Perhaps my cavatappo article recently posted on the arxiv got your
attention by chance?
But it has self intersections, so it would not be simple.
The torus was fun, but then I needed a connection with Italy to give a
popular talk at my university after they gave me a research award so I
pulled the torus apart into a helical pasta surface: Unfortunately the
video did not capture my Maple presentation
Your remarks on Wikipedia about simple closed geos existing on the
ellipsoid of revolution only for a sufficiently oblate ellipsoid caught
my attention, so I immediately went to our library to look up your
reference to Klingenberg, but there is no real discussion, just a few
teasing remarks. Is there a simple way to derive that condition on the
ratio of the two ellipsoid parameters?
Thanks for contacting me.
From: Charles Karney <charles.karney@sri.com>
Sent: Tuesday, February 25, 2014 8:55 AM
To: Robert Jantzen <robert.jantzen@villanova.edu>
Subject: Re: Geodesics
image of library slip (see above)
Bob,
Here is the proof that oblate ellipsoids have additional simple closed
geodesic if a/b > 2. The equator is a closed geodesic, of course. The
Gaussian curvature, K, on the equator is 1/b^2 (= 1/a * a/b^2), so
Gauss' equation for perturbations about the equator is d^2 m/ds^2 +
m/b^2 = 0, i.e., a simple harmonic oscillation with wavelength 2*pi*b.
This is the wavelength for geodesic intersecting the equator at a
shallow angle. The wavelength increases to 2*pi*a as the angle
increases to 90deg. So if a/b > 2, there is an equatorial angle which
allows the geodesic to close (with no self-intersections) after two
equatorial oscillations. (In the figure, I chose a/b = 7/2, so there
are also geodesics which close after 3 oscillations.)
A couple of titbits about the Math. Dept. at Princeton:
(1) Tukey, in later life, was a furniture delivery man. My wife
(sometime in the '80s) bought a drop-leaf table from Elizabeth Tukey and
she and John Tukey came to deliver it.
(2) Attached is a copy of a library card from Darboux, Leçons sur la
théorie générale des surfaces, Vol 3, in the PU math library. In
addition to two Princeton names, Eisenhardt and Spivak, you will see
Levi-Civita and Hopf (Heinz, I think).
(3) Eisenhardt appears in a less flattering light in
https://blogs.princeton.edu/reelmudd/2011/04/printecons-jewish-students-from-the-days-of-scott-fitzgerald-to-the-center-of-jewish-life/#more-49
More about my involvement with geodesics later...
--Charles
From: Robert Jantzen <robert.jantzen@villanova.edu>
Sent: Tuesday, February 25, 2014 9:09 AM
To: Charles Karney <charles.karney@sri.com>
Subject: RE: Geodesics
HI Charles,
I see the name Lindquist, of Boyer-Lindquist coordinates for the Kerr
spacetime, Boyer was killed at 33 in one of the first mass murder events
in the US
http://en.wikipedia.org/wiki/Charles_Whitman
but Levi-Civita is penned in next to the date 1936... but I don't recall
reading anything about him visiting Princeton...but Wikipedia reminds me
In 1936, receiving an invitation from Einstein, Levi-Civita traveled to
Princeton,
United States and lived there with him for a year. But when the risk of
war in Europe again rose, he returned to Italy. The
1938 race laws enacted
by the Italian Fascist government deprived Levi-Civita of his
professorship and of his membership of all scientific societies.
Isolated from the scientific world, he died in his apartment in Rome in
1941.
Among his
PhD students
were
Octav Onicescu,
Attilio Palatini and
Gheorghe Vrânceanu.
Later on, when asked what he liked best about Italy, Einstein said
"spaghetti and Levi-Civita".[7]
http://en.wikipedia.org/wiki/Tullio_Levi-Civita
A good final quote…
I imagined the geodesic deviation equation at the equator was the tool,
I used that to analyze the geodesics on the torus outer equator. Thanks
for sharing that. I will have to play a bit more with this for fun.
bob
From: Charles Karney <charles.karney@sri.com>
Sent: Tuesday, February 25, 2014 9:45 AM
To: Robert Jantzen <robert.jantzen@villanova.edu>
Subject: Re: Geodesics
Yes, I rather thought that anyone with a cool name like Lindquist
has to
be famous. The connection with Whitman is jarring though...
On 2014-02-25 09:09, Robert Jantzen wrote:
> I see the name Lindquist, of Boyer-Lindquist coordinates for the
Kerr
> spacetime, Boyer was killed at 33 in one of the first mass murder
events
> in the US
>
>
http://en.wikipedia.org/wiki/Charles_Whitman
From: Robert Jantzen
<robert.jantzen@villanova.edu>
Sent: Tuesday, February 25, 2014 10:39 AM
To: Charles Karney <charles.karney@sri.com>
Subject: RE: Geodesics
Yes, Boyer was a visiting professor at Austin, where many famous
names in relativity passed through Texas, Roy Kerr himself, Raynor
Sachs, I would have to do some googling to reconstruct things,
Schucking perhaps, and John Wheeler retired there for a while before
returning to Princeton. The Texas Relativistic Astrophysics
Symposium just had its 50th anniversary
http://nsm.utdallas.edu/texas2013/
organized by Wolfgang Rindler (who has a famous GR book) over
in Dallas. A lot of Europeans came over to do GR here, the Air Force
was financing all kinds of abstract research at the time. I imagine
these names don't ring a bell for you as an outsider, apart from
Wheeler, maybe Kerr for the rotating black hole spacetime.
Your wiki article figures remind me of the "spherical" geodesics
around a black hole see figure 6 of the attached article., the
"spheres" are ellipsoids of revolution but I have not looked at
their actual induced metric geometry, gotta run to class.
https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid
You asked for feedback. The one thing I noticed as a relativistic
was that you never explicitly stated the line element for the
ellipsoid of revolution ds^2 = ..., which is the starting point for
all geometry. I was curious why?
bob
From: Robert Jantzen <robert.jantzen@villanova.edu>
Sent: Wednesday, February 26, 2014 3:31 PM
To: Charles Karney (charles.karney@sri.com)
<charles.karney@sri.com>
Subject: RE: Fine Hall library book slip [texas]
Attached file:
first-texas-symp-on-rel-astrophys.pdf
THE FIRST TEXAS SYMPOSIUM
ON RELATIVISTIC ASTROPHYSICS
Born at poolside on a summer afternoon, the idea for
a Texas-sized conference blossomed when it was realized
that the newly found quasars might be relativistically significant.
Engelberr L. Schucking
Hi Charles,
At the risk of seeming like a
spammer, I send this little jewel.
Incredibly the Texas story I
referred to the other day fell into my lap today, when I came
across the following Physics Today article, which has a differential
geometry related guy Alfred Schild at the heart of it (his book with
Synge: Tensor Calculus). And explains why Boyer was in Austin.
bob
From: Charles Karney <charles.karney@sri.com>
Sent: Thursday, February 27, 2014 10:57 AM
To: Robert Jantzen <robert.jantzen@villanova.edu>
Subject: Re: Geodesics
Dear Bob,
I first came to Princeton in 1977, so we would not have overlapped. I'm
still here working in what would have been called the RCA Research Labs
in your day. (It's the large building between Harrison St and
Washington Rd on the east side of Rte 1.)
I'm not sure I would necessarily characterize my work on geodesics as
"serious mathematical analysis". I was first drawn to the problem by
the (literal) geodesic application. I was annoyed that the Vincenty's
algorithm (which "everyone" was using) for the distance between points
on an ellipsoid (a) had limited accuracy (0.1mm) and no obvious way to
improve on this and (b) failed to converge for nearly antipodal points.
Along the way I discovered a wealth of 19th century papers on this
subject (by the likes of Gauss, Legendre, Bessel, Jacobi, Cayley,
Weierstass) which were now accessible thanks to Google Books, see
http://geographiclib.sourceforge.net/geodesic-papers/biblio.html
So my contributions are mostly just adapting this work to modern
computers. I particularly like Bessel's 1825 paper
https://arxiv.org/abs/0908.1824
He sews up the "direct geodesic problem", the formulation, series
expansions, and a cook-book recipe for its solution. His paper was a
very modern feel (despite his use of logarithms) and he discusses the
relative magnitudes of round off errors (the limited number of digits in
his log tables) and quantization errors (due to truncating the series).
There are, of course, other gems, Jacobi's study of caustics and his
solution of the problem on a triaxial ellipsoid (the first known
dynamical system with a "hidden" symmetry -- clearly a surprise to him),
the involvement of several Irish mathematicians in the study of
umbilical geodesics, Poincare's study of closed geodesics, and so on.
I've now included formulas for ds^2 in the Wikipedia article: biaxial in
terms of phi/lambda and beta/lambda and triaxial in terms of beta/omega.
The simple answer to why I hadn't included them is that I was following
Bessel's paper and this predated Gauss' Disquisitiones... which
introduced the study of the intrinsic properties of surfaces. This
probably also highlights the differences between my background (more
engineering oriented, the ellipsoid is really embedded in 3d) and yours
(relativity, so more used to handling coordinate systems abstractly).
Finally, no, I was not aware of your cavatappo paper (I had only seen
your paper on tori). Thanks for the links.
--Charles
From: Charles Karney
<charles.karney@sri.com>
Sent: Thursday, February 27, 2014 10:57 AM
To: Robert Jantzen <robert.jantzen@villanova.edu>
Subject: Re: Geodesics
Dear Bob,
I first came to Princeton in 1977, so we would not have
overlapped. I'm
still here working in what would have been called the RCA
Research Labs
in your day. (It's the large building between Harrison St and
Washington Rd on the east side of Rte 1.)
I'm not sure I would necessarily characterize my work on
geodesics as
"serious mathematical analysis". I was first drawn to the
problem by
the (literal) geodesic application. I was annoyed that the
Vincenty's
algorithm (which "everyone" was using) for the distance between
points
on an ellipsoid (a) had limited accuracy (0.1mm) and no obvious
way to
improve on this and (b) failed to converge for nearly antipodal
points.
Along the way I discovered a wealth of 19th century papers on
this
subject (by the likes of Gauss, Legendre, Bessel, Jacobi,
Cayley,
Weierstass) which were now accessible thanks to Google Books,
see
http://geographiclib.sourceforge.net/geodesic-papers/biblio.html
So my contributions are mostly just adapting this work to modern
computers. I particularly like Bessel's 1825 paper
https://arxiv.org/abs/0908.1824
He sews up the "direct geodesic problem", the formulation,
series
expansions, and a cook-book recipe for its solution. His paper
was a
very modern feel (despite his use of logarithms) and he
discusses the
relative magnitudes of round off errors (the limited number of
digits in
his log tables) and quantization errors (due to truncating the
series).
There are, of course, other gems, Jacobi's study of caustics and
his
solution of the problem on a triaxial ellipsoid (the first known
dynamical system with a "hidden" symmetry -- clearly a surprise
to him),
the involvement of several Irish mathematicians in the study of
umbilical geodesics, Poincare's study of closed geodesics, and
so on.
I've now included formulas for ds^2 in the Wikipedia article:
biaxial in
terms of phi/lambda and beta/lambda and triaxial in terms of
beta/omega.
The simple answer to why I hadn't included them is that I was
following
Bessel's paper and this predated Gauss' Disquisitiones... which
introduced the study of the intrinsic properties of surfaces.
This
probably also highlights the differences between my background
(more
engineering oriented, the ellipsoid is really embedded in 3d)
and yours
(relativity, so more used to handling coordinate systems
abstractly).
Finally, no, I was not aware of your cavatappo paper (I had only
seen
your paper on tori). Thanks for the links.
--Charles
Michael Spivak emails 2014 (going back to 2006)
From: Robert Jantzen <robert.jantzen@villanova.edu>
Sent: Tuesday, February 25, 2014 2:21 PM
To: Michael Spivak <puborperish@gmail.com>
Subject: Fine Hall library book slip
Attached file: see slip above
Dear Michael,
I thought you might be amused by this library slip from a book in
the Fine Hall library at Princeton. Your name is there with some
pretty interesting other names. It was sent to me today by a world
expert in geodesics on ellipsoids (for application to the Earth) who
somehow noticed my dilettante efforts with tori and cavatappi
helical surfaces [see links below].
Hope you are well.
bob
From: Michael Spivak <puborperish@gmail.com>
Sent: Wednesday, February 26, 2014 7:27 PM
To: Robert Jantzen <robert.jantzen@villanova.edu>
Subject: Re: Fine Hall library book slip
Thanks. As you can see, I realized immediately that it would be
hopeless for me to try reading it. Later on I bought a complete set
of the books, I think
a Dover edition, which I then found merely almost hopeless for me to
read.
Your name seemed to ring a familiar bell, and a bit of sleuthing
through your on-line files confirmed that I had the right person.
Now that I am at what can only
be called a "ripe old age", I keep encountering people from the
past.
From: Robert Jantzen <robert.jantzen@villanova.edu>
Sent: Thursday, February 27, 2014 6:07 AM
To: Michael Spivak <puborperish@gmail.com>
Subject: RE: 25 years ago or more ...
Hi Michael,
Sorry for not identifying myself---we had this exchange in 2006 (see
email below, I am an email packrat). I was a Princeton physics
undergrad 1970-1974 (where I used your "Calculus" book as an
entering freshman, later "Calculus on Manifolds") then sat in your
DG course at Berkeley around 1975-1976. I was in my twenties you in
your thirties, and wow, 40 years just went by like that. I am almost
62.
This past year by chance I became friends with a bright younger
faculty member in the Villanova business school who wanted to do
math in college but got diverted into finance by his Greek immigrant
dad, but still loves math and as a faculty member has been taking
math courses for credit (!) He gifted me your more recent book on
physics that connected me to you in 2006, out of the blue, without
me ever mentioning you to him. Unfortunately it looks like he is a
victim of vicious politics in the business school and will be
screwed out of getting his well-deserved tenure this year (these
idiot wall street wanabees perhaps feel threatened by his use of
more advanced math in his finance papers with a smart Russian at
Rutgers, or perhaps are just too stupid to appreciate him). So it
goes.
In my serendipitous encounter with Charles Karney (was joint
appointment at Princeton astrophysics and the Plasma Physics lab,
then joined SRI, a science research company nearby, wrote
https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid ) and
his ellipsoid geodesics, this library book slip came up that I sent
you and I was trying to explain to him why Boyer was in Austin to be
the first target of the first mass shooting in the US, and yesterday
I got an invitation to review a book proposal on "Einstein's Apple"
by Engelbert Schucking (book on homogeneous spacetimes) so I was
looking at his CV and found this 1989 Physics Today article he wrote
that explained why there were all these European imported
relativists at Austin and Dallas in those days (naturally interested
in differential geometry), including himself and Roy Kerr, whose
name you must know from rotating black holes (who became a friend of
mine through Roy's association with the International Center for
Relativistic Astrophysics in Italy created by Remo Ruffini who
worked with John Wheeler in the late 1960s early 1970s at Princeton,
and I have been going to Italy every year since 1979 through his
funding).
So I have attached this article, which you might enjoy. The 50th
Texas Symposium on Relativistic Astrophysics just took place in
December, organized by another European Texas import GR guy even
older than you: Wolfgang Rindler, of the famous Rindler coordinates
(later associated with Hawking radiation) who speaks fluent Italian
from his visits to an Italian GR guy in Rome earlier in life. I met
him again last summer in Italy.
Thanks for your quick response. Hope you are well in your aging
state. :-)
bob
bob jantzen
http://www.homepage.villanova.edu/robert.jantzen
http://www.drbobenterprises.com <<<< on-line somewhat
humorous but serious cuisine cookbook
http://www.icra.it/MG/mg13/ <<<< latest trienniel Marcel
Grossmann conference on GR and relativistic astrophysics (MG =
friend of Einstein)
-----Original Message-----
From: Publish or Perish, Inc. [
mailto:mailbox@mathpop.com]
Sent: Wednesday, February 22, 2006 10:25 PM
To: Robert Jantzen
Subject: Re: 25 years ago or more ...
Hi Bob,
Thanks a lot for writing. I've looked briefly at your websites and
can only say it's a good thing you're "not ambitious", or there
would have been so many things to mention that you might not have
been able to get them into a web page!
I think Phil's last name is something like Colella (don't know why,
just seems to have occured to me). He hung around the math common
room a lot, and I knew him quite well, and might even have asked him
a few things about physics. But learning physics (in my own
inimitable way) is something I've kept threatening to do for a long
time, and actually started about 2 years ago. Attached is something
I wrote up while in Tokyo. Unfortunately, after Tokyo I got bogged
down in extraneous things, though I am trying to get back to the
physics (although it might not be what physicists are willing to
call physics).
----- Original Message -----
From: "Robert Jantzen" <robert.jantzen@villanova.edu>
To: <mailbox@mathpop.com>
Sent: Wednesday, February 22, 2006 12:43 PM
Subject: 25 years ago or more ...
> Hi mike,
>
> The new issue of Practical Tex with your article
>
http://tug.org/pracjourn/2006-1/spivak/ led me to your website
>
http://www.mathpop.com/ inspired by my memories of a moment in
time
> around 1977-1978 when we had a mutual friend/acquaintance Phil in
the
> physics department (another Physics grad student at the time) who
also
> had a ponytail like me at the time, and now through the magic of
the
> internet, I can actually reach out and touch you, virtually
speaking.
>
> I was interested in differential geometry and classical general
> relativity and did my thesis with Abe Taub, and I had sat in on
your
> Differential Geometry lectures one semester during when I should
have
> been physically present in a Statistical Mechanics class I was
taking
> at the same time slot, but the opportunity warranted unusual
student
> behavior. (I first learned of you from the math major calculus
classes
> I took at Princeton in
> 1970-1971 where I used both the single variable and calculus on
> manifold texts, though I was a physics major there too). Somehow
later
> on I met you outside this context on campus at Berkeley while you
were
> exercising, I think you were into a weight training thing at the
time,
> which you explained to me. I cannot remember how Phil, whose last
name
> is now lost, fit into the story. I think you were trying to learn
some
> physics with him...?
>
> I also got somewhat involved in TeX (and got Barbara Beeton to
help me
> out with some Proceedings editing macros that have been in use for
> some time now), to the point of doing a wizzard class with Stefan
> Bechtolsheim whose
> 5 volume work sits on my shelf, but managed not to get sucked in.
When
> Abe Taub died I got his hardcover editions of Eisenhart's
Introduction
> to Differential Geometry and Continuous Groups of Motions. He had
been
> a grad student in that mix of mathematicians/physicists at
Princeton
> in the 1930s, a story which did suck me in to the extent that I
> volunteered to convert an inaccessible 600 some page oral history
> document to HTML for the entire world to see at its pleasure:
>
http://www.princeton.edu/mudd/math/
>
> But I am not ambitious and ended up in a comfortable teaching
position
> at Villanova University near Philly, which is not a bad place to
live.
>
> Thanks for investing your energy in TeX. I am sure many people
around
> the world appreciate it, even though most of them do not have the
> opportunity to say so.
>
> take care,
> bob
From: Michael Spivak <puborperish@gmail.com>
Sent: Thursday, February 27, 2014 12:38 PM
To: Robert Jantzen <robert.jantzen@villanova.edu>
Subject: Re: 25 years ago or more ...
I didn't remember this interchange at all, although I
definitely remember you. Turns out that just recently I
thought about Phil Colella, because I'm now working
on E&M. I looked him up and found that he's made it big
time in computational physics, and is now a member of the
Academy of Sciences, and sent him a letter,
which I'll forward to you, although I haven't gotten an
answer, so perhaps he's too busy or not interested
From: Robert Jantzen
<robert.jantzen@villanova.edu>
Sent: Thursday, February 27, 2014 3:24 PM
To: PColella@lbl.gov <PColella@lbl.gov>
Subject: FW: 25 years ago or more ... [copy of
email to Michael Spivak]
Hey Phil,
My memories of those days in Berkeley are pretty foggy,
but I still have a sort of shadowy image of you on
campus. I don't remember exactly how we met, but I do
remember I thought well of you.
Michael tried to copy me on your email but misspelled my
name...
Hope you are well after all these years.
bob
