AOn 1 Dec 2017 05:33:02 +0000,"Dabrowski, Andrew John" <[email protected]
<mailto:[email protected]>> wrote:
>
> On 11/29/2017 11:40 PM, Roger Hui wrote:
>> 2.5 Cantor Set
>>
>> Write a function to compute the Cantor set of order n, n>:0.
>>
>> Cantor 0
>> 1
>> Cantor 1
>> 1 0 1
>> Cantor 2
>> 1 0 1 0 0 0 1 0 1
>> Cantor 3
>> 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1
>>
>
> In Mathematica:
>
> cantor[n_] := If[n == 0, {1},
> cantor[n - 1] /. {0 -> Sequence[0, 0, 0], 1 -> Sequence[1, 0, 1]}]
I also tried this in Mathematica. The quickest way is to use one of the “higher
order” built-in (i.e., primitive) functions:
SubstitutionSystem[{1 -> {1, 0, 1}, 0 -> {0, 0, 0}}, 1, 2]
{{1}, {1, 0, 1}, {1, 0, 1, 0, 0, 0, 1, 0, 1}}
Or one can go back to more basic built-in functions, much as Andrew John
Dabrowski did:
cStep[lis_] := Flatten[lis /. {1 -> {1, 0, 1}, 0 -> {0, 0, 0}}]
NestList[cStep, {1}, 2]
{{1}, {1, 0, 1}, {1, 0, 1, 0, 0, 0, 1, 0, 1}}
Interestingly, Mathematica also has as a built-in function CantorMesh built,
which directly returns a graphic display of a collection of subintervals in the
construction of the Cantor ternary set, with the depth indicated by the
argument supplied to it. Thus
CantorMesh /@ Range[0, 2]
returns as output a list of images — the original unit interval, the 1st and
3rd thirds of that interval, and the 1st, 3rd, 7th and 9th ninths of that
interval. (I cannot include graphics here, but the output is show in my post
at
https://mathematica.stackexchange.com/questions/161024/extract-cell-indicators-from-meshregion/161098#161098
.)
——
Murray Eisenberg [email protected]
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 240 246-7240 (H)
University of Massachusetts
710 North Pleasant Street
Amherst, MA 01003-9305
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