While writing this novel, I was amused to encounter another work of fiction using the Glomar Explorer, though (luckily!) for a very different purpose: Ship of Gold, by Thomas Allen and Norman Polmar (Macmillan, 1987).
My thanks also to sundry CIA and KGB acquaintances, who would prefer to remain anonymous.
One informant I am happy to identify is Professor William Orr, Dept. of Geological Science, University of Oregon, my erstwhile shipmate on the floating campus SS Universe. The plans and documentation he provided on Glomar Explorer (now languishing in Suisun Bay, California, between Vallejo and Martinez — you can see her from Highway 680) were essential inputs.
The discovery of major explosive events on the seabed, referred to in Chapter 33, was reported by David B. Prior, Earl H. Doyle, and Michael J. Kaluza in Science, vol. 243, pp. 517–9, 27 January 1989, under the title ‘Evidence for Sediment Eruption on Deep Sea Floor, Gulf of Mexico.’
On the very day I was making the final corrections to this manuscript, I learned that there is now strong evidence that oil drilling can cause earthquakes. The October 28, 1989, Science News cites a paper by Paul Segall of the U.S. Geological Survey, making this claim in the October 1989 issue of Geology.
The report on the Neolithic grave quoted in Chapter 34 will be found in Nature, 276, 608, 1978.
Ralph C. Merkle’s truly mind-boggling paper ‘Molecular Repair of the Brain’ first appeared in the October 1989 issue of Cryonics (published by ALCOR, 12327, Doherty St., Riverside, CA, 92503) to whom I am grateful for an advance copy.
My thanks to Kumar Chitty for information on the U.N. Law of the Sea Convention, directed for many years by the late Ambassador Shirley Hamilton Amarasinghe. It is a great tragedy that Shirley (the hospitality of whose Park Avenue apartment I often enjoyed in the ’70s) did not see the culmination of his efforts. He was a wonderful persuader, and had he lived might even have prevented the U.S. and U.K. delegations from shooting themselves in the foot.
I am particularly grateful to my collaborator Gentry Lee (Cradle, the Rama trilogy) for arranging his schedule so that I could concentrate all my energies on the latest of my ‘last’ novels. . . .
Very special thanks to Navam and Sally Tambayah — not to mention Tasha and Cindy — for hospitality, WORDSTAR, and faxes. . . .
And, finally: a tribute to my dear friend the late Reginald Ross, who besides many other kindnesses introduced me to Rachmaninoff and Elgar half a century ago, and who died at the age of 91 while this book was being written.
Mandelmemo
The literature on the Mandelbrot Set, first introduced to the non-IBM world in A. K. Dewdney’s ‘Computer Recreations’ (Scientific American, Aug. 1985, 16–25), is now enormous. The master’s own book, The Fractal Geometry of Nature (W. H. Freeman, 1982), is highly technical, and much is inaccessible even to those with delusions of mathematical ability. Nevertheless, a good deal of the text is informative and witty, so it is well worth skimming. However, it contains only the briefest references to the M-Set, the exploration of which was barely beginning in 1982.
The Beauty of Fractals (H-O. Peitgen and P. H. Richter, Springer-Verlag, 1986) was the first book to show the M-Set in glorious Technicolor, and contains a fascinating (and often amusing) essay by Dr. M. himself on its origins and discovery (invention?). He describes later developments in The Science of Fractal Images (edited by H-O. Peitgen and Dietmar Saupe, Springer-Verlag, 1988). Both these books are highly technical.
Much more accessible to the general — though determined — reader is A. K. Dewdney’s The Armchair Universe (W. H. Freeman, 1988). This contains the original 1985 Scientific American article, with updates and information on software available for personal computers. I have been very happy with MandFXP, from Cygnus Software (1215 Davie St., P.O. Box 363, Vancouver BC, V6E 1N4, Canada), and have used this extensively on my AMIGA 2000. While making a TV documentary, ‘God, the Universe, and Everything Else’ for U.K.’s Channel 4, I had the rare privilege of showing Stephen Hawking some beautiful ‘black holes’ I had discovered, while expanding the set until it would have filled the orbit of Mars. Another supplier of M-Set software (for MAC and IBM) is Sintar Software (1001 4th Ave., Suite 3200, Seattle, WA 98154).
Needless to say, there are Mandelbrot ‘fan magazines,’ containing hints on speeding up programs, notes from explorers in far-off regions of the set — and even samples of a new literary genre, Fractalfiction. The newsletter of the field is Amygdala, edited by Rollo Silver, who also supplies software (Box 111, San Cristobal, NM 87564).
Undoubtedly the best way of appreciating the set is through the videotapes that have been made of it, usually with accompanying music. Most celebrated is ‘Nothing But Zooms’ from Art Matrix (P.O. Box 880, Ithaca, NY 14851). I have also enjoyed ‘A Fractal Ballet’ (The Fractal Stuff Company, P.O. Box 5202, Spokane, WA 99205-5202).
Strictly speaking, the ‘Utter West’ of the M-Set is at exactly–2, not –1.999 . . . to infinity, as stated in Chapter 18. Anyone care to split the difference?
I do not know if there have been any cases of Mandelmania in real life, but I expect to receive reports as soon as this book appears — and waive all responsibility in advance.
APPENDIX: THE COLORS OF INFINITY
In November 1989, when receiving the Association of Space Explorers Special Achievement Award in Riyadh, Saudi Arabia, I had the privilege of addressing the largest gathering of astronauts and cosmonauts ever assembled at one place. (More than fifty, including Apollo 11’s Buzz Aldrin and Mike Collins, and the first ‘space walker’ Alexei Leonov, who is no longer embarrassed at sharing the dedication of 2010: Odyssey Two with Andrei Sakharov.) I decided to expand their horizons by introducing them to something really large, and, with astronaut Prince Sultan bin Salman bin Abdul Aziz in the chair, delivered a lavishly illustrated lecture ‘The Colors of Infinity: Exploring the Fractal Universe.’
The material that follows is extracted from my speech; another portion appears at the beginning of Chapter 15. I’m only sorry that I cannot illustrate it with the gorgeous 35-millimeter slides — and videos — I used at Riyadh.
Today, everybody is familiar with graphs — especially the one with time along the horizontal axis, and the cost of living climbing steadily up the vertical one. The idea that any point on a plane can be expressed by two numbers, usually written x and y, now appears so obvious that it seems quite surprising that the world of mathematics had to wait until 1637 for Descartes to invent it.
We are still discovering the consequences of that apparently simple idea, and the most amazing is now just ten years old. It’s called the Mandelbrot Set (from now on, the M-Set) and you’re soon going to meet it everywhere — in the design of fabrics, wallpaper, jewelry, and linoleum. And, I’m afraid, it will be popping out of your TV screen in every other commercial.
Yet the most astonishing feature of the M-Set is its basic simplicity. Unlike almost everything else in modern mathematics, any schoolchild can understand how it is produced. Its generation involves nothing more advanced than addition and multiplication; there’s no need for such complexities as subtraction and — heaven forbid! — division, let alone any of the more exotic beasts from the mathematical menagerie.
There can be few people in the civilized world who have not encountered Einstein’s famous E = mc2, or who would consider it too hopelessly complicated to understand. Well, the equation that defines the M-Set contains the same number of terms, and indeed looks very similar. Here it is.
Z = z2 + c
Not very terrifying, is it? Yet the lifetime of the Universe would not be long enough to explore all its ramifications.
The z’s and the c in Mandelbrot’s equation are all numbers, not (as in Einstein’s) physical quantities like mass and energy. They are the coordinates which specify the position of a point, and the equation controls the way in which it moves, to trace out a pattern.
There’s a very simple analog familiar to everyone — those children’s books with blank pages sprinkled with numbers, which when joined up in the right order reveal hidden — and often surprising — pictures. The image on a TV screen is produced by a sophisticated application of the same principle.
In theory, anyone who can add and multiply could plot out the M-Set with pen or pencil on a sheet of squared paper. However, as we’ll see later, there are certain practical difficulties — notably the fact that a human life span is seldom more than a hundred years. So the set is invariably computer-generated, and usually shown on a visual display unit.
Now, there are two ways of locating a point in space. The more common employs some kind of grid reference — west-east, north-south, or on squared graph paper, a horizontal X-axis and a vertical Y-axis. But there’s also the system used in radar, now familiar to most people thanks to countless movies. Here the position of an object is given by (1) its distance from the origin, and (2) its direction, or compass bearing. Incidentally, this is the natural system — the one you use automatically and unconsciously when you play any ball game. Then you’re concerned with distances and angles, with yourself as the origin.