From: ROGER BAGULA Subject: Re: Relevance of the "Mengersponge"(= "Mengerschwamm")?? Date: Tue, 07 Mar 2000 21:23:06 GMT Newsgroups: sci.math,sci.fractals Summary: [missing] [MIME wrappers deleted --djr] The Menger cube is the 3d version of the Sierpinski carpet. Sierpinski sets are one of the universal symmetries of mathematics found from topology to orthogonal function theory: http://www.geocities.com/ResearchTriangle/Thinktank/7279/sier_space.html http://www.geocities.com/ResearchTriangle/Thinktank/7279/edgesetsier_.html http://www.geocities.com/ResearchTriangle/Thinktank/7279/2artcomb.html http://www.geocities.com/ResearchTriangle/Thinktank/7279/gen_siers_pgs.html http://www.geocities.com/ResearchTriangle/Thinktank/7279/pascal2nd.html You can also learn about dimensional spectra and prime numbers from the study of Sierpinski space and it's sets: http://www.crosswinds.net/~translight/sier_dirichlet.html There is a Sierpinski carpet 3d pyramid and it would seem that a Buckyball six sided 3d Sierpinski set is very likely as well. Like the Platonic solids of old, these are the fractal Platonic solids! Anyone minimizing their importance is showing his lack of understanding of mathematics. You asked for it! Wilhelm Sternemann wrote: > Hallo! > Can You agree with the following statements about the Mengersponge = > Menger's universal curve?: > In the study of mathematics you don't meet the Mengersponge (even > not in a lecture of Topology!) > It is an exotical object without importance for learning mathematics. > The concept of an universal curve has no applications. The discussion of > it begins and ends with the question of its existence. Its special > properties - for example: all point is ramification-point - are too > complex, to be of much interest. > Its existence is äquivalent to that of the Cantordust (Cantor`s > diskontinuum). > The Cantordust meets every student of mathematics. The special > properties are less complex and more interesting than the Mengersponge. > Its new role as a Fractal since Mandelbrot is total disparate from > its classical mathematical topological relevance. > The figures of iteration of the Mengersponge are very fascinationg as > threedimensional figures and could be used more in the mathematical > education. > > Please contradict to one or more of the statements, or can you find more > arguments and facts about or against the relevance of the mengersponge! > > Thank's > Wilhelm Sternemann -- Respectfully, Roger L. Bagula tftn@earthlink.net [lengthy sig deleted --djr] > Hadamard Self Similarity, Polynomials, Klein Groups > > and Banach Space > > by R. L. Bagula 17 Oct 1998 (C) > > In reading up on Banach space I ran into The Caratheodory-Fejer theorem and > Schur `s matrix operator mechanics. In it I saw the infinite version of the > matrix self similar Hadamard or Pascal modulo two matrix. I had been doing > work on getting a set of polynomials from the different scales of Hadamard > matrices and realized that a kind of orthogonality was involved. This article will > deal with that work and a speculation that results from it. > > I'll first go into the Hadamard matrices and the polynomials that result. The > first Hadamard matrix is: > > 1) H1={{0,1},{1,1}} > > And the polynomial that results is from the matrix: > > 2) M1={{x,1},{1,1+x}} > > which gives the polynomial: > > 3) P1(x)= -1+x+x^2 > > which has the golden mean and it's conjugate as roots. This shows a > resemblance to the Bonacci polynomial from Fibonacci series theory. The form > of the second matrix is: > > 4) H2={{0,H1},{H1,H1}} > > in the matrix self similarity manner of producing power of two larger matrices. > Adding a diagonal vector of x's gives M2 which then produces the polynomial: > > 5) P2(x)=1+x-4*x^2+x^3+x^4 > > The Eigenvalues that result are: > > 6) R2={-1,-1,(3+sqr(5))/2,(3-sqr(5))/2} > > When the same mechanics produces the third matrix: > > 7) H3={{0,H2},{H2,H2}} > > the polynomial: > > 8) P3(x)=1-x-13*x^2-8*x^3+20*x^4-13*x^6+x^7+x^8 > > This part was produced and checked on Mathematica and produced > Eigenvalues: > > 9) R3={(-1+sqr(5))/2,(-1+sqr(5))/2,(-1+sqr(5))/2,(-1-sqr(5))/2,(-1-sqr(5))/2, > > (-1-sqr(5))/2,(4+2*sqr(5))/2,(4-2*sqr(5))/2} > > The determinants are: > > 10) detH1=-1 > > 11) detH2=1 > > 12) detH3=1 > > The matrix self similarity operation like a continued fraction can be carried on > indefinitely or to an infinite limit. The resulting rows or columns can be used a > orthogonal basis vectors for an infinite dimensional Linear Algebra. We'll call > that matrix Hn as being 2^n by 2^n. > > Suppose that we have a Fourier basis set of: > > 13) vn={ exp(0),exp(i*t), exp(i*2*t),....,exp(i*2^n*t)} > > Then the new basis set produced from vn and Hn is: > > 14) vn'=Hn*vn > > By the rules of Linear Algebras the new basis set is also an orthogonal vector > basis set. Thus, we have a Hadamard Hilbert space and a Banach space built on > that. The resulting are best taken in the version based on: > > 15) H1'={{1,0},{1,1}} > > than the original one which was used for the polynomial: there are actually > four equivalent rotations of the matrix. The first few such vectors are: > > 16) v1=exp(0) > > 17) v2=exp(0)+exp(i*t) > > 18) v3=exp(0)+exp(i*2*t) > > 19) v3=exp(0)+exp(i*t)+exp(i*2*t)+exp(i*3*t) > > 20) v4=exp(0)+exp(i*4*t) > > As is well known, these vectors are equivalent to polynomials as well. In the > Banach space polynomial version: > > 16) v1=1 > > 17) v2=1+z > > 18) v3=1+z^2 > > 19) v3=1+z+z^2+z^3 > > 20) v4=1+z^4 > > I produced these from H3, but there is an infinite sequence of each of these > vectors. This is the kind of operator algebra that von Neuman visualized and > pioneered in the 1920's for quantum mechanics. It is this very orthonormal > vector set of polynomials that form the Sierpinski space first level set and it's > universality. This is the Sierpinski gasket class or Modulo 2 class. The Sierpinski > carpet or Modulo 3 class is built in a very similar manner with self similar > matrices as I have demonstrated in Pascal's triangles from sums. The infinite > Modulo n class is just the raw binomial polynomials. The matrix self similar > procedure at different modulo levels defines Sierpinski space. > > So far we have just real coefficients in the generator matrices. Since we are > actually dealing with complex functions, it should be possible to have complex > matrices as well. An example of this seems to be the Gray code version of > Pascal's triangle that Gary Adamson has produced and I have verified. In Galois > field terms the multiplication or ``and'' part is Pascal/Hadamard like and the > addition or ``xor'' part is Gray code like. They are real and complex parts of a > field structure to Sierpinski space. > > I came up with a conversion method for the n-ary case to get from one matrix > to the other: > > 21) v(i)'=mod(Max(v(i),v(i+1)),n) > > That produces an ``Xor''-like matrix from the Hadamard vectors using a fuzzy > logic operation. I haven't been able to check this method against M. C. Er's > n-ary Gray code method, but this rule is consistent with the binary rule and > works at the trinary level experimentally to produce a very Gray code pattern. > > So by now I have covered more than everything I said I would in the title > except Klein groups. Klein groups are a very strange type of group to those of > us used to linear groups. How can we get analytic functions from these bilinear > transform based groups? The best clue is found in the factorization of Banach > space functions using the Blaschke product . A Klein group analytic function can > be made into the Blaschke product form: > > 22) f(z)=(a(n)'/|a(n)|)*(a(n)-z)/(1-a(n)'*z) > > where a(n)' is the complex conjugate of a(n). If we let the Klein group be: > > 23) z'=(a1(n)*z+b1(n))/(c1(n)*z+d1(n)) > > Then, the solution for a(n) is: ( the character of the Klein group element) > > 24)a(n)=(c1(n)*b1(n)-a1(n)*d1(n))/c1(n)^2 > > This result comes from the realization all bilinears can be expressed as a form > like: > > 25) z'=a/(z+c)+b=(b*z+a+b*c)/(z+c) > > 26) detM=det{{b,a+b*c},{1,c}}=a > > Solution for this ``a'' give 24) which in turn allows us to express Klein groups > as analytic functions of the Blaschke product form. Since the Schur type of > functions are based on a Blaschke product form, a Banach space based on > vectors from these functions can be conceived. It was my thought that much as > we made infinite matrices in the Sierpinski space case above from 2by2 seeds, > we could use blocks of the Klein group matrices to produce infinite matrices > that have all the character of the original group. If we have a Klein four group > with a matrix like: > > 27) K10={{(a1(1)*z+b1(1))/(c1(1)*z+d1(1)),a1(2)*z+b1(2))/(c1(2)*z+d1(2))}, > (a1(3)*z+b1(3))/(c1(3)*z+d1(3)),a1(4)*z+b1(4))/(c1(4)*z+d1(4))}} > > If we let the number of a(n) from 24) represent these groups the 2 by 2 matrix > is: > > 28) K1={{a(1),a(2)},{a(3),a(4)}} > > and the first matrix self similar step is: > > 29) K2= {{a(1)*K1,a(2)*K1},{a(3)*K1,a(4)*K1}} > > in the four block form that we have been using with the Hadamard matrices. > The analytic Banach space basis vectors are: > > 30) vkn=Kn*vn > > I will call the resulting new kind of Banach space: Mumford space for the man > whose work introduced the world to Klein groups. This development is a > consistent algebra and forms a speculation about the nature of Klein groups on > the complex plane. > > So I have set Sierpinski sets firmly in functional space as a form of Banach > space. I have further generalized this procedure so as to produce analytic > functions for Klein groups in Banach space as well. I haven't been able to read > all the literature on Banach spaces: it wouldn't surprise me at all if I were just > rediscovering some genius' work that wasn't recognized for it's importance. If > so, I can only say I am doing my best to develop and follow the logical train of > the functions that I do know and I have made an effort to search out such > results in publications I can get my hands on here. To get this far has been very > much an up hill battle and the fight is far from done. The ``disjointness'' and > ``completeness'' of the Klein group functions has to be important. The integral/ > differential structure of both Sierpinski and Mumford spaces also has to be > explored. A new beginning for fractal theory is expressed in this short article. > > > > > > Last Modified: 17 October 1998 by R. L. Bagulaİ [HTML and other pointless additions deleted --djr] ============================================================================== From: Wilhelm Sternemann Subject: Re: Relevance of the "Mengersponge"(= "Mengerschwamm")?? Date: Thu, 09 Mar 2000 23:31:26 +0100 Newsgroups: sci.math,sci.fractals Hallo Mr Bagula Thank You for your answer! I will try to read and work in your sides! ROGER BAGULA wrote: > The Menger cube is the 3d version of the Sierpinski carpet. There exist more than one 3d version! Let I = [0; 1] and C be the one-dimensional Cantor dust You can take the 3d Cantor Dust CxCxC, the Menger cube (CxCxI)^(CxIxC)^(IxCxC) and the 3d Cantor foam (CxIxI)^(IxIxC)^(IxCxI) (where "^" means "intesection")! [deletia --djr]