Conference rusure::math

     Typos -- those should be 35's in the bottom row, not
     25's.  If n is a prime, then n will divide every element
     (except the 1's) of the row 1 n ... n 1 (the nth row).
     
     :-) Prove it.  What about the converse?
     
     Dan

    Re: .1
    
    My face is red!  But the point is still clear.
    
    Martin H.

    I have a dumb question.  What is meant by fractal in the base note?
    The figure shown can really only have the discrete topology for
    calculation of the Hausdorff dimension but putting a metric on this
    toplogy leads to infinite Hausdorff dimension (I think).  Or perhaps
    the definition is weaker in the base note and admitts any totally
    disconnected set?

    Any chance of somebody plotting this "fractal" on a smaller scale
    than Martin's ASCII entry on .0?  Maybe that will provide some insight.
    
    Note that this is in some respects the most natural number theory
    geometric figure, since you can generate it by taking
    
    		... 0  0  0  1  0  0  0  ...
    
    and pairwise XORing entries to get the next row, etc.  A bare-bones start.

      John

    I think there is something about this in an recent issue of
    "Mathematical Intelligencer" magazine, either it's the Jan. or the
    April issue, but it is definitely this year's.
    
    I've only flipped through the pages and didn't really read the article
    but I remember seeing plots of the Triangle mod 2 and other related
    stuffs. The author also discuss similar properties of some other
    number-theoretic functions.
    
    						David Ng

    re .0
    Do not mean to be rude here, but what is the big deal and what does
    fractal dimension have anything to do with it?
    
    Eugene

    re .6
    
    Well actually, when anyone says something is a fractal, I wonder
    what is meant because until recently I had no idea what was meant
    for the simple reason that no one I asked, including mathematicians,
    could tell me what a fractal set was.  I understand better now why
    no one answers the question but in the case of a set of discrete points
    I don't see how the usual understanding of fractal applies.  Now
    here is how I understand fractal sets:
    
    	a set whose Hausdorff dimension is strictly greater than its
    	topological dimension (Mandelbrot proposed this one);
    
    	a set having fractional Hausdorff dimension;
    
    	any highly irregular set e.g. some continuous nowhere
        differentiable curves might fit.
    
    The simplest example of a fractal set I know of is the Cantor set.
    The well known plots I have seen I only just found out are much
    more than the fractal - the colored part seems not to be the fractal
    set but rather the boundary curve is the fractal.
    
    The understanding of fractals I have is very slight based on the
    three notions above (which are the basic ideas found in "The Geometry
    of Fractal Sets" by Falconer) and there may be a satisfactory
    definition which still has not surfaced.  People still point to
    this and that and say it is fractal.  The question is, how can I
    look at something and tell whether it is fractal or not.  Is Peano's
    space filling curve fractal for example? (I bet it is!)
    
    CWS                                       

    Re: .3 and .7
    
    The reason I called it a fractal was it's resemblence to the 
    Sierpinski Arrowhead curve on page 142 of the Fractal Geometry
    of Nature.
    
    Re: .6
    
    I think it's interesting and I thought I would bring i up
    in this notesfile.  If you think it's trivial oh well.
    
    Martin H.

    re .8      
    
    I think it's interesting also.  I also eventually will find out in
    addition to the formal definition what people mean by fractals.
    
    But even if I thought it was trivial (which I don't because there
    could easily lurk something deeper in it), there are many "trivial"
    but quite beautiful things in mathematics.
    
    By the way, in the theory of design of experiments, there are Pascal
    triangles associated with constructions of finite geometries which
    are very likely to be systematic like your example. I will be
    interested to know if these Pascal triangles generate the same pattern.
    
    CWS

    I agree that mathematics needs a more formal definition of the
    term fractal. A good informal definition is an object which is
    composed of components which strongly resemble the parent.  By
    this definition fractals are not limited to curves on a plane,
    plants and landscapes have a strong fractal characteristic.
    
    About a year ago the IEEE Computer Graphics journal had an object
    called the sphereflake.  It was composed of a silver sphere which
    had smaller spheres floating around it.  Each smaller sphere had an
    identical arangment of children.  Each sphere was reflective and
    the whole object was a ray tracing nightmare.
    
    Martin H.

    re .8 .9
    I do not want to say that it is trivial.  What I am trying to say
    is that I miss the mathematical significance of that 1/0 Pascal
    triangle.  After a lot of effort, Gauss said that Fermat's last
    problem was no big deal because he could easily come up with problems
    that are just as difficult.  (Of course, Gauss was wrong. :-)  But what
    Gauss meant was that there are a lot of neat and difficult concepts
    in mathematics, but unless there is mathematical significance (whatever
    that definition is), it is no big deal.  What I mean in my original
    post is that I miss the mathmatical significance.
    
    Eugene
                    

    re. 10
    
    But the path of a particle in brownian motion is fractal according
    to Falconer in "The Geometry of Fractal Sets."  In fact, although
    self similarity exists in a few examples of fractal sets, there
    are known examples which have no self similarity such the one just
    mentioned.
    
    CWS 

    re .0
    
    Just in case you wanted to know...
    
    the number of odd binomial coefficients C(n,r) where n<2^k is 3^k
    
    the total number of binomial coefficients above is 2^(2k-1)+2^(k-1)
    
    (Consequently the density of 1s in the triangle in .0 is asymptotically
     0. Notwithstanding this there are infinitely many rows that are all
     1s.)