[seqfan] Re: A sequence from Johannes Kepler's "Harmonices Mundi"
bradklee at gmail.com
Fri Feb 17 19:30:18 CET 2017
Harmonices Mundi also contains the legendary third law for planetary
motion, which explains the geometric proportion of a planet's period to the
cube of semi-major axis. Like the later Planck, Kepler was working in an
empircal mode, from tables of observations. Today we have potential theory,
by which Kepler's third law is rigorously proven in a mathematical model
that approximates motion of the planets.
Users of OEIS may be interested in a challenge to accomplish exact
calculation of the period by ennumerating a convergent power series ( Cf.
A276816, A276817 ). Hint: for Coulomb potential, no precession occurs and
the radial period exactly equals the orbital period.
Entries A276816 and A276817 have been criticised as too difficult to read /
use. I am working dilligently on revisions, and preparing a chapter for my
dissertation to greatly improve quality. Until then, you might be
interested in a table from the next work, related to discussion of
87.969 224.701 365.256 686.98 4332.59 10759.2 30685.4 60189
=====Period Ratios. Upper: Harmonices (Log_2(Tb/Ta))_12. Lower: Ta/Tb===========
. 1(4/12) 2(1/12) 3 5(7/12) 6(11/12) 8(5/12) 9(5/12)
2.55 . 0(8/12) 1(7/12) 4(3/12) 5(7/12) 7(1/12) 8(1/12)
4.15 1.63 . 0(11/12) 3(7/12) 4(11/12) 6(5/12) 7(4/12)
7.81 3.06 1.88 . 2(8/12) 4 5(6/12) 6(5/12)
49.25 19.28 11.86 6.31 . 1(4/12) 2(10/12) 3(10/12)
122.31 47.88 29.46 15.66 2.48 . 1(6/12) 2(6/12)
348.82 136.56 84.01 44.67 7.08 2.85 . 1
684.21 267.86 164.79 87.61 13.89 5.59 1.96 .
This table is primarily of use in evaluating the validity of a commonly
applied approximation that describes the solar system comprised of eight
planets and the sun by a time-independent gravitational potential.
The upper triangle is more of a humorous calculation, as well as a play on
Kepler's quadrivium ideology. The numbers displayed are octaves plus number
of half steps. The computation assumes equal spacing between half steps,
and octave ratio exactly equal to 2. The error of harmonic approximation
for a period ratio is less than 0.05 in decimal notation ( compare entries
in the bottom right corner ).
Kepler's work was intended to be a synthesis of arithmetic, geometry,
music, and astronomy. This tree of rational numbers was developed in
congruence with application to scientific problems. Did Kepler doubt his
own thesis enough to believe that corrections to planetary motion could be
necessary at the order of 1/144? This is one of many questions that I think
a modern audience will struggle with, especially because the source text is
written in latin.
Even when a translation is available--
See page 163 :
--Kepler's perspective is difficult to understand. The work is complicated
by the religious history of "Sacrum Imperium Romanum", the Holy Roman
Empire. Notice that the publication date is within one or two years of the
start of the thirty year's war.
The list moderators are strictly enforcing rules that limit the talk, but
they say that discussion of other topics are appropriate to back-channel.
In particular I'm interested to hear perspectives that compare "Harmonices
Mundi" with 12th century monastic writing from Germany, such as
"Ysengrimus". I for one do not believe that it is possible to understand or
discuss "Harmonices Mundi" without considering the religious history, and
invite others to send me private messages regarding this topic, as it
apparently is not okay for discussion on the list.
Cheers and Joyous Research,
On Sat, Feb 11, 2017 at 5:34 AM, Peter Luschny <peter.luschny at gmail.com>
> We all know Stern-Brocot, Dijkstra, and the Calkin and Wilf tree.
> David Eppstein posted today on G+ this picture .
> And Eppstein added to Wikipedia: "Even earlier, a similar tree appears in
> Johannes Kepler's "Harmonices Mundi" (1619, volume=III, page=27)." 
> Hope you like it as much as I do.
>  http://www.ics.uci.edu/~eppstein/0xDE/CalkinWilfKepler.png
>  https://archive.org/stream/ioanniskepplerih00kepl#page/27/mode/1up
>  https://oeis.org/A002487
> Seqfan Mailing list - http://list.seqfan.eu/
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