**Sphere vs spheroid**- This polar view looks almost like the inside view of a sphere. When the marble is at 90° you get a cone as the second picture shows, and rings when it's straight ahead at 180°. The big picture has a marble almost at 180°.

Spherical billiards are much simpler because every ray "follows" a great-circle, and stays in a plane through the sphere's center. So to understand a sphere, it's sufficient to study the paths in a circle.

Let's think about what we'd see when we'd highlight our viewpoint, say, with a glow-worm camera. Every white blob then corresponds to a

*closed path*. There's a closed path straight ahead, one gives a triangle, another a square, and so on. Not only do you get a closed path for every

*regular polygon*, there's one for every

*star polygon*!

Every

*fraction*of 180° gives a closed path, but

*irrational*angles, e.g. sqrt(2), don't give closed paths. So we should see a gap for every irrational number. Actually, we only see gaps for some of them, most are covered up by the fact that i don't know how to render a pointlike highlight.

For a clearer view, have a look at

*Thomae's function*, which is exactly about this idea. And, you can see a distorted version of it, hidden in the original picture.

https://en.wikipedia.org/wiki/Thomae%27s_function

One might think that this polar view of a

*prolate spheroid*should be understood by looking at

*ellipses*instead of circles. But the poles are special points! At it, there are no rays that pass

*between*the

*foci*of the ellipse. And that means, none of these paths can belong to the oscillating class, they're all rotating ones. And those behave exactly like circular ones, only mildly distorted.

But when you move the camera away from the pole, even if only a tiny bit, things look much different! The breaking of rotational symmetry has a most dramatic effect: The pearls sitting on the peaks of Thomae's function get pulled apart into spirals! I made the marble in picture 3 transparent so we can see through the blobs.

I've got even more pictures at hand and stories to tell, stay tuned!

This is part 2 of a series of appendices to a post on Visual Insight:

http://blogs.ams.org/visualinsight/2015/04/15/sphere-in-mirrored-spheroid/

**top, Appendix 1**is my announcement on g+ here:

https://plus.google.com/+RefurioAnachro/posts/KbcDG6zbSEw

**next, Appendix 3:**

*"A prolate spheroid's domains"*

https://plus.google.com/+RefurioAnachro/posts/jUQWPxq9jUM

#mathematics, #billiards : #polar view of #spheroid , #EndlessReflections

‹

›

spheroid polar

3 Photos - View album

- Apr 16, 2015
- Dunno, I don't find the spherical situation boring at all - it helps me visualise and understand the situation. Of course the earlier spheroid looks more spectacular, agreed, but I have a weakness for simplicity... ;)Apr 16, 2015
- You're right +Patricia Ritter, i've fixed that comment. I often read the word "boring" as an in-joke to really mean "well-understood", in itself a notion with surprising depth ;^)Apr 16, 2015
- +Refurio Anachro Hardly anything is "well-understood" by me ;) but since you are doing these things for a wider audience, it's great that when asked you guys deliver even what seems trivial to you! Thanks!Apr 16, 2015
- You're welcome +Patricia Ritter. What was it that "trivial" means... I'm to lazy to look it up... Maybe it's a moving target, like "well-understood". Which can develop back
*and*forth, for me at least...

Or maybe "trivial" is "life belt", but only if you're not an expert (that would be almost everyone).Apr 16, 2015 - +David Roberts Sounds crazy but makes absolute sense :)

It (? stuff...) always seems to boil down to properly understanding how to organise logical thought, in other words to understanding the structure of the language most appropriate to the task... or even the meta-language... [late night ramblings]Apr 16, 2015 - >
*Another variant of this is due to Freyd: The purpose of category theory is "to show that which is trivial is trivially trivial"*

Right to the point +David Roberts!Apr 17, 2015

Add a comment...