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Different Models

See also:  -[zix teaching methods]- (teaching folder)
    
On this Page:  {Intro}
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               {Rational Fractals}
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Art and Maths

  plus.maths.org
     http://plus.maths.org/issue33/features/dartnell_art/index.html
     http://plus.maths.org/issue33/features/dartnell_motion/index-gifd.html
               (stealth things)

  Like Sand thru the hour glass
   http://plus.maths.org/latestnews/jan-apr08/sand/index.html

   PlanetPerplex.org
     http://www.planetperplex.com/

   
               
 



             
 



  Octorians/Octonions

  -[SOL-3 web: Corvalis]-

   http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/octonion/index.htm

   http://math.ucr.edu/home/baez/octonions/
         http://pass.maths.org.uk/issue32/features/baez/
         http://plus.maths.org/issue33/features/baez/index.html


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Rational Fractals

google:  "rational fractals"
  
  -[Fractal Domains - Macintosh grapher]-
  
  -[Bagula's article]-
       bagula-C-source-rational-fractals.txt
  
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  on the intersection beiser, etc curves
    http://advogato.org/person/nomis/

   Fractal 97 abstract on  Rational fractals (sort of)
          http://www.emergentis.com/abstr97.html
    
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Tree Formalism for Fractal Description 
S. Duval and M. Tajine 

We study a new notion of fractals based on the embedding of trees in a metric space (E, d). Actually, if R is a tree over an alphabet with arity (F, rho) then we associate to each symbol f of F a contracting mapping if rho(f)>0 and a compact of E otherwise. The embedding of R is obtained by the union of embeddings of its branches. To embed an infinite branch, we compose the mappings along it. We obtain, in this case, a point which belongs to E. To embed a finite branch, we successively apply the mappings along the branch to the leaf associated compact. The modeling of trees is done by using grammars of any order. The grammars of order 0, 1 and 2 generate respectively rational, algebraic, and functional fractals. More generally, a n-fractal is generated by a grammar of order n. This allows to classify the fractals according to the grammar order used. The rational fractals contain the IFS generated fractals. We can describe the high level operations most often found in object modeling. That way, given two fractals G and H, we model G u H, G x H, G + H and h(G) where x represents the cartesian product, + represents the sum of Minkowski and h is a homogeneous function in E. We are also able to describe the geometric simplicial complexes. The fractal geometry introduced here, is closely linked to discrete geometry in the sense that the model contains its own discretization. In fact, if a fractal G is the embedding of a tree R and R' is a tree obtained from R by truncation, then the embedding of R' is a discretization of G. It should be noticed that for the modeling of a segment, an adequate truncation permits to find the Bresenham's discretization. In the particular case of rational fractals, it is possible to effectively calculate the Hausdorff dimension. If G is a fractal obtained by the embedding of a rational tree R then the Hausdorff dimension of G is a function of the convergence radius of the generative series corresponding to R. 

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