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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