The world of newsletters is both growing in some ways and contracting in others. The Internet favours some - Longevity Report has blossomed this year because of feeds from that source. Rollo Silver has imported some fractal material into Amygdala (see below), but I found little there that can fill Fractal Report. However many people on the Internet (myself included) have reduced our reading time of paper based material.

Word processing has made it possible for many more people to produce newsletters. Roger Bagula has produced some very good colour pictures having spent very little on equipment, although he probably has a very low circulation to service. (Average colour printers produce about 3 pages/min as against 15 for a laser or even more for a photocopier.) However, the more newsletters there are, the fewer potential subscribers there are for any given one, therefore the economics are not good. Fractal Report and Longevity Report have been fortunate because of Terra Libra - several of our readers took up their LMS commission newsletter scheme, and this has produce a valuable subsidy, not just from these subscriptions but more so form the second level subscriptions that resulted.

One of the main problems is the cost of advertising, which is why I support newsletter listing schemes such as Fact Sheet Five and New Hope International. Also mentioning of contemporaries can also benefit newsletter production as a hobby. Some people have actually written in to say that this feature is the one bit of Fractal Report that they like the best!

As ever, I need to push for good quality articles. This time we have quite a lot of descriptive material. There is a problem with program listings in that we actually need someone to find out some basically new type of fractal. We have the "Take each point and iterate it" type of fractal, eg Mandelbrot, Julia sets, or "the take each point and jump to another" type, eg Martin's Mappings, IFS. By incorporating different equations into one program, mixed in different ways, we can get all sorts of different patterns, this seems to be what Roger Bagula does.

No doubt what would take the public's imagination would be a "fractal flyby" type of program where one could fly over the surface of a large fractal object. However maybe desktop computers aren't fast enough for this quite yet, but they could be by the end of the decade. The there could be a new flurry of interest. No doubt the flyby engine could be made so that people could type in formulae of their own design to make appropriate objects. Such a system would require the ability to store a lot of data and to move it about fast, or a fast enough processor to generate the pictures in real time as the "camera" is moved by the operator. Considering how long it takes VistaPro to generate images on a Pentium this may be some way off.

However a fractal 3D flyby is hardly the sort of program one can type in over a time span of a couple of hours - which is what Fractal Report readers want. More likely, it would be supplied on a CD-ROM. But there would be scope for Fractal Report to publish formulae for interesting objects - rather as we have published Fractint formulae.

Alternatively or as well someone may invent some totally different form of fractal from the iteration or transformation types mentioned above.

Whatever the future brings, I intend to carry on with Fractal Report. Maybe those that stay with me will be in for a few surprises yet, but do keep those articles coming in, even if your subject isn't one of the grand ideas presented above!

Announcements

News of Contemporaries

Roger Bagula's The Fractal Translight Newsletter has continued its monthly appearance, although it is soon to go bi-monthly as it has been losing money. Mr Bagula continues to lambast those who won't answer his letters and discuss his theories. He printed what presumably Dr Michael W. Ecker (REC) intended to be a personal letter. In it Mike described Roger as "a pathological character filled with delusions of grandeur", and described a letter Roger had written to him as "gratuitously insulting". I sent Roger another personal letter suggesting that he turned to his religion (he is a "Born-Again Christian") to see where he is going wrong in his personal relations with respect to his hobby. This got reproduced in full. I tried to be logical about what is happening, but it obviously didn't work.

Mr Bagula describes his relationship with the religious and medical establishments in an article Systematic Religious Discrimination in the United States. Clearly he is at variance with the establishment, which is interesting as he once worked for the FDA. This is the government medical enforcement agency often linked with Nazi Germany by its detractors! Unfortunately to fit in the article the print is so small and distorted as to be very difficult to read. It is shoved right next to Henon Gumowski Map but there is 4" left at the bottom of blank space.

But there is still something of interest to those just interested in fractal iterations rather than mental attitudes (some say illness). The Bagula colour printer will probably need a new blue refill soon as the front page is in blue, purple and magenta. Many fractal enthusiasts are likely to complain as these fascinating images give no clue as to their origin.

Useful Tip for Fractint on a Budget

Bagula has a go at Fractint admitting it is "an educational tool" but says it can't be used for his "astonishing" results. He does provide some formulae to try, though. He says that the save to disk graphics modes are excellent on a CGA XT machine as he can then get his Amiga to display the picture in all its glory. This is a very useful tip to those on a budget - Amigas are obsolete and cheap, so are XTs. You could probably get yourself given a CGA XT if you are lucky, and Amigas are about �150 to �250 in our local paper, and are probably a good deal less in the big cities.

Gary W. Adamason contributed an article on Polynomials, Gary Brunwell speculated on a numbering system that would be used by an alien with 12 fingers and there were the usual collection of Mr Bagula's own programs.

[Fractal Translight Newsletter $20/yr ($50 overseas) from R.L. Bagula 11759, Waterhill Road Lakeside CA 92949 USA] Send $5 if you want a sample issue.

Commercial Brain Computer Interface

Third Millennium Magazine [PO Box 1452, Surfers' Paradise 4217, Queensland, Australia] had an interesting article concerning controlling computers by brain waves. Advanced Neurotechnologies, of Colorado Springs, USA, has launched the system commercially (but no cost or address details were given.) It uses brain waves from 0.5 to 40Hz from sensors attached to the scalp. The company has produced several games to demonstrate the system and train people to generate the appropriate waves.

Amygdala

Amygdala had 391 subscribers for its September issue, no 35. Much of this issue was taken up with prints of Internet postings. I have used this myself to good effect in Longevity Report, but regretfully I have found few postings that seem to fit in with the aims of Fractal Report. However on page 3 of Amygdala there is a useful list of web pages following a report on a Fractal "Multi-User-Dungeon" in the web. There is a very useful article by Jesse Jones on fractal drawing styles. In a concise manner 10 different methods of colouring the point in an iteration type fractal are given. No mention of Roger Bagula, despite Bagula's comments on Rollo Silver who was linked with Dr Ecker and myself as partners in some plot to deny understanding of his (Bagula's) vision of the future.

REC

Dr Michael Ecker made no mention of Roger Bagula in REC 70. Instead, we got some high speed prime number generators, pan-digital numbers, a contest to suggest questions to which someone cannot truthfully reply "no", humour and computer error messages, a program to generate all polynomial fractals at once by Malcolm Lichtenstein (who also writes for Roger Bagula!!!) and symmetric circular and polynomial designs.

Fractal dreams

by Dr Clifford Pickover

The book Chaos in Wonderland just came out in paperback, and I thought some of you would like a computer recipe from the book. The book is a blend of science fiction, graphics, mathematics, astronomy, computer graphics, and fractals to introduce the reader to chaos science -- the science behind many intricate, unpredictable patterns in mathematics and nature.

In the book, status in the alien's society is based on the beauty of their fractal dreams. The following steps are required to create the swirling patterns:

x = 0.1; y = 0.1; /* starting point */

DO 10 Million Times

xnew = sin(y*b) + c*sin(x*b)

ynew = sin(x*a) + d*sin(y*a)

x = xnew; y = ynew; PlotDotAt (x, y)

END

The values of the real number constants a, b, c, and d may be chosen at random in a range (-3 < a,b < 3) and (0.5 < c, d < 1.5 ). These simple systems generate information as the system evolves. To see the patterns unfold, use the rules and starting conditions, repeat the equations over and over again, stand back, and watch the visually exciting behavior evolve on the computer screen. Each new value of x and y determines the position of points on a plane.

To produce the King's beautiful fractal dream, use

the following constants: (a = -0.966918, b = 2.879879, c = 0.765145, and d = 0.744728). The picture boundaries are (-1.86 < x < 1.86) and (-1.51 < y < 1.51) . The Lyapunov exponent, which is explained in detail in the book, characterizes the degree of chaos in the pattern. For the King's dream, the value of the Lyapunov exponent is 0.48. If you magnify the center of the pattern, you will find additional intricate plumage.

Below is more information on the book, for those interested. Perhaps your library has the book.

Pickover, C. (1995) Chaos in Wonderland: Visual Adventures in a Fractal World. St Martin's Press: New York. ISBN 0-312-12774-X

In addition to giving computer recipes, the book also includes a novella describing the adventures of two scientists exploring a world filled with fractal spiders, zinc-oxide ants, and mathematician creatures with semiconductor heads. Status in their society is determined by the beauty of their fractal dreams. The biology, history, and sociology of the civilization is also described.

Some Topics:

- A survey of board games played on fractal playing boards, with comments from some of you.

- A list of the "100 Strangest Mathematical Titles Ever Published", as suggested by readers.

- Computer programs for building compelling graphic representations of globular clusters (in astronomy)

- A list of the 15 Most Famous Transcendental Numbers

- A discussion of Apocalyptic Powers, that is, numbers of the form 2**i, which contain the digits "666".

- The usual assortment of strange fractal graphics with computer recipes.

Reviews:

"To sum it up: it's all fun with fractals."

- Los Angles Times

"Pickover skilfully introduces some of the important concepts of chaos theory (attractor, fractal, Lyapunov exponent) .... All is intriguing.... Ingenious... Extraordinary."

- Choice Magazine, 1995

"Clifford Pickover does a wonderful job of presenting the very complicated topic of fractals. The images are gorgeous."

- American Scientist

"Pickover has published nearly a book a year in which he stretches the limits of computers, art, and thought."

- Los Angeles Times, 1995

"Pickover's book does for the theory of chaos and fractals what Abbott's Flatland did for higher-dimensional geometry."

- Mathematical Reviews, 1995

"A courageous experiment in imaginative mathematical exposition. Intriguing. Radical."

- New Scientist, 1995

"Stir together a mixture of fractals, chaos, computer graphics, and science fiction. What do you get? You get a dazzling introduction to chaos science by Clifford Pickover, IBM's indefatigable computer scientist. Dr. Pickover alleges that his gorgeous swirling art, generated by the strange attractors of two simple formulas are the dreams of limbless, brainy creatures living below the icy surface of Ganymede, Jupiter's largest moon.

'Things flow about here!' exclaimed Alice, in the queer little shop behind the mirror; but never in Lewis Carroll's wildest imagination could she have dreamed of the flowing patterns in Pickover's Wonderland."

- Martin Gardner, Scientific American

"A fascinating project, a worthy addition to Flatland and The Planiverse!"

- Arthur C. Clarke

"Pickover weaves a fascinating and entertaining tale of fact and fiction in _Chaos in Wonderland_. While learning of the intricacies of the world of the Latoocarfians, we are skilfully introduced to the mathematics of chaos. His mixture of fantasy and mathematics creates an eeriness of reality that is both informative and thought provoking."

- Theoni Pappas, author of The Joy of Mathematics

"A leading expert in computer visualization, Pickover now enthrals us with his art, mathematical games, and science fiction. This latter- day Lewis Carroll introduces us to alien creatures with computer brains and mathematical souls. Their social status is based on the beauty of geometrical patterns communicated to one another with infrared beams. You will be delighted at the mathematical wonderland Pickover provides using media more suited to us humans."

- Prof. J. C. Sprott, author of Strange Attractors

Keys to Infinity

by Clifford A. Pickover

This book is dedicated to all those who do not have a book dedicated to them.

"Science is not about control.

It is about cultivating a perpetual

condition of wonder in the face

of something that forever grows

one step richer and subtler

than our latest theory about it.

It is about reverence, not mastery."

- Richard Powers, The Gold Bug Variations

Preface

"We live on a placid island of ignorance in the midst of black seas of infinity, and it is not meant that we should voyage far." - H. P. Lovecraft

"The heavens call to you, and circle about you, displaying to you their eternal splendours, and your eye gazes only to earth." - Dante

"I could be bounded in a nutshell and count myself a king of infinite space." - Hamlet

I think all of us first become interested in the concept of infinity early in childhood. Perhaps our initial fascination starts when we hear about large numbers, or outer space, or death, or eternity, or God. For example, when I was a boy, I often visited my father's library to examine his large collection of old books. The one that stimulated my early thoughts about infinity was not a mathematics book, nor a book on philosophy, nor one on religion. It was a history book published in 1921 titled The Story of Mankind. In the book, Hendrik Willem Van Loon starts with a little parable next to a sketch of a mountain:

High up in the North in the land called Svithjod, there stands a rock. It is a hundred miles high and a hundred miles wide. Once every thousand years a little bird comes to the rock to sharpen its beak. When the rock has thus been worn away, then a single day of eternity will have gone by.

This idea of eternity -- a temporal infinity -- is enough to start any child wondering about the inexhaustible fabric of numbers, space, and time.

A few years later, I wandered through my father's library and was rewarded with another book by Van Loon titled The Arts, published in 1937. I pulled the dusty book from the shelf and was delighted to find the following philosophical gem, a conversation between a student and a wise, old teacher:

"Master, will you not tell us what the highest purpose may be to which mortal man may aspire?"

A strange light now came into the eyes of Lao-Kung as he lifted himself from his seat. His trembling feet carried him across the room to the spot where stood the one picture that he loved best. It was a blade of grass, for within itself it contained the spirit of every blade of grass that had ever grown since the beginning of time.

"There," the old man said, "is my answer. I have made myself equal of the Gods, for I too have touched the hem of Eternity."

Lao-Kung, like many of the ancient philosophers and writers, considered the concept of God to be intimately intertwined with the infinite. For example, facing this page is a view of Heaven from Dante's Divine Comedy showing the numbers of angels increasing to infinity the higher one ascends. St. Augustine believed not only that God was infinite, but also that God could think infinite thoughts. According to Augustine, God "knows all numbers".

Augustine's works, along with the simple Van Loon quotes, provided a seed in an early childhood from which my interest in infinity and large numbers grew, and in particular provided an early stimulus for Keys to Infinity.

Infinite Worlds

"The trouble with integers is that we have examined only the small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions." - Ronald Graham

Prepare yourself for a strange journey as Keys to Infinity unlocks the doors of your imagination with thought-provoking mysteries, puzzles, and problems on topics ranging from huge numbers to life itself. Each chapter is a world of paradox and mystery.

For example, consider my favourite chapter Welcome to Worm World which describes the evolution of huge worms on checkerboard worlds. Readers of all ages can study their behaviour with just a pencil and paper. (Growing international interest in this topic has led to a recent short publication in Discover Magazine.) In Welcome to Worm World you'll be among the first to learn about the Internet Superhighway WormWorld Tournament where researchers around the world competed to find the longest evolving worms. How do the worms' behaviour change as Worm World grows to the size of our universe?

Consider each chapter as a launch-pad for thinking and experimenting. For example, in Ladders to Heaven you are asked to imagine what it would be like to climb an incredibly long ladder stretching from the earth to the moon. You can only use ropes and other mountain climbing gear. Impossible, you say? Read further and find out what scientists have to say about such a gargantuan task.

In The Leviathan Number, you'll learn about a monstrous number so large as to make the number of electrons, protons, and neutrons in the universe pale in comparison. (It also makes a googol -- 1 followed by 100 zeros -- look kind of small.) In this chapter, you'll learn about large numbers beyond the ability of humans to grasp or compute, apocalyptic numbers, superfactorial functions, apocalyptic powers.... What can we know about numbers too large to compute or imagine?

In Fractal Milkshakes and Infinite Archery you'll learn about a bubbly froth lurking in the fabric of our number system. The foam is comprised of an infinite regression of circles known as Ford circles.

In Slides in Hell, you'll be asked to descend immense porous slides with zany mathematical properties. Want to gamble from which hole in the slide you'll fall?

Have you ever dreamed of playing God, simulating life or preventing cancer? Then the chapter Creating Life Using The Cancer Game is for you.

Want to fly through immense grids of dots -- as big as the universe -- with startling properties? Then take a look at Grid of the Gods.

Hop aboard a flying saucer stealing humans from earth, and compute the sex of the one-billionth abductee in Alien Abduction Algebra.

Keys to Infinity is for anyone who has pondered the immensity of numbers, dreamed of daring challenges, and wondered about the infinitely small. I hope that Keys to Infinity will stimulate creative thinking, enhance computer programming skills, and suggest the usefulness of simple mathematics for solving curious, practical, or mind-shattering problems. BASIC and C source programs are included for those of you who own computers. Some of the larger programs are gathered together in an Appendix at the end of the book.

My Keys

"I looked round the trees. The thin net of reality. These trees, this sun. I was infinitely far from home. The profoundest distances are never geographical." - John Fowles

To help you on your journey, I offer various keys:

1. Essays on all of the previously mentioned topics and more, everything from vampire numbers to the loom of creation.

2. Puzzles, such as the fiendishly difficult "Cyclotron Puzzle", with hints to remind you there are often more ways of looking at the world than are immediately obvious.

3. Quotations from novelists, philosophers, and famous scientists.

4. Program Codes, so you can experiment further using personal computers as an aid to your pencil and paper explorations.

5. Fractal and other images of infinity to stimulate your imagination. (Fractals are intricately shaped objects that reveal infinite detail as they are continually magnified.)

Some of the topics in the book may appear to be curiosities, with little practical application or purpose. However, I have found all of these experiments to be useful and educational, as have the many students, educators, and scientists who have written to me during the last few years. It is also important to keep in mind that throughout history, experiments, ideas and conclusions originating in the play of the mind have found striking and unexpected practical applications. I urge you to explore all of the topics in this book with this principle in mind.

As in all my previous books, you are encouraged to pick and choose from the smorgasbord of topics. Many of the articles are brief and give you just a flavour of an application or method. Often, additional information can be found in the referenced publications. In order to encourage your involvement, computational hints and recipes for producing some of the computer-drawn figures are provided. For many of you, seeing pseudocode will clarify concepts in ways mere words cannot.

I have created all of the computer graphics images in Keys to Infinity and have provided a brief description of the colour plates towards the end. The book chapters are arranged somewhat randomly to retain the playful spirit of the book, and to give you unexpected pleasures. Some of the more technical chapters are placed at the end. Throughout the book, there are suggested exercises for future experiments and thought, and directed reading lists. Some information is repeated so that each chapter contains sufficient background information, and you may therefore skip chapters. The basic philosophy of this book is that creative thinking is learned by experimenting.

Perhaps I should say what this book is not about. It does not contain the standard number-crunching problems found in scientific texts -- most often these do not stimulate creativity, nor do they have artistic appeal. Also, the problems and topics in this book are not of a "linear" variety, where variables are fed into an equation and a succinct answer is returned. In fact many of the exercises are of the "stop-and-think" variety, and can be explored without using a computer.

The book is not intended for mathematicians looking for a formal mathematical treatise. Various books in the past have given fascinating accounts of infinity in mathematics, culture and art. For example, Eli Maors' To Infinity and Beyond and Rudy Rucker's Infinity and the Mind describe the history of number theory and various ideas connected to the concept of infinity. Their topics include: number series, prime and irrational numbers, Cantor sets, and non-Euclidean geometries. They also discuss infinity in the Kabbalist and Christian concepts of God, and also astronomers' evolving concepts of the size and structure of the universe. Two other useful books are Stan Gibilisco's Reaching for Infinity and Ray Hemmings' and Dick Tahta's Images of Infinity (see General Reading).

Since there have been so many excellent books on the subject of infinity, my current book, Keys to Infinity, attempts to provide unusual views on the way that the human mind makes sense of the world through the use of computer tools, games, puzzles, numbers, and mathematical relations. Many chapters directly touch on the concept of infinity, while others are meant to stimulate readers' minds in a more general sense regarding the unlimited extent of time, space, or quantity. I leave more direct discussions of infinity in number theory and culture to my predecessors.

Anti-omniscience

"The universe is not only stranger than we imagine, it's stranger than we can imagine." - Arthur C. Clarke

Keys to Infinity emphasizes creativity, fun, and expansion of the mind. For most chapters, no specialized knowledge is required. As I just mentioned, even though there are many chapters with mathematical ideas and computer programming hints, almost all problems are of the "stop-and- think" variety that do not require programming or sophisticated mathematics to allow you to explore and imagine.

Many of the questions I pose in the book are unanswered. Some may be unanswerable. As Stanford psychologist Roger Shepard recently noted at a Sante Fe Institute workshop on the limits of scientific knowledge, even if our computers and mathematical tools continue to improve, we may not understand the world any better. He says, "We may be headed toward a situation where knowledge is too complicated to understand." As Princeton astrophysicist Piet Hut has pointed out, the structure of the physical universe may represent the ultimate limit on human knowledge. John Horgan (Scientific American) believes that particle physicists may never be able to test theories that unify gravity and the other forces of nature because the predicted effects become apparent beyond the range of any conceivable experiment.

Finally, Ralph Gomory, the former director of research at IBM, and who is now president of the Alfred P. Sloan foundation in New York City, believes that our educational system does not place enough emphasis on what is unknown or even unknowable. To solve this problem, the Sloan Foundation may start a program on the limits of knowledge. Hopefully Keys to Infinity will stimulate students in thinking about both the unknown and unknowable.

The Electric Smorgasbord

"There was from the very beginning no need for a struggle between the finite and infinite. The peace we are so eagerly seeking has been there all the time." - D. T. Suzuki

In many chapters of Keys to Infinity I quote colleagues from around the world who have responded to my questions, and I thank them for permission to reproduce excerpts from their comments. My questions were sent through electronic mail and often posted to electronic bulletin boards. Two common sources for such information exchange were "rec.puzzles" and "sci.math" -- electronic bulletin boards (or "newsgroups") that are part of a large, worldwide network of interconnected computers called Usenet. (The computers exchange news articles with each other on a voluntary basis.)¹

Some define "Usenet" as the set of people (not computers) who exchange puzzles, tips, and news articles tagged with one or more universally-recognized labels signifying a particular "newsgroup". There are thousands of newsgroups on topics ranging from bicycles, to physics, to music. Usenet started out at Duke University around 1980 as a small network of UNIX machines. Today there is no UNIX limitation, and there are versions of the news-exchange programs that run on computers ranging from DOS PCs to mainframes. Most Usenet sites are at universities, research labs, or other academic and commercial institutions. The largest concentrations of Usenet sites outside the U.S. seem to be in Canada, Europe, Australia and Japan.²

General Reading

"Mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite" - Paul Erdoes

1. To get a feel for Usenet's popularity, consider that around 40,000 messages are sent each day to 2,000 newsgroups in Usenet. The most popular newsgroup, news.announce.newusers, has approximately 800,000 readers. It is rumoured that a newsgroup called "rec.cliff-pickover" may be started for discussions on topics and problems discussed in my books.

2. Interestingly, the Internet (a major computer network linking millions of machines around the world) is growing at a phenomenal rate. Between January 1993 and January 1994, the number of connected machines grew from 1,313,000 to 2,217,000. Over 70 countries have full Internet connectivity, and about 150 have at least some electronic mail services. As I write this, there are about 20-25 million users of the Internet. (Source: Goodman et al. (1994) The global diffusion of the Internet. Commun. ACM 37(8):27.)

References

1. Maor, E. (1991) To Infinity and Beyond. Princeton Univ. Press: New Jersey.

2. Rucker, R. (1983) Infinity and the Mind. Bantam: New York.

3. Hemmings, R. and Tahta, D. (1984) Images of Infinity. Leapfrogs Insight Series: Vermont.

4. Gibilisco, S. (1990) Reaching for Infinity. TAB Books: Pennsylvania.

5. Gamow, G. (1988) One, Two, Three... Infinity. Dover: New York

6. Horgan, J. (1994) Anti-omniscience. Scientific American. August, 271(2): 20-22.

7. Kasner, E., and Newman, J. (1989) Mathematics and the Imagination. Tempus: Redmond, Washington.

8. Schroeder, M. (1991) Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Freeman: New York.

Contents of Keys to Infinity

1. Too Many Threes

2. Ladders to Heaven

3. Infinity Machines

4. Infinity World

5. Grid of the Gods

6. To the Valley of the Seahorses

7. The Million-Dollar, Trillion-Digit Pi Sequencing Initiative

8. Infinite Chess

9. The Loom of Creation

10. Slides to Hell

11. Alien Abduction Algebra

12. The Leviathan Number

13. Welcome to Worm World

14. Fractal Milkshakes and Infinite Archery

15. Creating Life using the Cancer Game

16. No Zeros Allowed

17. Infinite Star Chambers

18. Infinitely Exploding Circles

19. The Infinity Worms of Callisto

20. The Undulation of the Monks

21. The Fractal Golden Curlicue is Cool

22. The Loneliness of the Factorions

23. Escape from Fractalia

24. Are Infinite Carotid-Kundalini Functions Fractal?

25. The Crying of Fractal Batrachion 1,489

26. Ramanujan, Infinity, and The Majesty of the Quattuordecillion

27. Recursive Worlds

28. Chaos in Ontario

29. Cyclotron Puzzles

30. Vampire Numbers

31. Computers, Randomness, Mind, and Infinity

Appendix of Longer Programs, Description of Colour Plates, Acknowledgements, About the Author

Fractal Resources on the Internet

FTP's

Fractint 19.2

ftp://ftp.internexus.net/pub/Graphics/Fractals/frain192.zip

ftp://oak.oakland.edu/SimTel/msdos/graphics/frain192.zip

PostMan

dutepp0.et.tudelft.nl /pub/import/dutecai/PUBLIC/local/postman

WWWs

Site on 3D/4D Mandelbrot set

http://www.dtek.chalmers.se/Datorsys/Project/qjulia/index.html

Terry's Web page with new fractal type + C source code

http://www.teleport.com/~neighbor/semiints.shtml

Quaterion Web info

http://www.krs.hia.no/~fgill/quatern.html

http://www.dtek.chalmers.se/Datorsys/Project/qjulia/index.html

http://speedracer.nmsu.edu/~jholder/fractals/keen.html

http://ccrma-www.stanford.edu/~stilti/images/chaotic_attractors/nav.html

http://www.omnigroup.com/People/bungi/julia.html

Site on Landslide graphic competition

http://www.LANDSLIDE.COM/houdini

Textures, icons, etc.

http://www.itw.com/~imagesys

General fractal generating programs

http://www.acs.oakland.edu/oak/SimTel/win3/graphics.html

http://spanky.triumf.ca/www/fractint/fractint.html

http://einstein.et.tudelft.nl/~reinoud/

Interactive IFS application

http://www.cosy.sbg.ac.at/rec/ifs

Calculus Web site

http://www.ozemail.com.au/~dihoyer

Fractal images/art

http://www.swcp.com/artvark/index.html

http://www.primenet.com/~lkmitch/

Neural Nets versus

Digital Computers

by Dr Art Quaife (Trans Time Inc)

In a Cryonet message, Joseph Strout quotes a FAQ that states:

"In principle, Neural Nets can compute any computable function, i.e., they can do everything a normal digital computer can do."

I believe that the first half of this statement is FALSE, while the second half is TRUE. The two statements are NOT equivalent. To support my claim, I will give more background than I gave in a previous post on this subject.

Recursive functions: Abstract models of computation.

In the 1930's, the two most important developments of this century in mathematical logic occurred. They were:

1. Proof of the justly celebrated Goedel incompleteness theorems, and the related limitative results of Church, Tarski, and Turing.

2. Finding an adequate definition of the informal notion of "algorithm". At about the same time (ca. 1936), the formal definitions of Church's lambda-definable functions, Post's notion of 1-definability, Herbrand-Goedel-Kleene systems of equations, and Turing machines were developed. These notions were all soon proved to be equivalent; the Church-Turing thesis is that they each capture the intuitive notion of "effectively computable function."

Further, each of these definitions led to the description of a *universal* model -- a model that could compute every function that any instance of the definition could compute.

Since that time, many other formulations of the notion of "computable function" have been offered; all of them have turned out to be provably equivalent to those listed above. Note that these are all ABSTRACT models of computation. The question of whether any of these models could be implemented on a physical device is quite a different matter. In fact, the first digital computer was not produced for another decade.

Of the computation models listed above, the Turing machine is closest to describing an actual physical computer. But an essential feature of a Turing machine is that it has an INFINITELY long tape. The tape is used both for input and for scratch memory. If the machine T is designed to compute the particular recursive function f, the amount of tape required to compute a particular value f(n) is finite, but as n varies, in general there is no upper bound on the amount of tape required -- hence the requirement of an infinite tape.

Definitions

A function that maps words over an alphabet into words over an alphabet is *partial computable* (partial recursive) iff it is partial Turing-computable (equivalently, lambda-definable, or any of the other provably equivalent definitions.) "Partial" means that the function may be undefined for some inputs.

A model computer is *computationally universal* iff it will compute every partial computable function.

Neural nets

In 1943, McCulloch and Pitts¹ offered nerve nets as a formal model of the nervous system. Later researchers developed a number of different formulations of essentially the same model.

Researchers also focused on formulating these machines as "acceptors", wherein a particular machine (or net) would signal either "yes" or "no" to finite input tapes over an alphabet, thus determining a set of tapes that the machine accepted.

Kleene² showed that the set of input tapes accepted by his finite automata, and also the McCulloch and Pitts nerve nets, is the set of REGULAR sets of tapes, a set which he defined. Later Medvedev³ extended this equivalence: Automata defined by semigroups, and Turing machines with FINITE tapes, also accept exactly the class of regular sets of tapes.

It is easy to specify McCulloch and Pitts neurons that act as the logic gates AND, OR, and NOT. So we can use neural nets to design any logic circuit, presumably including a Cray computer. This confirms the claim that neural nets can do anything that digital computers can do.

But finite automata are considerably weaker in computational power than Turing machines with an infinite tape. Regular sets have fairly simple structures. In particular, the complement of a regular set is regular, whereas of course the complement of a recursively enumerable set need not be recursively enumerable. The set of tapes {01, 0011, 000111, . . . } (n 0's followed by n 1's) is NOT a regular set over the alphabet {0, 1}, and hence there is no finite automaton that accepts only its members. But it is easy to design a Turing machine that accepts it.

Let a *language* be a set of words over a given alphabet. In order of increasing power of the machines, we have the Chomsky hierarchy:

Computing Model Language Class Accepted

Finite automata Regular languages

Pushdown automata Context-free languages

Linear bounded automata Context-sensitive languages

Turing machines Recursively enumerable languages

Physical computers

Real world digital computers, such as PCs or Crays, are often thought of as being able to compute all recursive functions. But of course that is not true. Every machine that has ever been built has only FINITE memory. That means that for any given piece of hardware, there are recursive functions f for which the shortest program to compute f is too large to even load into the hardware's memory. There are other recursive functions g whose program will load into memory, but for some n will run out of memory in trying to compute g(n).

"Potentially universal" physical computers

Think of your desktop computer as consisting of a central processing unit (CPU) together with a finite amount of random access memory (RAM). Imagine that you live next to a RAM factory. When the machine runs out of memory in a particular computation, it signals "add on another memory cell", which is a register large enough to hold any symbol in the alphabet. Then such an extensible machine is "potentially universal".

But suppose a particular computation requires a GOOGOLPLEX of memory cells to complete it. This number of cells is VASTLY more than the number of atoms in the known universe, and clearly as a matter of physics, our RAM factory is long ago "out of stock".

No physical computer can compute all computable functions!

Thus an actual hardware machine, loaded with an appropriate program, is closer to realizing a finite automaton than a Turing machine. Indeed the Medvedev result cited above shows that the hardware machine is equivalent to a Turing machine with some FINITE tape.

There are many other more recent textbooks such as ^4,5 that cover most of the above ideas. See also the breezier^{6, pg. 164}, where Dewdney (who used to write the Computer Recreations column in Scientific American) states: "Moreover, as we shall show now, neural nets are no more powerful than finite automata, the humblest inhabitants of the Chomsky hierarchy."

Conclusion

Neural nets are equivalent in computational power to finite automatons. You can do anything with neural nets that you can do with conventional von Neumann digital computers. But neural nets are not computationally universal.

I wish to thank Prof. Marvin Minsky for several very helpful comments on an earlier draft of this article.

References

[1] McCulloch, W.S. and Pitts, W.A. Logical Calculus of the Ideas Immanent in Nervous Activity, Bulletin of Mathematical BioPhysics 5: 115-133 (1943).

[2] Kleene, S.C. Representation of Events in Nerve Nets and Finite Automata. Automata Studies, C.E. Shannon and J. McCarthy editors, Princeton University Press (1956).

[3] Medvedev, Y.T. On the Class of Events Representable in a Finite Automaton. Reprinted in Sequential Machines, E.F. Moore editor, Addison-Wesley (1964).

[4] Davis, M.D., and Weyuker, E.J. Computability, Complexity, and Languages. Academic Press (1983).

[5] Lewis, H.R., and Papadimitriou, C.H. Elements of the Theory of Computation. Prentice-Hall (1981).

[6] Dewdney, A.K. The Turing Omnibus: 61 Excursions in Computer Science. Computer Science Press (1989).

Future Health

Clifford A. Pickover

Pickover, C. (1995) FUTURE HEALTH, Computers and Medicine in the Twenty-First Century. St. Martin's Press: New York.

ISBN 0-312-12602-6.

This collection considers the tremendous effects that computers will have on medicine and medical service in the next century. The book also gives a sampling of state-of-the-art application of computers in medicine. The chapters describe:

o futuristic operating rooms

o the challenges of future medical schools in preparing 21st-century physicians

o futuristic fractal models in pathology

o the use of new medical imaging technologies

o the use of electronic gophers to obtain medical information

o digital dentistry

o the use of artificial intelligence in medical diagnosis

o computer conferencing for medical consulting

o bloodless robotic surgery

o making solid models from medical images

o futuristic examinations rooms and much more....

PREFACE

"I tied a cord to the upper part of the spine, where it is firm and less flexible and, pulling it straight to the ceiling, fastened the end of it to a hook in the wall."

- Bernard Albinus (1696-1770)

My interest in human anatomy began in early childhood. I remember going into my father's study and gazing at the anatomical works of Bernard Siegfried Albinus, the greatest descriptive anatomist of the eighteenth century. In 1725, after Albinus found a fresh skeleton of a fully grown male "with all the tendons, ligaments, and cartilage attached", he became determined to make careful drawings of the body and skeleton for use by both artists and anatomists. He preserved the soft parts by soaking them in vinegar. One of his first drawings is shown facing this page in the book.

What would have Albinus thought of today's medical images? With the aid of computers, new anatomical maps have emerged just in the past few decades which render his noble efforts obsolete. Since their rapid growth following the Second World War, computers have changed the way we perform scientific research, conduct business, create art, and spend our leisure time. They're also playing increasingly important roles in medicine. For much of the twentieth century, the X-ray had been the preferred imaging tool allowing doctors to probe the mysteries of the human body and the disease process. New technologies, however, began to enter clinical use in the 1970's and 80's, allowing physicians to visualize the interior of the human body with unprecedented clarity. These new, exotic-sounding technologies included: computed tomography (CT), magnetic resonance imaging (MRI), positron emission tomography (PET), ultrasound, video thermography, superconducting quantum interference devices (SQUIDs), and digital subtraction angiography (DSA).

Let's first consider the well-known diagnostic images provided by CT scans and MRI images. The first CT scanner was installed in a Wimbledon, England hospital in 1971. The scanners have since improved in various ways, and today these marvellous machines can scan the body at a resolution of 1.5 millimetres. To produce these high-resolution views, computers determine the density of each point by processing information from all the X-rays passing though the point. Shades of grey or colour may be assigned to the density values, thus creating a high-resolution image of a slice of the human body. Computers may also subsequently process the 2-D slice information to build 3-D models of the human anatomy. Using today's technology, it's possible to build virtual reality systems allowing students and physicians to use computers that enable them to "walk" around and study these 3-D models.

Today, magnetic resonance imaging machines are also very popular diagnostic devices. The very first animal to be studied by an MRI machine, in 1973, was a four-millimetre clam. Today, there are much larger machines which can accommodate a human body. As with the CT images, computers are also used to produce MR images. To accomplish this, a computer processes information on the alignment of magnetic fields in tissues after they are subjected to radio waves. Interestingly, in 1982 there were only four commercially available MR units in the United States. In 1990, there were over 2000 machines. Several features make MR preferable to CT scans in many situations. Aside from the fact that MR does not produce harmful ionizing radiation, it also produces a natural contrast between static and flowing matter and can provide better contrast resolution than CT or ultrasound. Stephen Hall in his book Mapping the Next Millennium (Random House) has suggested that, in the 21st century, the chest X-ray will become obsolete, supplemented by whole-body magnetic resonance imaging maps. Everyone will have their bodies scanned, digitized, processed by a computer, and kept on permanent record. Physicians will use this record to search for diseases and note changes in a person's body through time.

I'd like to mention a few of my favourite current applications of computers in medicine before allowing the chapter contributors to take you further in the future. Let's start by imagining a slightly decayed, 2000-year-old mummified child being probed by the most advanced computer instrumentation and computer graphics equipment. This is not some grotesque scene from a Steven King novel, rather this is precisely what an interdisciplinary research team at the University of Illinois is excited about. In the early 1990s, using two supercomputers (a CRAY 2S and a Connection Machine CM-2) to construct 3-D video animations from 2-D CAT scan slices, David Lawrence unravelled the identity of the small Egyptian child within the mummy's ancient wrappings. In a related project, Ray Evenhouse used various computers to carefully reconstruct the mummy's skull from head scan information. He built the flesh back on the skull, and then, with a special computer program used in updating old photos of missing children, he aged the mummy's face to 18 years old. Finally, Evenhouse produced physical 3-D models of heads at various ages.

The researchers' combined tests -- including radiography, CAT scans, 3-D imaging, and wood, textile, resin and insect analyses -- shed light not only on the mummy, but on mummification during Egypt's Roman period. Researchers concluded that the mummy is an 8 year old who died of unknown causes around A.D. 100. Without even removing the mummy's wrappings, they can tell that at least three organs were left inside the body.

Here are just a few additional interesting applications of computers and technology to medicine:

VIRTUAL CORPSES:

Researchers at the University of Colorado are creating "virtual cadavers" to help train medical students. These 3-D computer models are created in a sequence of steps. First, human cadavers are deep frozen in gelatin. With an extremely sharp carbon-diamond blade, researchers then shave off 1/2 mm sections from the cadaver. Each cross-section is photographed and digitally stored at a resolution of 1500x1200 pixels in 24-bit colour. The models are then stored on an indexed video-disc catalog developed by Interactive Education Inc. (Raleigh, North Carolina). Currently, the developers have interactive 3-D models of the human thorax, including the lungs, heart, arteries, veins, etc. Users can perform simulated dissections or surgery.

VIRTUAL INJECTIONS:

Dr. George Shelplock of Indiana University has recently created a multimedia anaesthesia training program for the Macintosh called Brachial Plexus Blocks. The program allows anaesthesiology residents to practice injecting a local anaesthetic into the brachial plexus, a group of nerves that innervate the arm. By manipulating a simulated needle, and by clicking on an X-ray vision option that causes the skin to dissolve away, users can gain valuable insight into how to best administer the anaesthesia.

ROBOT SURGEONS:

Robot surgeons are already operating in the 1990's. A 250-pound robot named Robodoc was the first robot to perform surgery on a human. Currently Robodoc routinely operates on arthritic dogs at the veterinary clinic of Hap Paul, head of Robodoc development at Integrated Surgical Systems in Sacramento, California. Robodoc carves out the cavity in the bone where an implant will be inserted. Robodoc is about 10 times as accurate as a human holding a drill. Sensors monitoring pressure on the drill bit will stop Robodoc if it were to start cutting into soft tissue. Perhaps Robodoc will aid orthopaedic surgeons by the end of the decade who already do 160,000 hip replacements in the United States each year.

PAIN-INDUCING PATTERNS:

Computers are being used to produce patterns which help physicians diagnose problems of the brain. Here is some background information. In 1984, several British researchers discovered that some people find a certain pattern of stripes painful to look at; moreover, stripe viewing apparently induced headache attacks in some subjects with histories of headaches. In 1989, researchers in the U.S. further demonstrated that this kind of pattern appears to help in distinguishing those people who suffer from migraine headaches from other types of headaches. Migraine sufferers, when presented with this pattern will find the pattern extremely objectionable and attempt to avert their gaze, while people who do not suffer from this type of headache will have relatively little difficulty looking at the pattern. The test pattern was designed for use by physicians as one part of an overall diagnostic test, and can be used to help distinguish migraine from non-migraine headache sufferers. It must however be used with caution, as the pattern is capable of triggering migraine headaches in some people. Certain patients with epilepsy may also suffer seizures after looking at the pattern.

Despite the potential for triggering migraine headaches, the interesting pattern has been published in the journal Brain in 1984. You can program this pattern on your computer using the following hint: it resembles a circle filled with alternating black and white vertical stripes. At a viewing distance of 43 cm, this grating has a spatial frequency of 3 cycles/degree of visual arc, and a Michelson contrast of about 0.7.

VISIBLE HUMAN PROJECT:

As an extension to the Virtual Corpse project just discussed, researchers at the University of Colorado Health Science Center in Denver are also excited about their project (started in 1991) to create the ultimate digital model of the human anatomy. This complete, 3-D model will contain high-resolution images and information concerning every cubic millimetre of a male and a female corpse. Medical students, for example, could use this data to precisely visualize the locations of blood vessels within the brain or nerves within the spine. Powerful graphics computers can then use this data to draw realistic 3-D renderings, animations, and magnifications from any angle, of any part of the body. This Visible Human Project, as it is called, involves the capturing of image data from medical image scans (CT and MRI) of human cadavers, as well as digitization of cryosection photographic data. No doubt, in a few years, you will be able to take a simulated submarine ride through the heart and aortic arch, much like the scientists did in the famous science fiction tale Fantastic Voyage.

BRAIN PANCAKES:

Surgeons at the Johns Hopkins School of Medicine in Baltimore, Maryland have implanted small, drug-filled, pancake-shaped wafers in the brain. After removing a tumour, they leave behind several wafers in the skull which release drugs over four weeks as the wafers dissolve. The surgeons hope to prevent tumour recurrence. Robert Langer, a professor of biomedical engineering at the Massachusetts Institute of Technology, created the wafers, and he notes that it is easy to control the drug dose if you know how fast the wafers dissolve. Another of Langer's wafers makes use of a maze of tiny tunnels within the wafer to allow drug molecules to slowly escape over months of time. Experiments are underway to determine how effective the brain pancakes are in preventing tumour recurrence.

Future Health takes many of these ideas forward, giving an account of the state of the art, and speculating on advances in the 21st Century. It therefore includes a range of topics which should interest students, health care professionals, biologists, physicians and a general audience fascinated by speculations on unusual technologies and the future of medical care and education. The book consists of two parts. In Part 1: Managing Information, contributors describe the challenges of future medical schools in preparing physicians in the 21st Century. They also discuss the importance of computer science in medicine, the concept of hybrid computational-physician leaders, the use of electronic gophers to obtain medical information, the use of artificial intelligence in medical diagnosis, the use of operating rooms in the 21st Century, and the use of computer conferencing. In Part II: Technological Breakthroughs authors discuss a range of techniques which will have increasing use in the 21st Century, for example: digital dentistry, robotic surgery, new medical imaging technologies, and even the use of computers in pathology. Most of the ideas expressed in this book are practical and are either currently being implemented or will be implementable within the next decade or two. My goal therefore is to provide information which students, lay people, scientists, politicians, and physicians will find of practical value today as they make personal, educational, and policy decisions about their needs in the coming decades.

This book's cover image showing an three-dimensional x-ray computed tomography scan (CT scan) of the head is courtesy of Court B. Cutting, M.D., a craniofacial surgeon at New York University, and Alan Kalvin, Ph.D., a research staff member at the IBM Watson Research Laboratory. Dr. Kalvin used the software IBM Visualization Data Explorer to create the brain cross-section. (I used custom graphic programs to enhance the coloration and set the figure on a background of mist and stars.)

Clifford A. Pickover

Yorktown Heights, New York

SOME RESOURCES

o Hall, S. (1992) Mapping the Next Millennium Random House: New York.

o West, B. (1990) Fractal Physiology and Chaos in Medicine. World Scientific: New Jersey.

o Computers in Biomedical Research, an official publication of the American Medical Informatics Association. (ISSN 0010-4809, published bimonthly by Academic Press, 6277 Sea Harbor Drive, Orlando, FL 32887-4900)

o American Medical Informatics Association, an association dedicated to the development and application of medical informatics in the support of patent care, teaching, research, and health care administration. They are also interested in computer applications in medical care. Available journals: Journal of the American Medical Informatics Association, MD Computing, Computers and Biomedical Research. (AMIA, 4915 St. Elmo Ave, Suite 302, Bethesda, Maryland 20814, e-mail: [email protected])

o The Society for Computer Applications in Radiology, an organization for professionals who realize that computers have become an indispensable part of daily activities in medical imaging. State-of-the-art practice includes: computer generated and enhanced images, radiology information management, image transmission and display, and decision support systems. Membership benefits include a subscription to The Journal of Digital Imaging. Contact: SCAR, PO Box 8800, 4750 Lindle Road, Harrisburg, PA 17105-8800.

o The BioMoo center is a virtual facility used by hundreds of biologists to communicate, collaborate, and design electronic tools to do science. This ambitions attempt to create a virtual reality research center on the Internet is complete with "labs", "offices", "meeting rooms", and even a "café". (Essentially it's a software program running on a computer at the Bioinformatics Unit of the Weizmann Institute of Science in Jerusalem.) To visit BioMoo, you may telnet to: bioinfo.weizmann.ac.il 8888 or 132.76.55.12 8888. At the BioMoo welcome screen, type "connect guest". For more information, see: Anderson, C. (1994) Cyberspace offers chance to do virtually real science. Science. May 14, 264: 900-901.

o The book series on Mathematical Biology and Medicine promotes interdisciplinary approaches in biology and in medicine. The book series includes topics such as: cardiac modelling, computer models in medicine, epidemiology, physiology, models of tumour growth, and genome research. The Journal of Biological Systems covers similar topics. (Publisher contact for book series and journal: World Scientific Publishing, Suite 1B, 1060 Main St., River Edge, New Jersey 07661. Editorial Contact: R. V. Jean, Department of Mathematics and Computer Science, Universite du Quebec, 3000 Avenue des Ursulines, Rimouski, Quebec, Canada G5L 3A1.)

o There are currently numerous health products available for your PC. I do not endorse any of these, but wish to stimulate your imagination and give examples of the kind of "futuristic", inexpensive PC applications available today. For example, using inexpensive "3-D Body Adventure" software, one can walk through a 3-D spinal cord, study evolution, rotate a skull, and more (Knowledge Adventure, 4502 Dyer St., La Crescenta, California 91214). PharmAssist, from Software Marketing Corporation (602-893-3377) is a program providing facts on thousands of prescription and nonprescription drugs. HealthSoft (800-795-4325) publishes similar software, as well as the Family Health Guide and Medical Dictionary. The Family Doctor on CD-ROM is available from Creative Multimedia (503-241-4351). The Mayo Clinic offers the Family Health Book and Heart Book on CD-ROM, from Interactive Ventures (612-686-0779). HealthDesk allows you to track your family's medical history and assess hereditary risk factors (800-578-5767). Medical insurance claim forms can be organized with ClaimPlus from Te Corp (800-725-2645) or MedSure from Time Solutions (800-552-3302). DynaPulse from Pulse Metric (800-927-8573) allows you to effectively monitor your pulse rate and blood pressure. SimHealth from the Markle Foundation (800-824-2643) lets you simulate various health care plans. (Much of this product information comes from: Soviero, M. (1994) The digital doctor makes house calls. Popular Science. April, page 53.)

Pickover, C. (1995) FUTURE HEALTH, Computers and Medicine in the Twenty-First Century. St. Martin's Press: New York.

ISBN 0-312-12602-6.

o futuristic operating rooms

o the challenges of future medical schools in preparing 21st-century physicians

o futuristic fractal models in pathology

o the use of new medical imaging technologies

o the use of electronic gophers to obtain medical information o digital dentistry

o the use of artificial intelligence in medical diagnosis

o computer conferencing for medical consulting

o bloodless robotic surgery

o making solid models from medical images

o futuristic examinations rooms

and much more....

Table of Contents

Preface

PART I. MANAGING INFORMATION AND SERVICE

Chapter 1.

Preparing Future Physicians: How will Medical Schools Meet the Challenge?

- David Kaufman, Ed.D., Director, Medical Education Unit

- Ms. Grace Paterson, M.Sc., Coordinator, Medical Informatics Dalhousie University

Chapter 2.

Just How Many Patients Can Fit in an Exam Room?

- Risa B. Bobroff

- Ronda H. Wang Baylor College of Medicine

Chapter 3.

Computers and Medicine: Advancing the Field

- Christopher Galassi, MD, MS

Methodist Hospital of Indiana

The Future of Computer Conferencing for Medical Consulting - W. R. Klemm, DVM, Ph.D

- J. R. Snell, DVM, MS

Department of Veterinary Anatomy and Public Health Texas A&M University

Chapter 5.

The Impact of Gophers on Biomedical Science

- Tim Littlejohn, Ph.D.

Department de Biochimie Universite de Montreal

PART II. TECHNOLOGICAL BREAKTHROUGHS

Chapter 6.

The Future of Computers in Pathology

- Gabriel Landini, Dr. Odont, PhD

- John W. Rippin, PhD. FRC Path. Oral Pathology Unit The University of Birmingham

Chapter 7.

Bloodless Robotic Surgery

- John R. Adler, M.D.

- Achim Schweikard, Ph.D.

Dept of Neurosurgery, Stanford University Institut für Informatik, Technische Universität München

Chapter 8.

Medical Images Made Solid

- Peter J. de Jager and Johan W.H. Tangelder

Delft University of Technology

Chapter 9.

Computer-Assisted Dental Care: Dentistry Goes Digital- Allan G. Farman, PhD (odont.), MBA Professor, Radiology & Imaging Sciences Division

- William C. Scarfe, BDS, MS

School of Dentistry University of Louisville

Chapter 10.

Medical Imaging and the Futures of Computers in Medicine

- Michael de la Maza

Artificial Intelligence Laboratory

Massachusetts Institute of Technology

- Deniz Yuret

Artificial Intelligence Laboratory

Massachusetts Institute of Technology

Fractal Underwater Images

by Loretta DeMars

Underwater Photographer - Phoenix, Arizona USA

Nature continues to surround us with its beauty and also with its answers to the puzzles of life. When do we really stop, look, and see it? For me, that started 8 years ago. As a SCUBA diver since the early 1980's, I was no longer able to leave the beauty of the sea behind when I completed each dive. I learned underwater photography so that I could safely bring the sea creatures with me onto land. The textures, patterns, and rhythms of the sea were what I captured first. After reading James Gleick's book Chaos¹ in 1990 I pointed my camera at undersea images with a different lens over my mind's eye. Often I stopped short (well as quickly as anyone can stop short in a 1 knot current passing along a wall in Beqa Lagoon, Fiji) and did a focus, point, click, shoot, capture image sequence at formations I had never seen before. After 5 years of collecting these slides, I want now to bring some of these images of the sea to the community of mathematicians and scientists among you.

The first image (Figure 1) is a feather star. It is also known as a crinoid (Class Crinoidea in the Group Phylum Echinodermata).² This feather star, according to Paul Humann, is one of the most ancient of animals and is even referred to as "living fossils". They anchor themselves to the reef and even move about it with jointed legs called cirri. The arms are fragile and can be broken off. Fortunately, broken arms can be regenerated.

The second image (Figure 2) is a soft coral. It is of the family of Nephtheidae and the order Alcyonacea². The soft corals are thick trunked with many branches at the end of which are polyps (octal) that clump on the branch tip. The coral trunk and its branches increase in volume when feeding. The embedded spicules are very visible through the translucent trunk membranes. Touching the coral produces a stinging sensation and this wards off predators, and protects many species capable of living among the stinging branches.

The third image (Figure 3) is a common sea fan. It is in the family called Gorgoniidae. The stems and branches of the gorgonian sea fan are formed on a central skeleton. This particular example grows in a single plane. It positions itself perpendicular to the prevailing currents to maximize the nutrients it gathers from the sea. This image was taken in the Caribbean Sea where this sea fan (and its symbiotic flamingo tongue) are common.

The fourth image (Figure 4) is an unknown classification to me. I took the photo because it looked to me like an excellent example of branching. I came home to compare it to the pattern of an electrical discharge by Oscar Kapp on page 310 of James Gleick's book Chaos¹. What do you think?

I offer these images to you, the scientists to enjoy and to determine if there is

need for future examples from the sea.

Loretta can be reached at the email address: [email protected] or the earth address - PO Box 10651, Phoenix Arizona 85064, USA. You can also view some of her photography on the Internet through the URL <http://www.cybermart.com/demars/>

Her screensaver (22 images) is commercially available by direct purchase through the Internet connection to Second Nature. The home page is <http://www.secondnature.com/nature.html>

References

1. J. Gleick Chaos Making A New Science Penguin Books 1987

2. P. Humann Reef Coral Identification New World Publications 1993

3. J. Gleick and E. Porter Nature's Chaos Penguin Books 1990

(editorial note - if you wish to study these pictures in detail it is recommended that you get colour slides or prints from the author.)

[images not on Internet version, sorry]

Various Internal Perturbations

of the Mandelbrot Set

by Malcolm Lichtenstein

SCREEN 12

sc=2.8

FOR cy=-2.5 TO 2.5 STEP 5/(51*sc)

d=d+1: c=0

FOR cx=-3.3 TO 3.3 STEP 5/(51*sc)

c= c+1

if instat then end

REM 'r' fed back to real x and imag y

REM ---------------------------------

x=0: y=0: k=0: r=0

WHILE r<4 AND k<16

k=k+1

xx=x*x-y*y+cx+r

y=2*x*y+cy+r

X=xx

r=SQR(x*x+y*y)

WEND

PSET(c,d),k*(r>4)

REM 'r' to real x only

rem ------------------

x=0: y=0: k=0: r=0

WHILE r<4 AND k<16

k=k+1

xx=x*x-y*y+cx+r

y=2*x*y+cy

X=xx

r=SQR(x*x+y*y)

WEND

PSET(c+70*sc,d),k*(r>4)

REM 'r' to imag y only

rem ------------------

x=0: y=0: k=0: r=0

WHILE r<4 AND k<16

k=k+1

xx=x*x-y*y+cx

y=2*x*y+cy+r

X=xx

r=SQR(x*x+y*y)

WEND

PSET(c+140*sc,d),k*(r>4)

REM x outcome to real x, y to imag y

rem --------------------------------

x=0: y=0: k=0: r=0

WHILE r<4 AND k<16

k=k+1

xx=x*x-y*y+cx+x

y=2*x*y+cy+y

X=xx

r=SQR(x*x+y*y)

WEND

PSET(c,d+60*sc),k*(r>4)

REM x outcome to real x

rem -------------------

x=0: y=0: k=0: r=0

WHILE r<4 AND k<16

k=k+1

xx=x*x-y*y+cx+x

y=2*x*y+cy

X=xx

r=SQR(x*x+y*y)

WEND

PSET(c+70*sc,d+60*sc),k*(r>4)

REM y to imag y

rem -----------

x=0: y=0: k=0: r=0

WHILE r<4 AND k<16

k=k+1

xx=x*x-y*y+cx

y=2*x*y+cy+y

X=xx

r=SQR(x*x+y*y)

WEND

PSET(c+140*sc,d+60*sc),k*(r>4)

rem x/y to real x, y/x to imag y

rem ----------------------------

x=0: y=0: k=0: r=0

WHILE r<4 AND k<16

k=k+1

xx=x*x-y*y+cx+x/(y+.001)

y=2*x*y+cy+y/(x+.001)

X=xx

r=SQR(x*x+y*y)

WEND

PSET(c,d+120*sc),k*(r>4)

rem y/x to real x, x/y to imag y

rem ----------------------------

x=0: y=0: k=0: r=0

WHILE r<4 AND k<16

k=k+1

xx=x*x-y*y+cx+y/(x+.001)

y=2*x*y+cy+x/(y+.001)

X=xx

r=SQR(x*x+y*y)

WEND

PSET(c+70*sc,d+120*sc),k*(r>4)

REM standard Mandelbrot for

rem comparisons

rem ---------------------------------

x=0: y=0: k=0: r=0

WHILE r<4 AND k<16

k=k+1

xx=x*x-y*y+cx

y=2*x*y+cy

X=xx

r=SQR(x*x+y*y)

WEND

PSET(c+140*sc,d+120*sc),k*(r>4)

next cx

next cy

beep

end

Fractint Formulae

by Roger Bagula

monkey(XAXIS) {; rlbagula

;

z = Pixel: ;

z = 1/(z*sin(1/z)+pixel)

z = Sqr(z)

LastSqr <= 4

; Use LastSqr instead of recalculating

}

monkey2(XAXIS) {; rlbagula

;

z = Pixel: ;

z = 1/(sqr(z)*sin(1/sqr(z))+pixel)

LastSqr <= 16

; Use LastSqr instead of recalculating

}

monkey2i(XAXIS) {; rlbagula

;

z = Pixel: ;

z = (sqr(z)*sin(1/sqr(z))+pixel)

LastSqr <= 16

; Use LastSqr instead of recalculating

}

monkey3i(XAXIS) {; rlbagula

;

z = Pixel: ;

z = (z*sqr(z)*sin(1/(z*sqr(z)))+pixel)

LastSqr <= 16

; Use LastSqr instead of recalculating

}

monkey4i(XAXIS) {; rlbagula

;

z = Pixel: ;

z = (sqr(sqr(z))*sin(1/sqr(sqr(z)))+pixel)

LastSqr <= 16

; Use LastSqr instead of recalculating

}

F_turtle(XAXIS) {; R.L.Bagula

; Classical fractal showing LastSqr speedup

z = Pixel, z = Sqr(z+1/z): ; Start with z**2 to initialize LastSqr

z = z + Pixel

z = Sqr(z)

LastSqr <= 16

; Use LastSqr instead of recalculating

}

F_turtle2(XAXIS) {; R.L.Bagula

; Classical fractal showing LastSqr speedup

z = Pixel, z = Sqr(z+1/z): ; Start with z**2 to initialize LastSqr

z = z + Pixel

z=sqr(z+1/z)

LastSqr <= 16

; Use LastSqr instead of recalculating

}

a_sct(XAXIS) {; R.L.Bagula

; Classical fractal showing LastSqr speedup

z = Pixel, z = Sqr(z+sqr(pixel)/z): ; Start with z**2 to initialize LastSqr

z = z + Pixel

z=sqr(z+sqr(pixel)/z)

LastSqr <= 16

; Use LastSqr instead of recalculating

}

atest(XAXIS) {; R.L.Bagula

; Classical fractal showing LastSqr speedup

z = Pixel, z = Sqr(z+1/z): ; Start with z**2 to initialize LastSqr

z = z + 1/Pixel

z = Sqr(z)

LastSqr <= 16

; Use LastSqr instead of recalculating

}

Editorial comments:

The front page picture shows Monkey2i with floating point activated. x is from -5.77 to 5.77 and y -4.33 to 4.33. (Colours are inverted in all these images so that they print better. Coordinates are rounded)

Zooming into this produces some interesting effects. If you magnify the centre image and then zoon into the centre of the green structure on the right you get

x is from -2.88 to 2.88 and y -.22 to .22

Zooming in on one of the arms gives:

x is .05101 to .05287, y from .00060 to .00059 (suggest you find it by zooming) Note the little black objects, all similar but slightly different, lying in the "boxes". It would be interesting if it was possible to make Fractint automatically pan along these arms. One can do it manually by making the zoom boz full size and moving it slightly off centre.

Zooming deeper still into the structure, we find yet a smaller arm, still with the blobs growing inside it (they show as white as the next two pictures are inverted):

further along:

Looking at the edge we see this:

...as we move along the arm ...

... the blobs are slightly different.

It would also be interesting if we could somehow program Fractint to select a "blob" and then make a "film" by recording it as a frame of an .avi file and then moving to the next blob and recording that and so on until it reached some length determined by the user. Such .avi files could then be played as animations which may show movement in the blob that could be quite interesting.

here we see the blobs being "made" in the tentacles...

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