ph@miro.Berkeley.EDU (Paul Heckbert) (11/15/89)
There's an interesting opinion piece on the hype and publicity regarding
fractals in the current issue of "Mathematical Intelligencer":
Steven Krantz
"Fractal Geometry"
The Mathematical Intelligencer, Vol. 11, No. 4, Fall 1989.
which you should be able to find in a nearby college library.
To quote some of Krantz' more provocative statements:
"Hailed as a lingua franca for all of science, the theory of
fractals is said by some to be the greatest idea since calculus.
...
One notable difference between fractal geometry and calculus
is that fractal geometry has not solved any problems.
It is not even clear that it has created any new ones."
The journal also printed a rebuttal by Mandelbrot, who basically
defends his work as highly regarded, but does not address Krantz'
contention that the study of fractals has been unscientific.
Check it out!
Paul Heckbert, CS grad student
508-7 Evans Hall, UC Berkeley INTERNET: ph@miro.berkeley.edu
Berkeley, CA 94720davidsen@crdos1.crd.ge.COM (Wm E Davidsen Jr) (11/15/89)
In article <19544@pasteur.Berkeley.EDU>, ph@miro.Berkeley.EDU (Paul Heckbert) writes: | The journal also printed a rebuttal by Mandelbrot, who basically | defends his work as highly regarded, but does not address Krantz' | contention that the study of fractals has been unscientific. Thanks for the posting. I don't buy the argument that a study is unscientific if it doesn't solve problems. And I think the problem it has raised is "how does this relate to the real world?" I will try to find the article if I can. -- bill davidsen (davidsen@crdos1.crd.GE.COM -or- uunet!crdgw1!crdos1!davidsen) "The world is filled with fools. They blindly follow their so-called 'reason' in the face of the church and common sense. Any fool can see that the world is flat!" - anon
musgrave-forest@CS.YALE.EDU (F. Ken Musgrave) (11/16/89)
In article <19544@pasteur.Berkeley.EDU> ph@miro.Berkeley.EDU (Paul Heckbert) writes: >There's an interesting opinion piece on the hype and publicity regarding >fractals in the current issue of "Mathematical Intelligencer": > > Steven Krantz > "Fractal Geometry" > The Mathematical Intelligencer, Vol. 11, No. 4, Fall 1989. > >which you should be able to find in a nearby college library. >To quote some of Krantz' more provocative statements: > > "Hailed as a lingua franca for all of science, the theory of > fractals is said by some to be the greatest idea since calculus. > ... > One notable difference between fractal geometry and calculus > is that fractal geometry has not solved any problems. > It is not even clear that it has created any new ones." > >The journal also printed a rebuttal by Mandelbrot, who basically >defends his work as highly regarded, but does not address Krantz' >contention that the study of fractals has been unscientific. Fractal geometry is a very new and general field, and to date largely ill-defined. One could say that this is symptomatic of its generality. At any rate, detractors such as Krantz should help to develop its form and definition - such dialectic is necessary to determine the true import of fractal geometry. Mandelbrot deigned to address Krantz's specific inaccurate claims against himself and fractal geometry, not his grandiose statements such as you quote above. One should not bother, probably, to try to rebut statements such as "...the emperor has no clothes". It is clear that Krantz, with credentials far inferior to Mandelbrot's, has a personal vendetta to pursue (his motivation for this is evident in his article and is not ill-founded), and it is only of dialectical interest to give credence to his rhetoric. But it is of great dialectical interest! What is the power, and what are the limitations of fractal geometry as a language for the description of Nature? It is far too soon to tell, as Krantz, Mandelbrot, and Kadanoff all agree. In the meantime, the quality of mathematics, science, and art associated with fractal geometry will vary widely. Such public conversations as Krantz's will serve to keep researchers who touch upon the field, honest. That fractal geometry recommends itself to the senses, both trained and untrained, is not to be helped and is indeed to many of us a powerful indi- cation that it is somehow essential to Nature. This aspect of fractals will serve to keep them in high public profile to have them appear to be "hyped" as compared with other mathematics and science. Some mathematicians and scientists will therefore feel compelled to discharge their slings and arrows at "the emperor" to 'keep him down to size'; this is human nature and even an indispensable dynamic to intellectual inquiry. Let the controversy rage on! *===============================================================* F. Kenton ("Ken") Musgrave arpanet: musgrave-forest@yale.edu Yale U Depts of Math and CS (203) 432-4016 Box 2155 Yale Station Primary Metaphysical Principle: New Haven, CT 06520 Deus ex machina
billd@fps.com (Bill Davids_on) (11/16/89)
In article <1619@crdos1.crd.ge.COM> davidsen@crdos1.UUCP (bill davidsen) writes: >In article <19544@pasteur.Berkeley.EDU>, ph@miro.Berkeley.EDU (Paul Heckbert) writes: >| The journal also printed a rebuttal by Mandelbrot, who basically >| defends his work as highly regarded, but does not address Krantz' >| contention that the study of fractals has been unscientific. > > Thanks for the posting. I don't buy the argument that a study is >unscientific if it doesn't solve problems. And I think the problem it >has raised is "how does this relate to the real world?" I will try to >find the article if I can. Here is another excerpt from Krantz for those of you who may have trouble finding a copy: There is an important issue implicit in this discussion that I would now like to examine. A famous counterexample (due to Celso Costa) in the theory of minimal surfaces was inspired by the viewing of a Brazilian documentary about samba schools -- it seems that one of the dancers wore a traditional hat of a bizarre character that was later reflected in the shape of the example. I once thought of an interesting counterexample by lying on my back and watching the flight of seagulls. Whatever the merits of samba dancers and seagulls may be, they are not scientists and they are not mathematicians. Why should fractal geometers be judged any differently? I think this guy just wants to get a little fame for himself. Maybe he's just annoyed that there is so much emphasis on fractals these days (actually he alludes to this in the next to last paragraph where he talks about how getting money to buy hardware to do fractals is easier than getting money to study algeraic geometry). I still think he's a dweeb for writing the paragraph I just quoted. It was completely uncalled for. The editor's note says that Krantz originally submitted this paper to the American Mathematical Society (AMS) and the editor asked for changes (which were made). Upon second review, he decided that it was still too strong and requested more changes and Krantz declined and complained to the Council of the AMS because he wasn't getting published as previously agreed. Krantz had distributed copies of the paper after the first revisions to several mathematicians (including Mandelbrot). It ended up never getting published by the AMS and the Mathematical Intelligencer picked it up along with Mandelbrot's rebuttal. I do think it's good that someone's watching the fractal geometers and not allowing them to get away with just showing pretty pictures but Krantz has crossed the line from being a skeptic to being a jerk. --Bill Davids_on
brucec@demiurge.WV.TEK.COM (Bruce Cohen;685-2439;61-028) (11/17/89)
In article <3775@celit.fps.com> billd@fps.com (Bill Davids_on) writes: >I think this guy just wants to get a little fame for himself. Maybe >he's just annoyed that there is so much emphasis on fractals these >days (actually he alludes to this in the next to last paragraph where >he talks about how getting money to buy hardware to do fractals is >easier than getting money to study algeraic geometry). I still think >he's a dweeb for writing the paragraph I just quoted. It was >completely uncalled for. Now I'm sorry I let my subscription to the Intelligencer lapse; sounds like some real juicy flammage. I think what you're seeing here is the classic antipathy of "pure" mathematicians for "applied" mathematics. The quotes are because I don't think anyone can really tell where the line is drawn; maybe there aren't any real distinctions in the mathematics, and all you can say is that there are pure and applied mathematicians. In any case, I believe that the Intelligencer is where I saw the quote: "Applied mathematics is bad mathematics." I *believe* the quote was attributed to Paul Halmos, but I won't swear to it; if I'm wrong, be gentle with me. "Small men in padded bras don't look the same falling from high places." - R.A. MacAvoy, "The Third Eagle" Bruce Cohen brucec@orca.wv.tek.com Interactive Technologies Division, Tektronix, Inc. M/S 61-028, P.O. Box 1000, Wilsonville, OR 97070
glenn@eos.UUCP (Glenn Meyer) (11/17/89)
musgrave-forest@CS.YALE.EDU (F. Ken Musgrave) writes: > It is clear that Krantz, with credentials far inferior to Mandelbrot's, >has a personal vendetta to pursue (his motivation for this is evident in his >article and is not ill-founded), and it is only of dialectical interest to >give credence to his rhetoric. > But it is of great dialectical interest! What is the power, and what >are the limitations of fractal geometry as a language for the description of >Nature? It is far too soon to tell, as Krantz, Mandelbrot, and Kadanoff >all agree. > In the meantime, the quality of mathematics, science, and art associated >with fractal geometry will vary widely. Such public conversations as Krantz's >will serve to keep researchers who touch upon the field, honest. > That fractal geometry recommends itself to the senses, both trained and >untrained, is not to be helped and is indeed to many of us a powerful indi- >cation that it is somehow essential to Nature. This aspect of fractals will >serve to keep them in high public profile to have them appear to be "hyped" >as compared with other mathematics and science. Mr. Musgrave: 1. Credentials do not a valid argument make. If that were the case, the "crystal spheres" theory of stellar mechanics, propounded by so many eminent pre-Renaissance astronomers dating back to Ptolmey, would never have been so thouroughly discredited. 2. To some of us, "that fractal geometry recommends itself to the senses" is NOT a powerful indication that fractals are somehow essential to Nature. The "looks like" approach to science has been a powerful persuader in the past -- it sustained Ptolmey's astronomy of cycles and epicycles for 14 centuries -- but has also sometimes been a great obstacle to the advance of knowledge. 3. Fractals don't necessarily recommend themselves "to the senses, both trained and untrained." For example, a geologist might look at fractal land formations and respond, "These don't look like land formations. Where are the strata?" 4. Fractals don't "appear" to be hyped, they are hyped; or rather, fractal images are used in hype. If you can use a computer to generate a structure that looks like something in Nature -- for example, a Purkinje cell in the human cerebellum -- then you in some eyes will appear to know something about Purkinje cells, how they work, and how they grew. I have seen fractals used in fund-raising presentations, for this reason. I have yet to see anyone verify, scientifically, that fractal imaging is anything but a visual tool, a paint brush with a mathematical basis. Glenn Meyer glenn%{eos,carma}@ames.arc.nasa.gov -- Glenn Meyer (glenn%carma@{io,aurora,eos,pioneer}.arc.nasa.gov) CARMA/Sterling Software NASA-Ames, M.S. 233-14, Moffett Field, Ca. 94035 Office telephone # 415-694-4804
hallett@pet3.uucp (Jeff Hallett x5163 ) (11/18/89)
In article <1619@crdos1.crd.ge.COM> davidsen@crdos1.UUCP (bill davidsen) writes: > And I think the problem it has raised is "how does this relate to > the real world?" But Bill, isn't that a problem with a whole bunch of studies? Someone thinks of a great new thing to study and people are forced to ask, "So what?" I've often bucked at people who want to study things for the sheer enjoyment of studying it - would anyone care to comment on the usefulness of study without a goal or use for the acquired knowledge? Granted I like to tinker with stuff for which I don't foresee a use. Maybe that's the issue; maybe the results cannot be foreseen at the start of the study? Who knows, maybe the study of fractals will eventually lead to warp drive or something... I guess I've just waffed on the issue and said nothing. :^) :^) I think fractals is useful in the sense that it gives us some insight on nature herself, but then again, it is good to avoid "mental masturbation" (IMHO). -- Jeffrey A. Hallett, PET Software Engineering GE Medical Systems, W641, PO Box 414, Milwaukee, WI 53201 (414) 548-5163 : EMAIL - hallett@gemed.ge.com Est natura hominum novitatis avida
markv@phoenix.Princeton.EDU (Mark T Vandewettering) (11/18/89)
In article <5594@eos.UUCP> glenn@eos.UUCP (Glenn Meyer) writes: [ remarks which are quite eloquently presented ] >2. To some of us, "that fractal geometry recommends itself to the >senses" is NOT a powerful indication that fractals are somehow essential >to Nature. The "looks like" approach to science has been a powerful >persuader in the past -- it sustained Ptolmey's astronomy of cycles and >epicycles for 14 centuries -- but has also sometimes been a great obstacle >to the advance of knowledge. Fractal geometry may be able to generate images which are convincing images of nature (and even this I might contest) but this in no way indicates anything about the mechanism that generates such phenomena. Hence, there use in computer graphics might be justified as an approach to generating convincing imagery, but as far as helping to understand real phenomena, I have yet to see convincing arguments that fractals are of any use whatsoever. I could draw a bunch of lines that looks like a bug, and that doesn't mean I understand anything about the nature of bugs. "Looks like" is not equivalent to "is like". >4. Fractals don't "appear" to be hyped, they are hyped; or rather, >fractal images are used in hype. Total agreement. Fractal imagery is now appearing on the cover of physics journals, computer journals, hell, just about everywhere. To be honest, they bore me, because basically they are the same, and on all scales :-) If I never saw another Mandlebrot set, or another mountain generated with binary subdivision, I would indeed be a glad human. Mark
hutch@fps.com (Jim Hutchison) (11/22/89)
In <1449@mrsvr.UUCP> hallett@gemed.ge.com (Jeffrey A. Hallett) asks: >In <1619@crdos1.crd.ge.COM> davidsen@crdos1.UUCP (bill davidsen) states: >> And I think the problem it has raised is "how does this relate to >> the real world?" >But Bill, isn't that a problem with a whole bunch of studies? Someone >thinks of a great new thing to study and people are forced to ask, "So >what?" I've often bucked at people who want to study things for the >sheer enjoyment of studying it - would anyone care to comment on the >usefulness of study without a goal or use for the acquired knowledge? Well, as for "study without a goal", it looks like the goal is to understand this "strange thing". The merit of that goal can probably be related to what you are trying to accomplish in the process. Anyway, enough of that, how about if I just tell you what i helped me to learn? Certain "fractal images" show the rate at which an equation converges or diverges from a given point. This was incredibly helpful to me when I was studying numerical methods. "See the black, bad numbers. see the red, good numbers." Then you go and change the algorithm a bit and the shape changes, because you caused the convergence to change. O.k., maybe you don't want to call *that* fractals. How about the pattern of nerve impulses in a properly functioning heart? There was a program on Chaos on a network educational TV program, in which they showed 2 different graphs of nerve activity in the human heart. The nice ordered pattern was fibrilation (bad), the disordered (looking) model was a normal functioning heart. Seems that ordered pulses are not all that productive, a nice sort-of-space-filling pattern is apparently much better. Sorry to quote from TV, but I have no serious interest in medicine. -- /* Jim Hutchison {dcdwest,ucbvax}!ucsd!celerity!hutch */ /* Disclaimer: I am not an official spokesman for FPS computing */
prem@geomag.fsu.edu (Prem Subrahmanyam) (11/22/89)
In article <4158@celit.fps.com> hutch@fps.com (Jim Hutchison) writes: >There was a program on Chaos on a network educational TV program, in which >they showed 2 different graphs of nerve activity in the human heart. The >nice ordered pattern was fibrilation (bad), the disordered (looking) model >was a normal functioning heart. Seems that ordered pulses are not all that Actually, it was the brain that followed this pattern. The fibrillating heart was chaotic, with a normal heart having a normal rhythm. The brain was coked up to make a repeatable "waveform", otherwise, it was intensely random in neural activity. By the way, although fractals are a part of chaotic study, chaotic study is not just fractals. ---Prem Subrahmanyam (prem@geomag.gly.fsu.edu)
Jim.McNamee@p7.f12.n376.z1.FIDONET.ORG (Jim McNamee) (11/22/89)
> The fibrillating heart was chaotic, with a normal heart having a > normal rhythm.
Actually, the "normal" heart produces an ECG having a 1/f power spectrum whereas the "sick" heart has a more traditional looking spectrum.
I have to disagree with Krantz's dim view of the utility of fractals. Fractals are making contributions to the way we describe and understand all sorts of natural phenomena, much the way calculus did. The fact that traditional mathematicians have shown little interest in this area neither proves or disproves its worth.
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Jim McNamee == ...!usceast!uscacm!12.7!Jim.McNameebradley@phillip.edu.au (Bradley White, Systems Programmer) (11/23/89)
In article <4158@celit.fps.com>, hutch@fps.com (Jim Hutchison) writes: > In <1449@mrsvr.UUCP> hallett@gemed.ge.com (Jeffrey A. Hallett) asks: >>In <1619@crdos1.crd.ge.COM> davidsen@crdos1.UUCP (bill davidsen) states: >>> And I think the problem it has raised is "how does this relate to >>> the real world?" > >>But Bill, isn't that a problem with a whole bunch of studies? Someone [stuff deleted] > > How about the pattern of nerve impulses in a properly functioning heart? > There was a program on Chaos on a network educational TV program, in which > they showed 2 different graphs of nerve activity in the human heart. The > nice ordered pattern was fibrilation (bad), the disordered (looking) model > was a normal functioning heart. Seems that ordered pulses are not all that > productive, a nice sort-of-space-filling pattern is apparently much better. > Sorry to quote from TV, but I have no serious interest in medicine. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ How true. I think you have your examples around the wrong way.. > > -- > /* Jim Hutchison {dcdwest,ucbvax}!ucsd!celerity!hutch */ > /* Disclaimer: I am not an official spokesman for FPS computing */ -- Bradley White. | Internet: bradley@phillip.edu.au Phillip Institute of Technology, | ACSnet : bradley%phillip.edu.au@munnari.oz Computer Centre, | Phone : (03) 468 2584 Plenty Road, | Bundoora, Victoria, Australia. |
zerr@cat50.CS.WISC.EDU (Troy Zerr) (11/23/89)
In article <5383@orca.WV.TEK.COM> brucec@demiurge.WV.TEK.COM (Bruce Cohen) writes: >I think what you're seeing here is the classic >antipathy of "pure" mathematicians for "applied" mathematics. The quotes >are because I don't think anyone can really tell where the line is drawn; >maybe there aren't any real distinctions in the mathematics, and all you >can say is that there are pure and applied mathematicians. In any case, I >believe that the Intelligencer is where I saw the quote: "Applied >mathematics is bad mathematics." >brucec@orca.wv.tek.com You seem to be unclear about what "Applied mathematics" is. . . The applied mathematician wishes to predict or explain some (usually) physical situation by constructing a mathematical model and using the techniques of mathematical proof to derive certain properties of this model -- which are then reinterpreted in a physical context. (This definition is woefully incomplete, and excludes entire fields such as numerical analysis -- but it suffices for the present discussion.) For example, suppose I wish to know the shape of a telephone wire suspended from two telephone poles. First I make some simplifying assumptions (e.g. the wire is one-dimensional and of uniform density) and then reinterpret these assumptions as a differential equation. I then solve the differential equation, and obtain an equation for the curve describing the shape that the (idealized) telephone wire will assume at equilibrium. Notice that once I have constructed (and justified the applicability of) the appropriate differential equation, the techniques used to derive a solution are as rigorous as those of one performing "pure" mathematics. (Theorem Proof Theorem Proof . . .) Constrast this with the activity of a prototypical "fractal geometer", the sort whose books are often seen on bookstore shelves. I program a computer to start with a triangle, choose a random (well, not really random since my computer isn't very creative) point in the triangle, move it above or below the plane of the triangle, and add new edges to form a pyramid. I then repeat the procedure with each of the (Triangular) faces of the pyramid, ad infinitum. (well, not really ad infinitum since my computer will only do finitely many calculations before I run out of coffee and doze off.) With suitably chosen random numbers, the resulting picture looks like an island. (Well, on my crude display it looks like an island . . . on a more refined display it looks like a bunch of spikes.) Am I to conclude that islands are constructed by some geological process of repeated subdivision? Of course not! All I have done is to produce a picture which looks (to me) like an island. Admittedly, I chose a particularly obnoxious example -- "fractals" may indeed be good models of some physical phenomena. I agree that computer graphics are capable of generating interesting mathematical QUESTIONS, and that Hausdorff measure and Hausdorff dimension are interesting on solely mathematical grounds. But most of what is popularized is more akin to "pretty pictures" with no real underlying motivation, attaching numbers to physical phenomena with no rigor or reason (What is the dimension of the coastline of Britain WHO CARES!), and attempting to ANSWER mathematical questions with computer graphics. (Question: is the "Mandelbrot Set" connected? Answer: Well, whenever I look at a picture of it generated by my computer, it looks connected, so it must be so.) Compare the methods of the Applied Mathematician and the Fractal Geometer. The applied mathematician (after providing adequate justification for his model) gives proofs for his (or her) assertions just as any other mathematician would. Our fractal geometer (whose books reach the populus and form their impression of mathematics) regards precise and well-founded definitions as an inconvenience and rigouous proofs as esoteric and boring. While our portrait of this unfortunate soul is not applicable to every fractal geometer, that it is applicable at all is unfortunate indeed. The debate here is not between "pure mathematics" and "applied mathematics", nor is it between "good mathematics" and "bad mathematics." It is between what is mathematics and what is not mathematics. Mathematics, simply defined, is what a mathematician does; and what a mathematician does and what our prototypical fractal geometer does differ dramatically. -Troy Zerr University of Wisconsin Department of Mathematics Madison, Wisconsin zerr@math.wisc.edu
Jim.McNamee@p7.f12.n376.z1.FIDONET.ORG (Jim McNamee) (11/23/89)
Mark T Vandewettering writes:
"Fractal geometry may be able to generate images which are convincing images of nature (and even this I might contest) but this in no way indicates anything about the mechanism that generates such phenomena."
In the strictest sense, that's true. But it should be pointed out that the same criticism must be leveled against Euclidean geometry. Only man and a few uninteresting natural phenomena rely on that branch of mathematics to create the world you see.
"... but as far as helping to understand real phenomena, I have yet to see convincing arguments that fractals are of any use whatsoever. I could draw a bunch of lines that looks like a bug, and that doesn't mean I understand anything about the nature of bugs."
You may not understand anything about the nature of bugs but you may learn how they came to have that particular appearance. And if form and function follow hand in hand you may also come to understand how bugs work.
"To be honest, [fractal images] bore me, because basically they are the same, and on all scales. If I never saw another Mandlebrot set, or another mountain generated with binary subdivision, I would indeed be a glad human."
When he was governor of California, the famous 'naturalist' and former U.S.president said, "A rock is a rock. A tree is a tree." Sounds as though the two of you take similar pleasure in the beauty of Nature.
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Jim McNamee == ...!usceast!uscacm!12.7!Jim.McNameethomson@cs.utah.edu (Rich Thomson) (11/25/89)
In article <5383@orca.WV.TEK.COM> brucec@demiurge.WV.TEK.COM (Bruce Cohen) writes: >I think what you're seeing here is the classic >antipathy of "pure" mathematicians for "applied" mathematics. And in article <3842@puff.cs.wisc.edu> zerr@schaefer (Troy Zerr) writes: [ Long discussion about what applied mathematics is, focusing on the concept of rigorous proof as a means of validating research in applied mathetmatics -- his example uses differential equations to calculate the form of a suspended telephone wire, e.g. a catenary ] ] Constrast this with the activity of a prototypical "fractal ] geometer", the sort whose books are often seen on bookstore shelves. ] I program a computer to [ create fractal images via typical random ] subdivision method ]. ] Am I to conclude that islands are constructed by some geological ] process of repeated subdivision? Of course not! All I have done is to ] produce a picture which looks (to me) like an island. Are we to conclude that at the subatomic level differential equations represent the structure of the telephone wire? The example you've used (random subdivision) may produce a similar structure, at a gross level, to an island in the same way your differential equation provides a gross level description of the spatial configuration of a suspended telephone wire. I could use the same reasoning to argue that smooth curves (i.e. polynomials) have absolutely nothing to do with nature because nature isn't smooth! Then I could argue that fractal descriptions are better because, while they are not perfect models, are better models than smooth analytic functions. But of course, this would all be meaningless because a model is only good until you find a better one and the process evolves an ever-refining picture (note that the evolutionary development of ideas is sometimes discontinuous: ex. relativity). ] Admittedly, I chose a particularly obnoxious example -- "fractals" ] may indeed be good models of some physical phenomena. Perhaps you should take a look at "fractals" from another perspective. Look at the work done in sting rewriting systems. First, go read Alvy Ray Smith's article[1]. Then, after you've familiarized yourself with the basic notion, take a look at the work that Aristid Lindenmayer has done with string rewriting systems to model growth. Another direction one can go with "fractals" is into the embryonic chaos theory that has been worked out by people like Devaney in his book[2], or start with Gleick's book _Chaos: The Making of a New Science_. You may find out that self-similar curves (in particular, the Cantor Set) have more relevance to the real world than what you thought. I agree that there is alot of people crunching on Mandelbrot set algorithms, etc, without understanding the underlying mathematics and theories. I feel that this comes from the fact that the theories are currently inaccessible to the layman in terms of well-reasoned and practical explanations. Go ahead and TRY and read Devaney's book and you'll see what I mean. We tried to use it here at the University of Utah's CS department and found it woefully painful to read and, more importantly, understand. Perhaps we're not die-hard mathematicians, but we're not stupid either; with out little group we had a good coverage of areas of knowledge that helped us out, but I don't think any of us (there are about 6) would have gotten much by reading the book alone. Mandelbrot's work has been more popularized, but he does tend to bathe in his own "glory" and his books (I found) didn't exhibit a uniform formalism or theoretic explanation; I found them to be useful introductory surveys. Things are improving, though currently the trend is to publish books that show you how to generate all these "fractal" images without understanding the ideas behind them. Personally, I feel it is more useful to study the works of the emerging "chaos theory" and things like Rene' Thom's catastrophe theory (as outlined in [3]). If you're looking for real-world modelling applications using fractal methods, check out the course notes from this year's SIGGRAPH course taught by Prusinkiwiecz (I hope I spelled that right) and Hanan[4] [1] "Plants, Fractals, and Formal Languages", Computer Graphics 18:3, 1984 [2] _An Introduction to Chaotic Dynamical Systems_, 2nd. ed., Robert L. Devaney, Addison-Wesley, 1989 [3] _Structural Stability and Morphogenesis_, Rene' Thom, 197? [4] _Lindenmayer Systems, Fractals and Plants_, P. Prusinkiwiecz & J. Hanan, 1989 ] -Troy Zerr ] University of Wisconsin ] Department of Mathematics ] Madison, Wisconsin ] zerr@math.wisc.edu -- Rich Rich Thomson thomson@cs.utah.edu {bellcore,hplabs,uunet}!utah-cs!thomson "Tyranny, like hell, is not easily conquered; yet we have this consolation with us, that the harder the conflict, the more glorious the triumph. What we obtain too cheap, we esteem too lightly." Thomas Paine, _The Crisis_, Dec. 23rd, 1776
markv@phoenix.Princeton.EDU (Mark T Vandewettering) (11/25/89)
After my last batch of flaming hate mail, I decided not to further stretch my bulging mailbox, but after a Thanksgiving turkey dinner, all seems better, so once more into the breach.... Excuse the confusing quotations here I have to sort them out myself.... In article <1989Nov24.114609.8837@hellgate.utah.edu> thomson@cs.utah.edu (Rich Thomson) writes: >In article <5383@orca.WV.TEK.COM> brucec@demiurge.WV.TEK.COM > (Bruce Cohen) writes: >] Am I to conclude that islands are constructed by some geological >] process of repeated subdivision? Of course not! All I have done is to >] produce a picture which looks (to me) like an island. Here we hit what I believe is the crux of the problem. Typical fractal mountains are not a model of physical terrain. They are a description of terrain. A model implies that some physical process is being estimated in order to produce behavior/appearance similar to something observed. A description is a qualitative "looks like" method. It says nothing about the physical processes which actually cause things like mountains to form. Researchers have noted that forces such as erosion play key roles in shaping the appearance of terrain. Some recent research papers have been written to demonstrate how fractals can be modified with forces such as erosion. These approaches are justifiably models, because they seek to provide a physical explanation for the formation of terrain. One might argue that beginning computer graphics wasn't modeling a scene, because there was little physical theory behind the generation of images. With the advent of radiosity and backward ray tracing, as well as improved surface descriptions, more and more computer graphics seek to model the interplay of light between objects in a scene. >Are we to conclude that at the subatomic level differential equations >represent the structure of the telephone wire? The example you've used >(random subdivision) may produce a similar structure, at a gross level, to >an island in the same way your differential equation provides a gross level >description of the spatial configuration of a suspended telephone wire. There is one small difference: caternaries have been discovered by a sound analysis of forces involved on the wire. The macroscopic properties of the wire are completely the result of relatively few forces, which can be analyzed analytically. No such basis has ever been demonstrated in fractal geometry. We rely on a qualitative and subjective approach (does this look right?) to model natural objects. >Another direction one can go with "fractals" is into the embryonic chaos >theory that has been worked out by people like Devaney in his book[2], or >start with Gleick's book _Chaos: The Making of a New Science_. You may >find out that self-similar curves (in particular, the Cantor Set) have more >relevance to the real world than what you thought. I would not recomment Gleick's book, as I think it causes far more confusion than it clears up. I don't believe that Gleick's understanding of fractals, chaos and non-linear dynamics is quite up to scratch. It was, however, my initial impetus to seek out other books and study the topic more, so it might be valuable in this regard. The bibliography is adequate at the very least. Devaney's book actually grinds into some of the theory, but I agree that it is at times difficult to read. I can't decide whether it is due to poor presentation, organization or what. Still, it seems moderately complete, and proves many of the theorems etc. just hinted at by Gleick. >show you how to generate all these "fractal" images without understanding >the ideas behind them. Personally, I feel it is more useful to study the >works of the emerging "chaos theory" and things like Rene' Thom's >catastrophe theory (as outlined in [3]). If you're looking for real-world >modelling applications using fractal methods, check out the course notes >from this year's SIGGRAPH course taught by Prusinkiwiecz (I hope I spelled >that right) and Hanan[4] Thanks for the bibliography.
fournier@cs.ubc.ca (Alain Fournier) (11/25/89)
It is comforting to see that now and then there are controversies on the net that lead to a thoughtful discussion and do not dissolve into invectives. At least fractals are of some use in this respect. While I have a serious allergy to fractal hype (which is endlessly itch-producing) and think that one of the most curious properties of fractals is to make hitherto reasonable individuals take leave of their senses and dignity, I can't resist some comments on Troy Zerr's article. I am not a mathematician (and I try to avoid playing one on TV), but the definition of Mathematics as something a mathematician does seems to me to be too circular for comfort. More importantly, the nice description of an applied mathematician at work (and I believe that is what they do) leaves us with an interesting unstated conclusion: how does the applied mathematician verify that the curve obtained (presumably a catenary) is a valid model for the wire; why, by looking at it, no less. It is not within the realm of the mathematician (and actually not even of the physicist) to prove that a given equation or set of equations IS the REALITY, only that it fits well the facts from an economical set of assumptions. They just compare it to experience, and looking at things is part of the experiment, especially is shape modelling is the game. Another important point, often overlooked by non-informaticians, is that an algorithm is not an explanation or a model. As a simple example, assume that I want to write a program that simulates the way various objects are submerged by rising water. If my objects are given by their individual positions in 3D space, I have no choice but to sort them by altitude if I want to output, pictorially or otherwise, the order in which they are submerged. I can pick various algorithms to sort, some of the most efficient of which work recursively (most graphics types will bucket-sort, but no matter). In doing so, do I claim or imply that the water find the objects by recursive subdivision? Of course not. By the same token using recursive subdivision to produce "terrain looking" objects does not imply that there is a tectonic phenomenon acting in a similar way. The issue of whether the simulated stochastic process (for instance fractional Brownian motion) is a legitimate model of terrain is another, more difficult, question. How long is the coast of Britain? I think that is an interesting question, and to show that in its naive form it is an ill-posed question is one of the most striking results of Mandelbrot's work. As questions go, I really do not care too much about how many finite simple groups there are, but I understand if some people do. There is no conclusion, except the obvious: nothing is so simple. The beautiful is not always the popular, the popular is not always the trivial, the trivial not always the unimportant, and the unimportant can be sometimes beautiful.
SL195@cc.usu.edu (A banana is not a toy) (11/28/89)
OK. So fractals haven't found a great use YET. But remember- they are a part of chaos. Chaos, when the mathematics behind it are slowly worked out, (admittedly (probably) many year hence) promises techniques to help solve nasty non-linear differential equations. The mathematics behind fractals fits *nicely* into renormalization group theory. I don't think something that "happens to fit" like they do is insignificant. How long was quantum mechanics and relativity scoffed at before it was accepted. -- All comments are my own, and many must be taken with a :-) =============================++++++++++++++++++++============================= | Demetrios Triandafilakos | James Knowles | "Remember, always remember,| | Shire of Cote du Ciel | BITNET: SL195@USU | my son -- a banana is not | | Principality of Artemesia | INTERNET: | a toy." | | Kingdom of Atenvelt | sl195@cc.usu.edu | - The Wise Guru | =============================++++++++++++++++++++============================= Be all that you can be - see your local SCA Knight Marshal now.
markv@phoenix.Princeton.EDU (Mark T Vandewettering) (11/28/89)
In article <119.256E54C5@uscacm.UUCP> Jim.McNamee@p7.f12.n376.z1.FIDONET.ORG (Jim McNamee) writes: > When he was governor of California, the famous 'naturalist' and former U.S.president said, "A rock is a rock. A tree is a tree." Sounds as though the two of you take similar pleasure in the beauty of Nature. Well, gosh guy, considering that I was raised as a country boy in the greenest state I can think of (good ole Oregon), I think I am as much in tune with the beauty of this fair world of ours as the next person. I like trees, and mountains, far more than I like fractals. I also like people who put line breaks in their postings. :-) >Jim McNamee == ...!usceast!uscacm!12.7!Jim.McNamee Mark VandeWettering (prevent run-on thoughts, hit some [returns])
mcdonald@aries.uiuc.edu (Doug McDonald) (11/29/89)
In article <119.256E54C5@uscacm.UUCP> Jim.McNamee@p7.f12.n376.z1.FIDONET.ORG (Jim McNamee) writes: > > Mark T Vandewettering writes: > >"Fractal geometry may be able to generate images which are convincing images > of nature (and even this I might contest) but this in no way indicates > anything about the mechanism that generates such phenomena." > > >"... but as far as helping to understand real phenomena, I have yet to >see convincing arguments that fractals are of any use whatsoever. I > could draw a bunch of lines that looks like a bug, and that doesn't >mean I understand anything about the nature of >bugs." > There are things in nature that are indeed of a fractal nature. The most obvious is the scale of fluctuations in the density of fluids near critical points. The realization that the amount of fluctuation scaled fractally gave rise to the theory that correctly predicted the equations of state, and thence to a Nobel Prize. This is actually an easy experiment: mix equal volumes of water and triethylamine. Watch what happens as you change the temperature through 15 degrees C. It turns a milky while. I do it as a lecture demo. (Don't try this at home folks, at least indoors. Triethylamine stinks something awful and is mildly poisonous.) Doug McDonald
eugene@eos.UUCP (Eugene Miya) (11/29/89)
In article <14203@cc.usu.edu> SL195@cc.usu.edu (A banana is not a toy) writes: >OK. So fractals haven't found a great use YET. Thank you for your honest open thoughts. 8) >of chaos. Chaos, when the mathematics behind it are slowly worked out, >(admittedly (probably) many year hence) promises techniques to help solve >nasty non-linear differential equations. Oh?! Well this will be news 8). Promises? Scouts honor and hope to die? As Bronowski said, "There is a difference between knowledge and certainty." There are no royal roads to mathematics. (guess who said that.) Somehow, I do not think computer graphics is a royal road; my classes in 1977 on catastophe theory have taught me that. Anyway, enough of this. please. Let's move onto other things. Another gross generalization from --eugene miya, NASA Ames Research Center, eugene@aurora.arc.nasa.gov resident cynic at the Rock of Ages Home for Retired Hackers: "You trust the `reply' command with all those different mailers out there?" "If my mail does not reach you, please accept my apology." {ncar,decwrl,hplabs,uunet}!ames!eugene Support the Free Software Foundation (FSF)
chasm@attctc.Dallas.TX.US (Charles Marslett) (11/29/89)
In article <119.256E54C5@uscacm.UUCP>, Jim.McNamee@p7.f12.n376.z1.FIDONET.ORG (Jim McNamee) writes: [in reference to a comment that the author could well live quite happily never seeing another fractal display or mountain image generated by binary subdivision.] > When he was governor of California, the famous 'naturalist' and former U.S. >president said, "A rock is a rock. A tree is a tree." Sounds as though the two >of you take similar pleasure in the beauty of Nature. > Jim McNamee == ...!usceast!uscacm!12.7!Jim.McNamee Actually, I take exception to the implication that nature is really computer generated graphics of any kind -- fractal or otherwise. It seems that he who can't tell the difference between a mathematical construct and real mountains is the one with a limited perception of nature's beauty. Charles Marslett chasm@attctc.dallas.tx.us
roland@cochise (11/30/89)
Allow my 2 pfennige to the discussion on the scienceness of fractals:
In article <3842@puff.cs.wisc.edu> zerr@schaefer (Troy Zerr) writes:
] Am I to conclude that islands are constructed by some geological
] process of repeated subdivision? Of course not! All I have done is to
] produce a picture which looks (to me) like an island.
Have you ever thought of the possibility that fractals might be not a model
of _physical_ processes, but a model of _perceptional_ processes ?
It is quite plausible to assume that human (visual(?)) apperception is
indeed based on repeated subdivision ( it is well known since ancient :-)
times that perceptional measures all are logarithmically scaled ).
So fractals _are_ SL195@cc.usu.edu (A banana is not a toy) (12/01/89)
>>of chaos. Chaos, when the mathematics behind it are slowly worked out, >>(admittedly (probably) many year hence) promises techniques to help solve >>nasty non-linear differential equations. > > Oh?! Well this will be news 8). Promises? Scouts honor and hope to die? > As Bronowski said, "There is a difference between knowledge and certainty." Considering that a nobel-prize material has been brought forth with the help of insights that are gained from chaos, **** Y E S ***** > There are no royal roads to mathematics. (guess who said that.) > Somehow, I do not think computer graphics is a royal road; my classes > in 1977 on catastophe theory have taught me that. What did I ever say in relationship to computer graphics???? I'm talking about *chaos*. Fractals are but a mere subset of chaos. -- All comments are my own, and many must be taken with a :-) =============================++++++++++++++++++++============================= | Demetrios Triandafilakos | James Knowles | "Remember, always remember,| | Shire of Cote du Ciel | BITNET: SL195@USU | my son -- a banana is not | | Principality of Artemesia | INTERNET: | a toy." | | Kingdom of Atenvelt | sl195@cc.usu.edu | - The Wise Guru | =============================++++++++++++++++++++============================= Be all that you can be - see your local SCA Knight Marshal now.
bdb@becker.UUCP (Bruce Becker) (12/04/89)
In article <119.256E54C5@uscacm.UUCP> Jim.McNamee@p7.f12.n376.z1.FIDONET.ORG (Jim McNamee) writes: | | Mark T Vandewettering writes: | |"Fractal geometry may be able to generate images which are convincing images |of nature (and even this I might contest) but this in no way indicates anything |about the mechanism that generates such phenomena." | | In the strictest sense, that's true. But it should be pointed out that the | same criticism must be leveled against Euclidean geometry. Only man and a | few uninteresting natural phenomena rely on that branch of mathematics to | create the world you see. Excellent point, can be easily generalized ... -- ^^ Bruce Becker Toronto, Ont. w \**/ Internet: bdb@becker.UUCP, bruce@gpu.utcs.toronto.edu `/v/-e BitNet: BECKER@HUMBER.BITNET _/ >_ Ceci n'est pas une | - Rene Macwrite
mitchell@cbmvax.commodore.com (Fred Mitchell - PA) (12/27/89)
The study of Fractals is a new science. As with anything new, there will
always be controversy. And perhaps that is one of the greatest intellectual
galvanizers of our time.
Fractals make an interesting study as a 'pure science'. Their application
to nature makes them even more wonderous. It gives you the ability to
classify certain phenomena in natue that would have previously been considered
'formless', such as clouds or mountains. To that end, Fractals have become
one more (and very powerful) tool by which we can measure our world. A sort
of 'yardstick' :-)
As for explaining the mechanism of certain phenomena, it can give fantastic
insights. For example, the development of the human embryo (and embryos in
general)- we can begin to understand how a HUGE amount of information
can be encoded by such a small dataset (DNA). While fractals is not the
end-all and be-all in this case, it gives hints as to the nature of the
fantastic and arcane mechanism involved. Only hints, mind you! :-)
Calculus has had centuries to become developed, as had other areas of
mathematics. Fractals (and the study of chaos, in general) is cumbersome
and damn-near impossible without the aid of the computer (would you spend
half a lifetime calculating a single instance of the Mandelbrot set, say
1K by !k? :-). Computers, in their speedy and compact present form, is itself
a very recent phenomenom- so lets give ourselves TIME to learn
about this mystery we've uncovered!
-Mitchell
mitchell@cbmvax.UUCP
To Life, Immortal.rick@hanauma.stanford.edu (Richard Ottolini) (12/31/89)
In article <9144@cbmvax.commodore.com> mitchell@cbmvax.commodore.com (Fred Mitchell - PA) writes: >As for explaining the mechanism of certain phenomena, it can give fantastic >insights. For example, the development of the human embryo (and embryos in >general)- we can begin to understand how a HUGE amount of information >can be encoded by such a small dataset (DNA). B.S. Fractals, if a valid measure of nature, say how LITTLE information there is. Compilicated appearing patterns actually can be parameterized by very few numbers, hence its attractiveness to explanation and information compression. Mitchell appears to be jumbling several types of "new age" mathematics-- complexity theory, chaos theory, fractals ...-- each which has precise and different definitions and something different to say about nature. Some may provide USEFUL results and become parts of the scientist's toolkit, while others will remain mathematical amusements.
siona@ucscb.UCSC.EDU (siona) (12/31/89)
A fractal is merely a geometrical object generated through a recursive algorithm. As a concept, it is comparable in generality to that of geometrical objects generated through iterative algorithms, which forms the whole field of differential geometry. I believe that a very large class of forms can be generated by both recursive and iterative algorithms; the difference lies in how much information is necessary to specify the algorithm. We currently have a certain amount of prejudice towards the differential approach, because we have a three hundred year tradition of differential calculus, and most of physics is expressed in differential equations. However, we are beginning to develop mathematical models in such fields as biology, where a recursive description is often much simpler. -- Deirdre Des Jardins Internet: ucscb!siona@ucsc.UCSC.EDU
fournier@cs.ubc.ca (Alain Fournier) (01/02/90)
In article <10148@saturn.ucsc.edu> siona@ucscb.UCSC.EDU (siona) writes: > >A fractal is merely a geometrical object generated through a recursive >algorithm.... The properties of objects and of the algorithms used to generate them can be quite different. Defining a fractal as an object generated by recursion is wrong (even though Mandelbrot himself came close to this). It is enough to consider that a straight line segment can be generated by a recursive algorithm (and to call it fractal would annihilate the class of non-fractals) and that a Koch island can be easily computed and plotted by iteration (the fact that it won't draw all the points in any finite time is not an argument against the iterative method, since the recursive one has the same problem).
bseeg@spectra.COM (Bob Seegmiller) (01/04/90)
In article <9144@cbmvax.commodore.com> mitchell@cbmvax.commodore.com (Fred Mitchell - PA) writes: > [misc deleted] > >........- we can begin to understand how a HUGE amount of information >can be encoded by such a small dataset (DNA). Small dataset? Human DNA? Not hardly. I don't believe fractals have much to do with information storage in DNA, either. I'm curious where he gets this idea, as I haven't seen it in any of periodicals (SciAm,Science News) I read. I agree with a previous response in re fractals as being "small", and applied in recursively defined representations. I'm not familiar with any structures or systems (both physical -- a.k.a. tissues/organs) in living organisms that can be fractally described. (Please note emphasis on physical -- I'm not addressing neural organization, or behaviors of any of such systems -- that belongs in another news group -- simply what can be genetically constructed from DNA.) (Of course, I expect some will take this inch and stretch it a mile -- sigh.) > [misc deleted] > > -Mitchell > mitchell@cbmvax.UUCP > To Life, Immortal. Sorry to post this to comp.graphics, but I felt it had to be answered, and had waited for someone else to before I stepped in.
mitchell@cbmvax.commodore.com (Fred Mitchell - PA) (01/05/90)
In article <6780@lindy.Stanford.EDU> rick@hanauma.UUCP (Richard Ottolini) writes: >In article <9144@cbmvax.commodore.com> mitchell@cbmvax.commodore.com (Fred Mitchell - PA) writes: >>As for explaining the mechanism of certain phenomena, it can give fantastic >>insights. For example, the development of the human embryo (and embryos in >>general)- we can begin to understand how a HUGE amount of information >>can be encoded by such a small dataset (DNA). >B.S. Cough. >Fractals, if a valid measure of nature, say how LITTLE information >there is. Compilicated appearing patterns actually can be parameterized >by very few numbers, hence its attractiveness to explanation and information >compression. Seems like a matter of semantics, to me. The old "Half full" or "Half empty" approach. >Mitchell appears to be jumbling several types of "new age" mathematics-- >complexity theory, chaos theory, fractals ...-- each which has precise and >different definitions and something different to say about nature. The "jumbling" as you call it, was intentional. I was not trying to say anything specific- just that we should keep an open mind and look for relationships where we normally wouldn't. >Some may provide USEFUL results and become parts of the scientist's toolkit, >while others will remain mathematical amusements. It depends on your orientation. If you want to be analytical, sure, then its a matter of what TOOL you can apply to what specific problem. But one should also be able to take a couple of steps back and see the whole picture. What is the gist of what I am trying to say? Therein lies my message. I was trying to elict an appreciation for the BEAUTY of what we have to date and where they might take us. Unfourtunatly, some are unable to grasp that. -Mitchell mitchell@cbmvax.UUCP "The eyes are open, the mouth moves, but Mr. Brain has long since departed." - The Black Adder
platt@ndla.UUCP (Daniel E. Platt) (01/06/90)
In article <298@spectra.COM>, bseeg@spectra.COM (Bob Seegmiller) writes: > ... I'm not familiar with any structures or systems (both > physical -- a.k.a. tissues/organs) in living organisms that can be > fractally described. Retinal Vessels are close to being DLA which are fractal... most blood vessels have fractal character to them. Look up Physica D 38, 98 (1989) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- || 1(914)945-1173 || || Dan Platt 1(914)941-2474 || || Watson (IBM) PLATT@YKTVMV.BITNET || || ..!uunet!bywater!scifi!ndla!platt || || || || The opinions expressed here do not necessarily reflect || || those of my employer! || || || -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
eugene@eos.UUCP (Eugene Miya) (01/06/90)
In article <9215@cbmvax.commodore.com> mitchell@cbmvax.commodore.com (Fred Mitchell - PA) writes: >In article <6780@lindy.Stanford.EDU> rick@hanauma.UUCP (Richard Ottolini) writes: >>In article <9144@cbmvax.commodore.com> mitchell@cbmvax.commodore.com (Fred Mitchell - PA) writes: >>>As for explaining the mechanism of certain phenomena, it can give fantastic >>>insights. >>B.S. >Cough. I side with Rick on this one. >>Fractals, if a valid measure of nature, say how LITTLE information >>there is. >Seems like a matter of semantics, to me. The old "Half full" or "Half empty" >approach. I've used this argument before. Okay then let's say we are at 0.5 of capacity. It is not a matter of semantics in this case. See below. >>Mitchell appears to be jumbling several types of "new age" mathematics-- >>complexity theory, chaos theory, fractals >The "jumbling" as you call it, was intentional. I was not trying to say >anything specific- just that we should keep an open mind and look for >relationships where we normally wouldn't. The scientific community has historically been built upon skepticism. This can be hard on those not used to it (i.e., don't take any of this too personally [retorically]). It does a little observation (except in CS 8), proposes theory, but its key is TESTING, experimentation, and recently application. Some pseudo sciences, pre-sciences, etc. never get past the earliest stages. The sciences can keep an open mind; its very structure is self correcting. That does not mean that the bureacracy of science is wrong, but it does try to keep "on track." Witness the two directions cold fusion and warm superconductors: limited work versus HS students running to duplicate, now quiet applications being studied and sought. >>Some may provide USEFUL results and become parts of the scientist's toolkit, ^^^ and equally may not >>while others will remain mathematical amusements. ^^^^^^^^^ >It depends on your orientation. If you want to be analytical, sure, then its yes, science tends to be ^^^^^^^^^^ this is called "reductionism," seems to work well >a matter of what TOOL you can apply to what specific problem. But one should >also be able to take a couple of steps back and see the whole picture. What >is the gist of what I am trying to say? Therein lies my message. >I was trying to elict an appreciation for the BEAUTY of what we have to date ^^^^ >and where they might take us. Unfourtunatly, some are unable to grasp that. This is a matter of individual interpretation, and some set of aesthetic values. There is a concept of scientific elegance, but you are speaking of something different here. I refer you to James Burke's closing comment in his Connections series that of how science and technology work versus how the arts and humanities work. I was so impressed by that statement I contacted Lynn White of the UCLA History Dept who gave me Burke's address and number (fan communications). Individual interpretation has great limits (but noble), but knowledge which all can use is more powerful. No, Rick is able to grasp what you are saying (having conversed with him in person). The problem is that people in science tend to get a bit weary of fads. This is getting away from graphics, I leave it to poster to edit the Newsgroups line to some group like sci.edu or sci.math. Another gross generalization from --eugene miya, NASA Ames Research Center, eugene@aurora.arc.nasa.gov resident cynic at the Rock of Ages Home for Retired Hackers: "You trust the `reply' command with all those different mailers out there?" "If my mail does not reach you, please accept my apology." {ncar,decwrl,hplabs,uunet}!ames!eugene
bseeg@spectra.COM (Bob Seegmiller) (01/06/90)
>I'm not familiar with any structures or systems (both >physical -- a.k.a. tissues/organs) in living organisms that can be >fractally described. (Please note emphasis on physical -- I'm not >addressing neural organization, or behaviors of any of such systems -- >that belongs in another news group -- simply what can be genetically >constructed from DNA.) Before too much time goes by, I need to apologize to this forum: both someone at longs.lance.colostate.edu and Kevin_P_McCarty@cup.portal.com have pointed out (one referring to Mandelbrot's book) that both the circulatory and pulmonary system have fractal characteristics. I should have limited myself to Mr. Mitchell's statement on possible fractal encoding of genetic information and that the human genome was small. In the interim, I asked a local (UCSD) researcher about the size of the human genome: offhand number he gave me was 10^14 base pairs, but don't quote him (so I haven't). I'd thought it was big (> 10^9), but ... >whew< 'Nuff said. -- /---------------------------------------+-------------------------------------\ | Bob Seegmiller <The usual disclaimer> | ..................... | | "Till mermaids wake us..." - Eliot | ..................... | | UUCP: ...!nosc!spectra!bseeg | ..................... |