[seqfan] what should be called a "fractal" sequence?

jean-paul allouche jean-paul.allouche at imj-prg.fr
Mon Jul 20 19:38:39 CEST 2020

Dear all

There is --in the continuous case-- a classical confusion between 
"fractal" and
"self-similar". For example a straight line is trivially self-similar 
but certainly
not fractal. On the other hand a fractal need not be strictly 
self-similar (whatever
this means). Now the constant sequence 1 1 1 1 ... is certainly 
self-similar whatever
this means but we dont want to call it fractal. In the continuous case, 
B. Mandelbrot
himself was very reluctant to give a precise mathematical definition of 
a fractal: he said
that whatever the definition, he would probably find an object that does 
not fit in the
definition, BUT that he would like to call fractal. Finally he accepted 
a temporary
definition (Hausdorff dimension > topological dimension) but he quickly 
"or any Hausdorff-like dimension > topological dimension".

May be one should follow this famous point of view. In any case what has 
been described
in the previous mails could/should be called "self-similar" but not 
fractal... except if someone
proposes a Hausdorff-like dimension d_1 and a topological-like dimension 
d such that always
d_1 \geq d and defines fractal to be the case where the inequality is 
strict. Of course any
other Hausdorff-like dimension could be chosen.


Le 19/07/2020 à 09:47, Éric Angelini a écrit :
>> That definition is both too strong and too weak.
> ... thank you for this comment Franklin —
> but what would be a good definition of
> a fractal sequence?
> Best,
> É.
> Catapulté de mon aPhone
>> Le 19 juil. 2020 à 08:28, Frank Adams-watters via SeqFan <seqfan at list.seqfan.eu> a écrit :
>> That definition is both too strong and too weak.
>> It is too weak, because any infinitive sequence is fractal by that definition. (An infinitive sequence is a sequence of positive integers that contains every positive integer infinitely often.)
>> Too strong, because, for example in the current instance A087088, the transformation getting the original sequence back is not just taking a subsequence; it also involves an arithmetic operation (subtracting one). A087088 is infinitive, and hence fractal by that definition; but this would be based on a different transformation than the one given in the definition of the sequence.
>> Franklin T. Adams-Watters
>> -----Original Message-----
>> From: Éric Angelini <eric.angelini at skynet.be>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Cc: David Sycamore <djsycamore at yahoo.co.uk>
>> Sent: Thu, Jul 16, 2020 10:06 am
>> Subject: [seqfan] Re: Fractal sequence A087088
>> I’ve always used the definition:
>> Fractal seq S = seq S containing
>> an infinite amount of copies of S.
>> There are a lot of ways to show/
>> fix/decide/etc. how to highlight
>> a single copy.
>> à+
>> É.
>> Catapulté de mon aPhone
>>> Le 16 juil. 2020 à 16:11, David Sycamore via SeqFan <seqfan at list.seqfan.eu> a écrit :
>>> Coming a bit late to this discussion, I have a question concerning the definition of what is meant by a “fractal” sequence? Could there be more than one different interpretation of this term?
>>> According to one definition, currently described in oeis, a sequence is fractal if, when all first occurrences are removed, what remains is the original sequence (which means that it contains a proper subsequence identical to itself).
>>> Removal of all first occurrences from A087088 gives:
>>> 1, 2, 1, 3, 1, 2,1, 4, 2, 1, 3, 1...
>>> which is not the same as the original.
>>> Can anybody explain why A087088 is considered to be fractal?
>>> Best regards,
>>> David.
>>>>> On 14 Jul 2020, at 07:12, Allan Wechsler <acwacw at gmail.com> wrote:
>>>> If I were asked to write a sequence title from scratch, I think I would
>>>> dispense with "simplest", and say something like "Positive ruler-type
>>>> fractal sequence with 1's in every third position." I think only this
>>>> sequence satisfies that description.
>>>> A "fractal" sequence can be constructed upon any "skeleton" of 1's, as long
>>>> as there are an infinite number of entries that are _not_ 1.
>>>>>>>> On Mon, Jul 13, 2020 at 12:42 PM Neil Sloane <njasloane at gmail.com> wrote:
>>>>> Allan, that is an excellent point.  So maybe the sequence should say
>>>>> something like "simplest two-step-insertion fractal" ?
>>>>> Best regards
>>>>> Neil
>>>>> Neil J. A. Sloane, President, OEIS Foundation.
>>>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>>>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>>>> Email: njasloane at gmail.com
>>>>>> On Mon, Jul 13, 2020 at 12:37 PM Allan Wechsler <acwacw at gmail.com> wrote:
>>>>>> To return to the claim of "simplest" sequence with this property; we are
>>>>> in
>>>>>> the difficult position of trying to read the mind of the person who was
>>>>>> making that claim. I think they had some notion of "simplicity" in mind
>>>>> for
>>>>>> which the statement was arguably true, but as it stands it is hard to see
>>>>>> what that notion was. The point about the ruler functions is a strong one
>>>>>> -- A001511 can be given a homologous four-step definition exactly
>>>>> analogous
>>>>>> to the one given for A087088, using gaps of one undefined place instead
>>>>> of
>>>>>> two. One is simpler than two, isn't it?
>>>>>> But even A000027, the positive integers, displays the required property.
>>>>>> Remove the only 1; decrement all other entries; behold. In what sense is
>>>>>> A087088 simpler than A000027? I think the author(s) had some additional
>>>>>> constraints in mind. But if I were shown the title only, and asked to
>>>>>> reconstruct the sequence, I would probably produce A000027.
>>>>>> On Mon, Jul 13, 2020 at 12:05 PM Frank Adams-watters via SeqFan <
>>>>>> seqfan at list.seqfan.eu> wrote:
>>>>>>> This sequence and A163491 are ordinal transforms of each other.
>>>>>>> Franklin T. Adams-Watters
>>>>>>> -----Original Message-----
>>>>>>> From: Peter Munn <techsubs at pearceneptune.co.uk>
>>>>>>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>>>>>> Sent: Mon, Jul 13, 2020 9:01 am
>>>>>>> Subject: [seqfan] Fractal sequence A087088
>>>>>>> Hello seqfans,
>>>>>>> A087088 claims to be "the simplest nontrivial sequence" such that
>>>>>> removing
>>>>>>> every "1" gives the same result as adding 1 to every term. Ruler
>>>>>>> sequences, such as A001511, share this property, so does anyone have a
>>>>>>> clear idea how "simplest nontrivial" might be defined?
>>>>>>> And can anyone shed light on the reason its offset is 3? [1]
>>>>>>> Best Regards,
>>>>>>> Peter
>>>>>>> [1] Apart from the b-file, the rest of the sequence is written as
>>>>> though
>>>>>>> the offset is 1 (so formulas are strictly incorrect). The relationship
>>>>> to
>>>>>>> A244040 contributed by Edgar and Van Alstine is neatest with offset 1
>>>>> or
>>>>>>> offset 0. A relationship I discovered recently (comment in
>>>>>>> https://oeis.org/A024629) is clearly neatest if the offset is 1,
>>>>> whilst
>>>>>> my
>>>>>>> work on symmetry (https://oeis.org/history/view?seq=A087088&v=25) and
>>>>>> with
>>>>>>> A335933 suggests an OEIS-incompatible offset of 1.5 .
>>>>>>> As we are only now starting to refer from other sequences to terms of
>>>>>>> A087088, it seems a good time to settle on a good offset. Unless anyone
>>>>>>> knows a good reason for keeping it as 3, offset 1 seems better.
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