FNKWP@ALASKA.BITNET (02/08/90)
Homer Smith commented that Brooks and Matelski were 'on the verge'
with regard to the M-set. They got a bit farther than the verge, I think.
In a paper by them (The Dynamics of 2-Generator Subgroups of PSL(2,C), pub-
lished in Riemann Surfaces and Related Topics, Proceedings of the 1978
Stony Brook Conference, Princeton University Press, 1980) there is a clearly
recognizeable text-mode image of the M-set (page 71) composed of asterisks.
It seems (based on the information I have on hand) that Brooks/
Matelski published the first image of the M-set. It also seems clear that
Mandelbrot was the first person to try magnifying the set, and thus discover
its fractal nature. I do not know of any subsequent papers by Brooks/Matelski
on this topic--but I do not pretend to know the mathematics literature!
If it comes to controversy over discovery, in which claims of priority
are often given top ranking, we should all remember one thing--the interest
of most of us on Frac-L in the M-set derives from Dewdney's article in the
Aug '85 Scientific American, and thus (via Hubbard) from Mandelbrot's work.
We are Mandelbrot's intellectual heirs. Which is not to deny credit where
credit is due for the first image of the M-set--but nor should credit be denied
to the person who actually _explored_ the M-set, and then informed us of his
discoveries.
Ken Philip <fnkwp@alaska>S20081@MITVMA.BITNET (anthony taylor) (02/09/90)
what is does the acronym PSL stand for?
scott%tekcrl.labs.tek.com@RELAY.CS.NET (02/09/90)
> what is does the acronym PSL stand for?
It stands for "projective special linear".
The "special linear" group SL(2,C) consists of 2x2 matrices with complex
entries whose determinant is 1. GL(2,C) ("general linear") is the group
of 2x2 nonsingular complex matrices.
The projective groups are certain subgroups of the linear groups whose
elements are equivalence classes of matrices. An element of PGL(2,C)
is a set of matrices {M*c1 : for M in GL(2,C), c in C, and 1 the identity
matrix}. An element of PSL(2,C) is a set of matrices
{M*s1 : for M in SL(2,C) and s in C with s*s = 1}.
PSL(2,C) is (I believe) a "simple" group, meaning it can't be decomposed
into smaller groups in certain ways. Simple groups are analogous to prime
numbers, in that they're the basic building blocks for constructing other
groups.
For more info, check books on algebra and group theory.
-- Scott HuddlestonJTUCKER@NAS.BITNET (John Tucker) (02/09/90)
*cc: John Tucker The following is quoted from a `reply' essay (to a review essay by Steven G. Krantz of two books on fractals that immediately precedes it), entitled "Some `Facts' That Evaporate Upon Examination", written by Mandelbrot. (It appears in the Fall 1989 issue of THE MATHEMATICAL INTELLIGENCER, Vol. 11, No. 4, pp. 17-19): THE BROOKS-MATELSKI REFERENCE. Krantz tells us "the `Mandelbrot set' was NOT invented by Mandelbrot but occurs explicitly in the literature a couple of years before the term `Mandelbrot set' was coined." As is well known, I "invented" that set in 1979-80 and fully published it in 1980 (in a widely quoted paper in ANNALS OF THE NY ACADEMY OF SCIENCES). Indeed, this happened "a couple of years" before 1982, when the term was coined by A. Douady and J. H. Hubbard. .... ..., my thoughts "were provoked" over months and years, helped by long practice with analogous situations elsewhere. During 1979 and 1980, I lectured on [the Mandelbrot set] M at Harvard, M.I.T., an A.M.S. summer meeting on W. Thurston's work, and (most important, perhaps) at Orsay (Paris-Sud) and Bures (I.H.E.S.). There I spent many hours and many meals describing M to Douady in as great detail as requested. Our discussion made him drop what he had been doing before, and he and Hubbard have made major contributions to iteration theory. Subsequently I spoke at D. Sullivan's CUNY Seminar and at Princeton to J. Milnor and W. Thurston. The set in question was not credited to anyone else at the time. ... STEVEN G. KRANTZ REPLIES The ideas of Brooks and Matelski were presented at a conference in 1978, and well pre-date any contribution that Mandelbrot may have made in 1979-80. [Please note that the Brooks-Matelski paper was presented at a 1978 Stony Brook conference on Riemann Surfaces and Related Topics, and was published in the ANNALS OF MATH., STUDIES 97, Princeton University Press (1981). John R. Tucker, Staff Officer Board on Mathematical Sciences National Academy of Sciences Washington, D.C.]