September 14, 2009

Kepler the Musician - Music of the Spheres Part 4

"The heavenly motions...are nothing but a continuous song for several voices, perceivednot by the ear but by he intellect, a figured music which sets landmarks in the immeasurable flow of time." (John Banville: Kepler, Minerva 1990)



Once upon a time, there was a boy who grew up in 16th century Germany. His name was Johannes Kepler. Being weak physically, he decided to beat all the other boys in school with his smarts. So, a few years later, he joined the Tübingen University’s student body where he figured out Copernicus was right: the center of our planetary system was not Earth, but the Sun.

Back in those days, the study of astronomy and astrology was one and the same. In any case, Kepler ended up studying our planets and the stars quite closely.

Then one day, as he was giving a math class (see how wonderful math is?), he drew two circles with an equilateral triangle (3 sides are same length) in between and realized that the ratio of the two circles replicated the ratio of the orbits of Jupiter and Saturn.

He tested out his theory, convinced of this geometrical relationship but realize that it wasn’t perfect. His major breakthrough took place when he realized that his predictions for planetary movement turned out better if he used elliptical orbits rather than circular ones.

This came as a shocker, because hello, circle’s better than ellipse, right? However, after many nights of getting sick at even the mention of it, “the elliptical orbits eventually revealed a scheme of celestial harmony more subtle and profound than any that had gone before.” (1)

Ok. I’ll talk more about Kepler’s ellipses in the next blog post. Yes, this one will be 2 weeks from now (and not three, but last week was Labor Day, you know. . . and I felt compelled to follow it^^).

Hope you had fun reading this, and don’t forget to post comments if you wanna talk some more.

--Alessa

PS: I’ve had complaints about the lack of math in the past posts, so I’ve added some here. . .

Same story, more math:

Blah Blah Blah. . .

And so one day, as he was giving a math class, he drew tow circles with an equilateral triangle inscribed in them. Noticing that the ratio of the two circles represented the same ratio as the orbits of Jupiter and Saturn, he decided to test other planetary orbital relationships with various regular polygons.

If the circumscribed circle has a radius A and the inscribed circle a radius B, then for an n-sided regular polygon the ratio B/A = Cos[Pi/N].

Ex: If N = 3 (triangle), then the ratio B/A = 1/2.
If N = 4 (square), then the ratio B/A = 1/[Sqrt(2)].

By following this logic, it appears that the ratio of the planetary orbits around the sun approximate circles inscribed with either a triangle (Mercur/Venus, Venus/Mars, Jupiter/Saturn, Saturn/Uranus, Uranus Pluto), a square (Venus/Earth, Earth/Marsh, Uranus/Nepture) or a pentagram (Neptune/Pluto).

Etc etc etc. . .

Links:

Kepler and Math

Kepler and the Music of the Spheres

2 comments:

  1. those drawings are hypnotizing...

    ReplyDelete
  2. Haha, I guess they are, huh? I didn't notice till you mentioned it :)

    ReplyDelete