Cantor set plot, Dirichlet function plot

*To*: mathgroup at smc.vnet.net*Subject*: [mg23579] Cantor set plot, Dirichlet function plot*From*: David Ong <do226 at is2.nyu.edu>*Date*: Sat, 20 May 2000 17:44:34 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Thanks Andrzej, The code for the Dirichlet like function plot wasn't quite what I had expected. However, I am glad to have another method to deal with these sorts of functions. Here is some code from Gabriel Chjnevert's "Ploting Monster's of Real variables", which I got from math source. Now I am looking for a simple way to plot an approximation to the Cantor set and would welcome suggestions. n = 150; points = Union @@ Table[{p/q, 1/Denominator[p/q]}, {q, n}, {p, q - 1}]; ListPlot[points, PlotRange -> {0, .51}] David On Sun, 21 May 2000, Andrzej Kozlowski wrote: > Sorry, the definition should of course have been: > > f[x_] := If[IntegerQ[Numerator[Rationalize[N[x]]]], > 1/Denominator[Rationalize[x]], 0] > > on 00.5.21 0:07 AM, Andrzej Kozlowski at andrzej at tuins.ac.jp wrote: > > > This is of course a somewhat different matter. One can define a reasonable > > "simulation" of this function and in principle plot its graph but I think it > > will be hard to see anything interesting. The definition is simple enough: > > > > f[x_] := If[IntegerQ[Numerator[Rationalize[N[x]]]], > > Denominator[Rationalize[x]], 0] > > > > f is a good approximation to what you want: > > > > In[3]:= > > f[Pi] > > > > Out[3]= > > 0 > > > > In[4]:= > > f[1.4] > > > > Out[4]= > > 5 > > > > In[5]:= > > f[Sqrt[2]] > > > > Out[5]= > > 0 > > > > Unfortunately it is hard to get a meaningful graph. You could try using > > ListPlot and do something like: > > > > In[6]:= > > l1 = Table[{x, f[x]}, {x, 3.0, Pi + 0.1, 0.001}]; > > > > In[7]:= > > l2 = Table[{x, f[x]}, {x, Pi - 0.14, Pi + 0.1, 0.001}]; > > > > In[8]:= > > ListPlot[Union[l1, l2]] > > > > But the result does not seem t me to be very instructive. In principle I think > > what I wrote in my first reply still holds and I do not think computers are > > suitable tools for investigating this sort of phenomena. > > > > on 00.5.20 11:02 PM, David Ong at do226 at is2.nyu.edu wrote: > > > >> > >> Sorry, I misstated the function. > >> f(x)=1/q when x is an element of p/q in lowest terms and f(x)=0 > >> otherwise. This is a strange looking function because it is continous at > >> every irrational and discontinuous at every rational. > >> > >> On Sat, 20 May 2000, Andrzej Kozlowski wrote: > >> > >>> The problem with your question is that the concept of an "irrational number" > >>> does not really make sense in relation to a present day computer. Neither > >>> Mathematica not any other computer program can distinguish between rationals > >>> and irrationals and no sensible concept of an "irrational" number can be > >>> implemented. Of course you could invent a new Mathematica function, > >>> IrrationalQ, an tell Mathematica it should return True for some well known > >>> irrationals, e.g. Pi, E, Sqrt[2], this would not get you very far. It is > >>> well known that there can be no algorithm which would decide whether any > >>> given (constructible) real number is rational or not. One can easily > >>> generate arbitrary long sequences consisting entirely of irrationals, e.g., > >>> anything of the form p^(1/n) where p is a prime and n a positive integer, > >>> or any real number of the form (1-x^n)^(1/n), where x is any rational s.t. > >>> 0<x<1, and n a positive integer>2, but no computer can check this. > >>> > >>> However, in spite of all the above, it is very easy to plot your function. > >>> You simply take the union of the graph of 1/x and the real axis (you must > >>> exclude 0 since your function has no value there). This is as good an > >>> approximation as one can ever hope for! > >>> > >>> > >>> -- > >>> Andrzej Kozlowski > >>> Toyama International University > >>> JAPAN > >>> > >>> http://platon.c.u-tokyo.ac.jp/andrzej/ > >>> http://sigma.tuins.ac.jp/ > >>> > >>> on 5/20/00 4:10 PM, David Ong at do226 at is2.nyu.edu wrote: > >>> > >>>> Hi, > >>>> > >>>> Would anyone know of an easy way to plot some approximation of this > >>>> variant of the Dirichlet function? > >>>> f(x)=1/x if x is an element of the rationals and 0 if x is not an element > >>>> of the rationals. > >>>> > >>>> > >>>> > >>> > >>> > >> > > -- > Andrzej Kozlowski > Toyama International University > JAPAN > > http://platon.c.u-tokyo.ac.jp/andrzej/ > http://sigma.tuins.ac.jp/ >

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