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Think Globally

Differential geometry can show us the shortest route between two points.

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1.
David Bee
Brooklyn
March 21st, 2010
9:29 pm
BTW, for those interested, an inexpensive softcover book was published two months ago titled The Great Equations: Breakthroughs in Science from Pythagoras to Heisenberg. Written by Robert P. Crease, this 10-chapter book, which seems to be written at the level of Dr. Strogatz's essay, has its first chapter titled " 'The Basis of Civilization': The Pythagorean Theorem" and Chapter 4 titled " 'The Gold Standard for Mathematical Beauty': Euler's Equation", both of which were discussed here by Dr. S or in postings in recent weeks.

For more, goto:
http://www.amazon.com...

2.
joe.shuren
bouvet island
March 21st, 2010
9:29 pm
As the National Geographic Society explained to Times readers May 22, 1927, "apply your string to a globe and you will find that the flying distance from New York to Paris, via the Azores, would be 4,107 statute miles, whereas a course outlined by a string stretched tautly from New York to Paris across New England, Canada and Newfoundland, and south of Ireland the way Lindbergh flew, would be 3,633 statute miles. Lindbergh flew that way to save 473 miles. " Lindbergh in his book, "The Spirit of St. Louis," details how he by himself had to invent a method to plot a course on flat maps of compass directions to approximate the great circle--in the days before GPS and when "computers" meant women who computed by hand or adding machines.
3.
Sam Nead
UK
March 21st, 2010
9:29 pm
In the second video the handlebars are locked so the bike always goes straight ahead. It would be very interesting to see a similar video where the handlebars are locked at a small but definite angle (say 20 degrees?) off of straight. (Then the bike would always be veering right, say, and would describe a path of constant geodesic curvature.)
4.
zed00
Irvine, CA
March 21st, 2010
9:29 pm
Hmmm....well, I didn't find much of this article very "intuitive" or fully understandable. Beautiful, perhaps, if you say so, but still a bit too fanciful to grasp. I wonder if you might not do better by using, say, one of Big Red's practice basketballs as an example, instead of globes and planes and motorcycles. That would at least be a tad more topical...
5.
N A Fortis
Los Altos CA
March 21st, 2010
9:30 pm
Oh, Times! If Sunday comes, can Srogatz be far behind?

Happily not. And the presentation this time is again a pleasure to experience.
The videos are excellent. A little bit about spherical geometry and its importance
to celestial navigation would have been nice. Still of importance even with the advent of GPS.

All in all, the usual joy. Thank you Professor S.

Naf Los Altos CA

6.
Pilot411
Berlin DE
March 21st, 2010
9:31 pm
Very nice analysis, although most of the time airplanes do not use the great circle distance exactly, as wind and weather have to be taken into account. The change in wind and weather make flight routes interesting though, as the shortest distance between two points may not be a straight line. Anyone who has flown in winter from Asia to the US west coast would notice that normally a more southern path is taken, as it is the shortest "time wise", and thus can use less resources like fuel than taking the shorter distance great circle route.

For those that are interested in looking at great circle distances between major airports around the world:

gc.kls2.com/

7.
Richard
Bozeman, MT
March 21st, 2010
9:31 pm
The differential geometry of the tin can might be more complicated than the sphere, but the topology is much simpler. A confusion that will pursue these essays to the very end.
8.
b.u.m.
Oregon
March 21st, 2010
9:31 pm
Lucid essay and videos to reconnect an old fogy with 50 year old concepts of high school geometry. Thanks
9.
P Q Hanser
Newton, MA
March 21st, 2010
9:31 pm
Some of the illustrations remind me of the property of differentiable manifolds that they are locally Euclidean and thus permit the calculus. When I took the class that introduced such manifolds my professor used the example of an ant looking around and declaring his world flat, ie, Euclidean. Of course, people did the same thing.
Bravo on the series! Any chance we'll get to measure theory or categories somewhere down the road?
10.
dls
Santa Barbara, CA
March 21st, 2010
9:31 pm
Timely, as I sit here with my differential geometry book open and the "Tensor" page up on Wikipedia. Thank you for this!
11.
Christina
Boston, MA
March 21st, 2010
9:31 pm
Thank you for this column. I was a math major in undergrad, and if my differential geometry professor had been as clear and engaging as this post I probably would have fared much better in his course and stuck around for a doctorate. Alas, I am now a recreational mathematician while employed elsewhere. Keep it up!
12.
imperato
NYC
March 21st, 2010
9:31 pm
light travels the path of shortest space time interval in GR
13.
r
estad
March 21st, 2010
9:31 pm
Interesting article. One (slight!) shortcoming of your article is that you only make the distinction between zero curvature and and nonzero curvature. It might be nice to talk about the difference between positive curvature and negative curvature, with respect to topics like how "straight lines" contract or pull apart.
14.
Chavez
Manhattan, KS
March 22nd, 2010
8:14 am
Thank goodness, I was sure that you were going to use the geometry of spheres to make the claim that Obama's use of a social security number from a guy born in 1890 was not theft or fraud and certainly did not point to Obama's illegal alien status. You didn't do that. Thanks.
15.
George O'Conner
Paris
March 22nd, 2010
8:14 am
Non-Euclidean geometry was not only a mathematical revolution but also a more general theoretical revolution. It changed the world into a place where it is less clear to see what is the shortest path. If classical science was simpler and more linear, modern science and math are much more convoluted and it's very hard to know how should one proceed - are you walking towards a dead end, are you taking an unnecessarily long path, or are you really taking the shortest possible path; it's very hard to know when you're on strange shapes. A great discussion on this question of what is the correct path to take in science, and how to do good science:
http://www.pandalous.com...

Speaking of differential geometry and possibilities of shapes, Perlman who recently became the first person to receive the Clay's million dollar Millennium prize for solving the Poincare conjecture deserves our congratulations.
16.
tudza
USA
March 22nd, 2010
8:14 am
My father supervised a group of service people for various stores. He wanted a list of people arranged by who was closest to any given store. Since I found I could get a list of zip codes to latitude and longitude figures for free, I decided to use that and great circles to do the job.

Results were as expected for all but one location. My answer for one location flagged a person different from the one currently assigned. I flagged someone close by great circle terms, but it failed to take into account the fact that Lake Michigan was in the way of a land route.

The shortest distance between two points in not always a great circle if you have to drive.
17.
David
Melbourne, Australia
March 22nd, 2010
8:15 am
Not withstanding winds, jetsreams etc. I presume that modern airplanes don't fly these great circle paths as the earth is rotating as the plane is in the air? Depending on the direction of travel (east to west vs west to east) I presume adjustments are made to give the shortest time in the air (less fuel etc)?
18.
db
Florida
March 22nd, 2010
8:15 am
Thank you for providing this view of mathematics to the general public. I only wish more people would come down from the ivory tower every now and then in order to demystify their fields of expertise for the rest of us. Columns like this, in my view, are always welcome.
19.
Venky
India
March 22nd, 2010
8:15 am
Will a geodesic be present for any surface? That is, given two points on a surface, will it always be possible to get from one point to the other on a motorcycle with locked handlebars? I can see how this would be work for simple, regular surfaces, but not for complex ones. At least, not intuitively.
20.
Len Charlap
Princeton, N.J.
March 22nd, 2010
8:15 am
7. To a topologist a tin can is just a circle, but to a geometer it ain't.

9. You have to be a litle careful here. A differentiable manifold has no geometry and when you give it one, it usually isn't Euclidean or flat geometrically. The usual sphere is a differentiable manifold, but with its usual geometry, it ain't flat. In fact there is no way you can put a geomtry on it (deform it) so it will be flat. If you restrict yourself to compact or finite manifolds and nice geometries (riemannian) there are only finitely many toplogical types of flat manifolds in each dimension, 1 in dim 1 (the circle), 2 in dim 2 (the torus and the Klein Bottle), 10 in dim 3, 75 in dim 4, and that's all we know (I think, I may be out of date. I wrote a book on flat manifolds a long time ago).
21.
Shahab mohd Altaf
INDIA
March 22nd, 2010
8:18 am
THINK GLOBALLY, ACT LOCALLY !
The paths that adapt to the surface
whether curved or Flat forming
Great circles or straight lines
conform to Line, coordination and alignment
the essential features of symmetry
As starlight bends near the sun due to its gravity
so also parallel lines converge when the path
turns from a straight line to a curve in an Ellipse
there are two points, but many paths in-between but one Center of gravity
This proves that a straight line on a flat surface
and a great circle on a curved surface are same
as Earth is an ellipse,neither totally flat nor curved !
22.
Jerry
Rockville, MD
March 22nd, 2010
8:18 am
The bending of space-time by gravity is an interesting example of the result of imposing a basic constraint. A physics professor once explained to me that this explanation is a consequence of a basic physical assumption namely that matter has a positive mass and energy has mass of zero. The question is how can gravity bend light, which has a mass of zero. The only explanation consistent with the rest of physics is that the light does not bend but the space-time through which it travels does bend. This is not difficult to visualize when the space-time that you are talking about is a mathematical construct, but it is difficult to understand when you understand space to be an airless void containing nothing "bend-able." If light actually had a very small mass, its bending by gravity could be explained more simply without resort to curved space-time, but this would upset the consistency of physics.
23.
Minuteman
Boston, MA
March 22nd, 2010
8:18 am
Dear Professor Srogatz, another well done exposition. Perhaps you can connect this article with your discussion of complex numbers and give the often neglected Riemann some of the spotlight? Please?
24.
Dr. Mike Murray
Olney, Illinois
March 22nd, 2010
8:18 am
What a fine thing it is that The NYT is publishing these articles by Dr. Strogatz. I have purchased and viewed his Teaching Company course on 'Chaos' and highly recommend it. No prior knowledge of higher mathematics is necessary to understand and enjoy the course.
25.
Robert
Calistoga, CA
March 22nd, 2010
8:18 am
Does this mean that the NYTimes can be delivered to me faster than before by following a geodesic curve in the delivery route? I hope so! Thanks, great article!

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