3Blue1Brown explains topology

December 29, 2024 12:20 PM

3Blue1Brown (previously) has released a new video: "This open problem taught me what topology is".

This is an updated "2nd edition" for the 2016 video "Who cares about topology?"

posted by ShooBoo (9 comments total)

This was great. Thanks for posting.
posted by roolya_boolya at 4:02 PM on December 29

I always get sort of irrationally fidgety with self-intersecting forms, like it offends some part of my brain in an almost uncanny, body horror sort of way. Stellated polyhedra are bad for this, but somehow Klein bottles are worse. I'm getting over it in bits and pieces but it has honestly been a distraction sometimes when trying to learn a bit more about topology now and then. Brains, man.

Anyway, lovely video, 3B1B doing good work as always and this pulled together a couple more things that hadn't quite clicked for me before as far as some of the mappings/isomorphisms of these different ways of thinking about these point-pair relationships. I really like that visualization of the rectangle points as sets following those lines of self-intersection in the z-axis mountain objects growing out of the loops; I'm curious whether (as analogous to the example with the circle) there's also always a rectangle midpoint corresponding to every local maxima or "peak" of the mountain.
posted by cortex at 7:00 PM on December 29

I think this was my third viewing and am quietly pleased with myself that I could hold a couple sections in my head long enough for that brief 'got it' sensation... until the next fold or projection that spasm'd that poor old muscle.

Waiting for one on de Rham cohomology, as I deeply crave that whimpering brain melt.
posted by sammyo at 7:34 PM on December 29

Mindbendy! According to the timestamp, it completely lost me by 14 minutes.
posted by monocultured at 3:56 AM on December 30

Now I'm wondering (I mean like distracted wondering) whether there's an infinite number of squares inscribed on the curve. An infinite number of rectangles? What about regular polygons?
posted by Nancy Lebovitz at 4:27 AM on December 30

One bit the video touches on there is the idea that when there's a point of self-intersection at any given layer of a "mountain", that intersection continues to exist upward and downward some amount as you change the height of your slice, with the point pairs on the loop that define the respective 3D point moving smoothly by small degrees at the same time. As *soon* as you've got that situation for a mountain, you have an infinite number of rectangles, since that continuous line of intersections moving through 3D space has infinitely many points along it!

Other regular polygons is an interesting question and man I have no idea how to start thinking about it though my immediate gut instinct is that anything (other than maybe a regular triangle) would be far more restricted and most curves wouldn't accommodate anything beyond a rectangle. But maybe if we embrace the rectangle concept and say a regular but scaled polygon, like permit the aspect ratio to stretch in an arbitrary direction the way we're allowing a rectangle instead of a square?
posted by cortex at 8:25 AM on December 30

(One quick thought there, though—taken to an extreme, a regular n-sided polygon for large n approaches a true circle, and it's pretty trivial to establish that most arbitrary curves do *not* closely approximate a circle. Or, allowing for stretching along an axis, do not closely approximate an ellipse either. So there's *somewhere* between n = 4 and n = infinity where something changes from "can totally pull that off every time" to "can pull that off almost never". The question is, is that line drawn between n = 4 and n = 5, or further out; and maybe the question is where are a series of breakpoint lines drawn where the category of curves that fit the brief gets discretely smaller by some new constraint?)
posted by cortex at 8:31 AM on December 30

The ellipse is an example of a curve which doesn't have any inscribed regular n-gons for n > 4 (unless it is a circle). That is because all the vertices of a regular n-gon lie on a common circle, and a circle and ellipse intersect in at most four points.
posted by eruonna at 4:57 PM on December 30 [2 favorites]

Yes! And but then if you allow for scaling the regular n-gon along an arbitrary axis so by the same ratio that the ellipse scales away from an initial circle, you can then fit that n-gon to the ellipse after all, at least handwavingly analogous to the original problem's greater flexibility in fitting rectangles vs. squares. How much farther you can really go with it for arbitrary n-gons vs. arbitrary curves is obviously a way broader question though, so no idea how to proceed from there.

One idle thought in this neighborhood: I reckon you can fit a distended regular n-gon to a rectangle up to n=6, but after that (with the exception case of n=8) I think you must be boned? Can't see a way to get a regular heptagon to work, octagon's a funny case because it every other side ends up orthogonal, and then nonagons on the problem just gets worse and worse.
posted by cortex at 11:21 AM on December 31