### This is great, and captures a pretty common state of affairs for me :)

This is great, and captures a pretty common state of affairs for me :)

**Originally shared by Calvin and Hobbes**

The Hopf Fibration of S^3 is amazing and beautiful. Rather than describe it here, I'll point you to a lovely online reference with pictures and videos by Niles Johnson.

nilesjohnson.net/hopf.html

To understand it better (and fibrations in general), I recommend this talk by Niles too.

www.youtube.com/watch?v=QXDQsmL-8Us

It turns out there is an analogue of the Hopf fibration for H^3. In fact, there is not just one "fiberwise homogenous" fibration in the hyperbolic case. There is a 2-parameter family of them, plus one additional fibration that does not fit the family. As with S^3, fibers in the H^3 cases are geodesics. They are ultraparallel in fibrations from the family, and parallel in the exceptional fibration. I found the following dissertation by Haggai Nuchi a good intro and resource to help think about all this.

www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf

I'm posting three early pictures of H^3 fibrations below, labeled with t…

Here are three views of a hyperbolic helicoid, an idea suggested by Saul Schleimer. The first puts the helicoid axis on the vertical line through the origin. The other two apply a hyperbolic isometry to move the axis towards the right. Whenever the axis does not go through the origin, it will be an arc in the ball model.

What curve does a single ideal edge of the surface trace out on the ball boundary? I suspect it is a loxodrome, but I haven't proved this. (

en.wikipedia.org/wiki/Helicoid

en.wikipedia.org/wiki/Rhumb_line

Chess secretly makes more sense on a {5,4} hyperbolic chess board. http://blog.andreahawksley.com/non-euclidean-chess-part-2/

http://blog.andreahawksley.com/non-euclidean-chess-part-2

What's a record?

ReplyDeleteIt gets worse when you take relativity into account. The outer parts should appear to be foreshortened more than the inner parts, and the radius should remain the same! So what should a spinning disk look like at relativistic speeds?

ReplyDeleteBack in the day, PC floppies used to spin at a constant rate, but Mac floppies would spin slower as the head moved towards the edge of the disk so as to store date at a constant density and not waste disk.

ReplyDeleteMelinda Green, true to the spirit of this comic, I thought about your comment while trying to fall asleep last night :) The logic that foreshortening is a function of distance seems sounds, so how to resolve the paradox?

ReplyDeleteI think the resolution involves the following... All the points are undergoing a central acceleration, and different points along the radius are undergoing different accelerations. Since accelerations are involved,

generalrelativity will be relevant. That means spacetime measurements won't be flat. Our intuitive expectation for arclength at some radius, s = r * theta, doesn't hold, and circle circumferences will deviate from their Euclidean value of 2 * pi * r.That's as far as I got though. I can't offer specific details about what it would actually look like.

> Since accelerations are involved, general relativity will be relevant.

ReplyDeleteThere is no difficulty in using accelerations within special relativity. Any time you draw a curved worldline in a spacetime diagram you've drawn the path of an accelerating object. Nothing breaks as a consequence. Unless the disk has a very high mass, and hence a significant gravitational field, its entire story can be told in the context of special relativity.

ah, ok. Guess I'll be wide-eyed again tonight :) I will need to think on it more, but I just did a quick search for "spinning disk relativity" and this surfaced:

ReplyDeletehttp://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

A few quotes from that page:

"Einstein's 1916 paper on GR [5] makes no mention of elevators; instead, the Equivalence Principle is introduced via the rotating disk. Einstein reproduces Ehrenfest's argument, but with a different conclusion: since we are no longer assuming flat Minkowski space, Einstein asserts that geometry for the rigid rotating disk is noneuclidean. The Equivalence Principle now implies that geometry in a gravitational field will also be noneuclidean. (By "geometry", I mean spatial geometry, i.e., we're not concerned with the temporal components of the spacetime metric.)"

"This really has nothing to do with general relativity, which is a theory of gravity. If one draws planes of simultaneity at various events in a rotating system such as a disk, then one can form a coherent picture of those events. Much care must be taken with the mathematical details, but these are an expected part of applying the machinery of special relativity." (as Dan Piponi is pointing out)

Having not read Einstein's paper, I hadn't known he discussed a rotating disk. In any case, the link makes it sound like the "fate of the rigid disk" is a subtle problem that has been given much attention!

I'd never thought about this before. Now it's going to keep

ReplyDeletemeawake.