[seqfan] Re: A sequence from Johannes Kepler's "Harmonices Mundi"

Antti Karttunen antti.karttunen at gmail.com
Sun Feb 12 14:36:17 CET 2017


On Sun, Feb 12, 2017 at 11:32 AM, <seqfan-request at list.seqfan.eu> wrote:

>
>
> Message: 15
> Date: Sat, 11 Feb 2017 12:34:58 +0100
> From: Peter Luschny <peter.luschny at gmail.com>
> To: "seqfan at list.seqfan.eu" <seqfan at list.seqfan.eu>
> Subject: [seqfan] A sequence from Johannes Kepler's "Harmonices Mundi"
> Message-ID:
>         <CAMMbGZYKfrtSwZvvgjpzU7eL8cMnAG1PTxvUCDgFTqfZ6g0uAg at mail.
> gmail.com>
> Content-Type: text/plain; charset=UTF-8
>
> We all know Stern-Brocot, Dijkstra, and the Calkin and Wilf tree.
> David Eppstein posted today on G+ this picture [0].
>
> And Eppstein added to Wikipedia: "Even earlier, a similar tree appears in
> Johannes Kepler's "Harmonices Mundi" (1619, volume=III, page=27)." [1]
>
> Hope you like it as much as I do.
>
> Peter
>
> [0] http://www.ics.uci.edu/~eppstein/0xDE/CalkinWilfKepler.png
> [1] https://archive.org/stream/ioanniskepplerih00kepl#page/27/mode/1up
> [2] https://oeis.org/A002487
>
>
> ------------------------------
>
>
Please see also
http://oeis.org/search?q=kepler+tree&language=english&go=Search
Especially http://oeis.org/A086592 (and the links & references there).

The name adopted in OEIS for the Kepler's variant seems to be settled as
"Kepler's tree of harmonic fractions".

Mapping the unique fractions between such systems induces several
permutations of natural numbers, e.g.:
http://oeis.org/search?q=id%3AA258746%7Cid%3AA258996&sort=&language=&go=Search
but probably not all combinations are yet in OEIS, or then appear under
another guise.

Maybe we need an index entry for these fraction trees?



Best regards,

Antti


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