|
Steve Trettel
@
stevejtrettel
Providence, RI
|
|
Minnesotan. Topology Postdoc @Stanford. Space nerd, cook, language revitalization activist
|
|
|
262
Tweetovi
|
1.024
Pratim
|
620
Osobe koje vas prate
|
| Tweetovi |
|
Steve Trettel
@stevejtrettel
|
27 min |
|
Wow - beautiful!
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
20 h |
|
The clips I’ve posted of the earth are also 90deg
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
23 h |
|
Anyone interested in the Nil geometry facts I’ve been tweeting out this week, be sure to check out the work of @ZenoRogue if you haven’t already 😃 twitter.com/zenorogue/stat…
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
Yeah! For each radius of helix, there’s a sequence of distances along the z axis where they meet back up after one,two,three... rotations. If this were a string these would be the “nodes” of a fixed-frequency vibrational mode. In Nil - they are the conjugate points of the metric!
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
(These are the coordinates we actually compute the geodesic flow in because the formula is simpler! They are just harder to explain / motivate)
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
You can change coordinates to a model where the rotation symmetries act like normal rotations of R3. This makes the geodesics look more like actual helices, but the trade off is now the metric tensor looks complicated. Here’s a rendering of some geodesics in these coordinates. pic.twitter.com/P5Q53sqpym
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
This can be seen in our paper a little if you look at the picture of the balls in nil on pg2. The curves drawn on them are the lines of latitude, which are wiggly precisely because the rotational symmetry acts funny in these coordinates.
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
It fixes the z-axis, but instead of rotating the orthogonal planes like a rigid rotation of E^3, it actually preserves a collection of hyperbolic paraboloids - like Pringle chips- instead of flat planes.
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
It’s just an “artifact” of the coordinates which are used. In one since they are the natural ones, because they come from the Heisenberg group and the metric looks simple. But, as a trade off, the extra 1-parameter of “rotations” looks complicated in these coordinates.
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
Maybe- “the shape of continued fraction approximates of phi” could lead to a nice series of pictures. (Or maybe trash I’m tired af hah)
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
Fun idea! If you do the “shape of a rational number” this way - it should close up? And the golden ratio is the “least rational” so it’s limit is all over the place. 🤔
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
Whoa that thing is all over the place!
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
Whoops the day got away from me! We will talk about why the earth doesn’t shrink in size as it recedes into the distance (quite unlike what happens in Euclidean space) another day. Here’s a hint 😃 pic.twitter.com/z3QZEYlTZL
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
6. velj |
|
Does the limit of the circles trace out a cool curve?
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
5. velj |
|
Thanks! 😀
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
5. velj |
|
Haha! Yes Lacan said some pretty crazy things (I had a housemate in graduate school who was reading something in which he mentioned “cybernetic homology” and Klein bottles when talking about the mind). Beautiful animation though!!
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
5. velj |
|
Ok - gotta get through airport security. Will finish the story (4) in a while! Ask away I’ll try and get to your questions while waiting at my gate too :)
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
5. velj |
|
More rings appear as this process repeats: in this figure light reaches the earth along three kinds of paths: headin straight there(green), traversing one large spiral (yellow), or two smaller spirals (blue). This results in a central image surrounded by 1 large and 1 small ring pic.twitter.com/pQkCNOZC3k
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
5. velj |
|
As the earth continues to recede, this gap grows - and we see the ring shaped mirage expanding away from the earth with a wider and wider black annulus in between! pic.twitter.com/EfeZxideDT
|
||
|
|
||
|
Steve Trettel
@stevejtrettel
|
5. velj |
|
Looking closely at this, we notice we see the earth in the direction of the red geodesics, and again in the direction of the blue geodesics, but in between we see a thin black circle, of light rays escaping into space!
|
||
|
|
||