[seqfan] Twindragon & other nonconventional bases.

[Antti Karttunen]

On Sun, Sep 4, 2016 at 8:52 PM, <seqfan-request at list.seqfan.eu> wrote:

>
>
> Message: 3
> Date: Sun, 28 Aug 2016 19:51:36 +1000
> From: Kevin Ryde <user42_kevin at yahoo.com.au>
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: Project: sequences obtained from Gaussian
>         Integers via    Penney's binary method of encoding?
> Message-ID: <8760qlql07.fsf at blah.blah>
> Content-Type: text/plain
>
> antti.karttunen at gmail.com (Antti Karttunen) writes:
> >
> > (but please tell if it is much older and discovered in some other
> country!)
>
> Another early reference but which I haven't actually seen
>
>     Solomon I. Khmelnik, "Specialized Digital Computer for Operations
>     with Complex Numbers" (in Russian), Questions of Radio Electronics,
>     volume 12, number 2, 1964.
>
> > Or are there better ones for Gaussian Integers?
>
> A few sequences go by square spiral (like Ulam plotted primes on).
>
> > A066321 "Binary representation of base i-1 expansion of n:
>
> Existing sequences I know,
>
>     A066321    N on X axis, being the base i-1 positive reals
>     A066322    diffs (N at X=16k+4) - (N at X=16k+3)
>     A066323    N on X axis, number of 1 bits
>     A137426    dY at N=2^k-1 (step to next replication level)
>     A003476    boundary length / 2
>                  recurrence a(n) = a(n-1) + 2*a(n-3)
>     A203175    boundary length, starting from 4
>                  (believe its recurrence is true)
>     A052537    boundary length part A, B or C, per Gilbert's paper
>




> "Boundary length" is with each Gaussian integer as a unit square.
> (I think the diffs question in A066322 would be answered from Penney,
> with X axis as hex digits 0,1,C,D, something, something :-).
>
> The correspondence of base i-1 to twindragon interior unit squares means
> various dragon curve sequences (like A003476) probably have
> interpretations in i-1, but maybe at a stretch.
>
> > "base i-1" is not really a good term to search for.
>
> I've been happy enough with i-1 in non-oeis things.  A few people call
> it twindragon because the limit fractal is the same, but for integers I
> find that unclear.
>

I mean that it is almost futile to search with keywords like base i-1
entered into the OEIS search field. (Please try it: 4856 results found.) So
at least the (normally invisible) anchor-part of the index-entry should be
something searchable. Maybe #twindragon would work perfectly well there?

Meanwhile, I found this:

Andrew Vince, "Digit Tiling of Euclidean Space":
https://www.math.uni-bielefeld.de/~frettloe/papers/vince-digitt.pdf
(from year 2000 ?). Seems quite accessible to us laymen.



Best regards,

Antti



> Base i-m with other integer m is also possible, digits 0 to norm(i-m)-1.
> Gilbert's boundary calculation is for that general case.  I think the
> number of digits you then need in i-m to represent points relatively
> close to the origin becomes even bigger than say the 5 bits you noted
> for z=-1.
>
> http://www.math.uwaterloo.ca/~wgilbert/Research/GilbertFracDim.pdf
>
>
> ------------------------------
>
> Message: 4
> Date: Sun, 28 Aug 2016 20:00:18 +1000
> From: Kevin Ryde <user42_kevin at yahoo.com.au>
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: Project: sequences obtained from Gaussian
>         Integers via    Penney's binary method of encoding?
> Message-ID: <87y43hp619.fsf at blah.blah>
> Content-Type: text/plain
>
> graememcrae at gmail.com (Graeme McRae) writes:
> >
> > how far the "biggest" of these base(2+i) numbers can get from the
> > origin.
>
> I lately had a go at widest point of the r5 dragon (base 1+2*i).
> I started high digit and was able to have at each stage only 2 sub-parts
> as candidates to contain the widest.  But as a wild guess maybe seeking
> maximum hypotenuse would be more than 2 candidates.
>
>
>