster. Redundancy is the only solution they can think of, But I think that the fact that similar bases of DNA make similar amino acids may be much more important than redundancy. At least, in fractal compression, where all redundancy is avoided, it works!
Finally, I tried something really bold: I deleted one bit of a sector, so that all the following bits were one place to the left. The program wasn't able to determine that the file was wrong, and didn't gave any error message, but one of the corners of the image was mostly random squares. Still, I could see a few shapes that were just right, Evidently, the program divides big images into smaller rectangles and processes each one separately. And the data of each rectangle are more than one sector long.
In Fractal Report 26 you suggested that if a decompressor is able to tell whether the input is a sensible picture and not random noise, then the compression factor is not the greatest achievable. I think it's not true. The programs that store data without compressing are the ones that don't make any differences. Compressors are the ones that look for underlying structures. And anyway, if you can compress an image, by definition, it is not random noise, even if you can't notice the difference with your eyes.
The other suggestion, that a perfectly compressed message may be indistinguishable from random noise to a receiver without the decoding algorithm, I think it's probably true. When spies agree things like: "If I scratch my nose it means danger, they're just sending and receiving a message that's indistinguishable from random noise. It would be more difficult to find a way to compress any kind of message so that a receiver without the decompressor cant realize it's a message, but I'm sure there must be one.
Comment:
Thanks for an extremely interesting letter.
When one considers how many natural objects are clearly fractals (eg cauliflower) it is not an unreasonable supposition that DNA coding is a fractal process.
Therefore methods of taking data from disks of human genome code and converting them to images as suggested in Frontiers of Scientific Visualisation (C. Pickover S. Tewksbury [eds] Wiley, ISBN 0-471-30972-9) makes a lot of sense. The book talks of lines in five dimensional space mapped to perspective drawings. These may be academically correct and very scientific, but they don't look very interesting, and I personally think that another method may eventually prove more useful.
Dr Clifford Pickover writes:
I saw a review of my book in a UK journal:
Stewart, I. (1994) Beware the Fearsome Prohaptors, My Son. New Scientist. October 22. page 46.
"We read of the pi slaves, doomed forever to the computation of successive digits of pi; death fungi with their internal labyrinths inhabited by microscopic poodles; the fearsome Prohaptors with their fractal swords, and the neon bats that trace space-filling curves in the sky."
The article reviews the book, Chaos in Wonderland: Visual Adventures in a Fractal World.
Here are some references on fractals and DNA.
1. Jeffrey, H. (1994) Fractals and Genetics in the Future. In Visions of the Future: Art, Technology, and Computing in the Next Century, 2nd Edition. C. Pickover (ed) St. Martin's Press: NY ISBN 0-312-12212-8. $16.95 Other Topics: science museums of the future, classrooms of the future, the future of computer art, fractals and genetics, and much more.
"The computer revolution is moving so fast that even science fiction finds it hard to keep up. What mind-boggling developments in technology, art, music, mathematics -- even in such games as golf and baseball -- are about to transform our culture in irreversible ways? Hold fast to your seat when you read what this exciting volume has to reveal!" - Martin Gardner
2. DNA Vectorgrams, in: Pickover, C. (1990) Computers, Pattern, Chaos and Beauty. St. Martin's Press: NY.
3. Pickover, C. (1992) DNA and Protein Tetragrams: Biological Sequences as Tetrahedral Movements. March. J. MolecGraphics. 10(1): 2-6.
4. Pickover, C. (1984) Computer-drawn faces characterizing nucleic acid sequences,in Journal of Molecular Graphics. 2: 107-110.
5. Pickover, C. (1987) DNA Vectorgrams: representation of cancer gene sequences as movements along a 2-D cellular lattice, in IBM J. Res. Dev., 31: 111-119.
6. Pickover, C. (1990) On Genes and Graphics. in Speculations in Science and Technology, 12(1): 5-15.
7. For an entire book devoted to using graphical patterns to understand DNA, see: Pickover, C. (Summer, 1995) Visualizing Biological Information. World Scientific Publishing.
(should be a good one - see Fractal Report for further information nearer the publication date. - ed)
8. Pickover, C. (1984) Frequency representations of DNA sequences: Application to a bladder cancer gene, in Journal of Molecular Graphics. 2: 50.
Also provided is the genetic sequence for a human bladder cancer gene. Source: Reddy, E. (1983) Nucleotide sequence analysis of the T24 human bladder carcinoma oncogene in Science 220, 1061.
More Editorial Comment:
Any reader interested in DNA imaging experiments wanting the above sequence on disk, please write in. Only write if you are are able to write innovative programs and are seriously likely to be interested in spending time on it. The disk is just a sequence of letters like this
GGATCCCAGCCTTTCCCCAGCCCGTAGCCCCGGGACCTCCGCGGTGGGCG
etc etc.
The task is to produce innovative patters from the data. You do not need to know anything about biology. Regard it simply as an excercise in making pretty patterns.
If your result is "pretty" your alogorithm may also help biologists and you will have played a small but significant part in the human genome mapping programme!
It really doesn't matter what the example gene is for - you can't catch the disease by playing with the information on a disk! Do not regard it is being in bad taste to play with this information - it is possible that you may help!
Here is a program I tried and its result:
screen 12 cls window (-640,-240)-(32,240) pi=4*atn(1) print pi call setup dim a(2400,2) for m=1 to 2400 for n=m to 2400 let x=x+a(n,1) let y=y+a(n,2) p=point (x,y) pset (x,y), p+1 if instat then end next n next m sub setup rem -----
shared a() print "Initialising array." let a$="" open "sequence.dna" for input as #1 for n=0 to 2400 retry: if eof(1) then exit for let b$=input$(1,1) if b$>="A" and b$<="T" then let a$=a$+b$ if b$="A" then a(n,1)=1 if b$="T" then a(n,1)=-1 if b$="C" then a(n,2)=1 if b$="G" then a(n,2)=-1 else goto retry end if next n print a$ cls end sub
Readers may be interested in the following book that has just been published
Michael Batty and Paul Longley (1994) Fractal Cities, Academic Press, London and San Diego, CA
price $49.95 (ISBN 0-12-455570-5) hardcover
From the publishers description:
Fractal Cities is a pioneering study of the development and use of fractal geometry for understanding and planning the physical form of cities, showing how this geometry enables cities to be simulated through computer graphics. It shows how cities evolve and grow in ways that at first sight appear irregular, but which, when understood in terms of fractals, illustrate an underlying order that reveals their complexity and diversity.
The book contains sixteen pages of stunning computer graphics and explanations of how to construct them, as well as new insights into the complexity of social systems. The authors provide an intelligible and gentle introduction to fractal geometry as well as an exciting visual understanding of the form of cities, thus providing one of the best introductions to fractal geometry available for non-mathematicians and social scientists.
Fractal Cities can be used as a text for courses on geographic information systems, urban geography, regional science and fractal geometry. Planners and architects will also find that there are many aspects of fractal geometry in this book relevant to their own interests. Furthermore, those involved in fractals and chaos, computer graphics and systems theory will find important methods and examples which are germane to their work.
Michael Batty is Director of the National Center for Geographic Information and Analysis in the State University of New York at Buffalo and Paul Longley is Reader in Geography at the University of Bristol