Looking beyond the Lovecraftian mysticism, this is actually pretty fascinating in and of itself. I'm not a true mathematician (only a lowly electrical engineer with some training and study in mathematics), but I would not have expected that equality to hold for complex values, let alone quaternions, etc.
The gossipy narrative style of the article is kind of jarring for an article on a topic like this. It took several paragraphs before it touched on the matter.
I dunno about gossipy, but the narrative style is standard at Quanta. It's written for the subscriber who is reading for leisure, and wants a good story as well as some amount of technical depth, not for the HN reader who wants to quickly judge whether figuring this thing out is worth their time, and will abandon it if not.
I always wonder what a popular science/math magazine would look like if it were oriented towards hackers. In this I mean people who have little background in the field but also the type of person who is used to bluntness and knows to RTFM.
I would subscribe to one. Journal articles are often opaque to people who aren't already in the field, and popular science falls too often into the storytelling trap seen here.
For this particular article, I'm not sure a hacker version could be much better. I'm slightly familiar with this area of research, I'm not sure a more "true" explanation couldn't be done in much less than 20 pages of fairly hard maths, and I don't imagine anyone would want to chew through that.
You could trim this down, but I personally find the background as interesting as the result.
The gap is very hard to close because there is a chasm between the two. RTFM here means years (typically 5+ for research math) of very focused study to follow what it says. While the popular science tries to convey something that is meaningful to the majority of people without any background exposure. The gap between the two is huge.
> I always wonder what a popular science/math magazine would look like if it were oriented towards hackers. In this I mean people who have little background in the field but also the type of person who is used to bluntness and knows to RTFM.
I always wonder about what something in between popular science/math and academic journals would look like. Something oriented towards people that know a nontrivial amount but are not researchers. E.g., you can assume they know calculus and have a conceptual understanding of what things like manifolds and homotopy but are fuzzy on the details.
For a topic like in the article, you need a lot of words to give a very vague understanding if you are aiming for a general audience. Not clear it is worth it for the reader. For that more targeted audience it could be a lot shorter and give a little more detail.
Probably too small a market, but I would definitely enjoy that type of content a lot.
I hate reading quanta mag because of the narrative, and didn't have the patience to weed through this to learn something conceptual. But I'm glad there are people who enjoy it.
I hate slide decks like these, where every page in the PDF contains one more bullet point than the last one. Maybe I'm particularly bad at this, but I spend way too long scanning each page for the new information. Is it impossible to configure LaTeX to only produce the final animation step as a completed page and skip the intermediate ones?
The ideal way to read a deck like this, is to download the pdf and enter present mode on your pdf viewer so that you can click through each slide and immediately see where the new material has been added.
The first sentence should have been the ball-is-equal-to-egg explanation with mention of topology. Before that I had no idea what they were talking about.
P.s. I have to assume the rules forbid shapes with surfaces of zero thickness. Otherwise I can just smash a ball into an inner-tube. If the shapes have thickness mandated, what is it? Are the thickness of the surfaces a consideration when morphing from one shape to another? Is the surface thickness negative or positive from zero? All of these questions stem from my experience in 3D modeling where these parameters must be defined.
Have you heard the quip that in physics a cow and a point are equivalent? This is because the physicist cares only about the motion of the thing.
In topology, a doughnut and a coffee mug are equivalent (a mug has exactly one hole, in the handle where your fingers grab). Because the mathematician doesn't care about how hard, thick, or breakable it is; they only care about how complex the shape is. So throw out thickness, size, elasticity, etc.
To add on to what dullcrisp said, which is all correct, even spheres with thickness are “the same as” spheres of zero thickness from the perspective of homotopy theory. “Sameness” here means homotopy equivalence [1]. In fact the thin sphere is a deformation retract [2] of the thick one. The deformation pushes each point of the thick sphere along radial lines towards the thin sphere. Being a deformation retract implies the two spaces are homotopy equivalent.
>Infinitely more maps from spheres to telescopes means infinitely more maps between spheres themselves. The number of such maps is finite for any difference in dimension, but the new proof shows that the number grows quickly and inexorably.
Lost me at the end, but, don't inner-tubes have 2 holes (genus 2), topologically: one for inflation and one for the wheel to fit in. This makes them distinct from a torus (genus 1) and no homotopy exists between them.
You probably are aware, but just to make it clear for others: topologically a torus has one hole (genus 1). The inner void is not considered a hole.
Similarly, a 2-sphere (surface of solid sphere in 3d) has 0 holes (the inner void is not considered).
You are right, but I am pretty sure they just meant the inner tube ignoring the puncture for inflation.
Usually people just use donut/bagel for an illustration of a torus. Not sure why they used inner tube here - maybe to make it clear it is just the surface?
Readit News
I would subscribe to one. Journal articles are often opaque to people who aren't already in the field, and popular science falls too often into the storytelling trap seen here.
You could trim this down, but I personally find the background as interesting as the result.
Expensive.
For a topic like in the article, you need a lot of words to give a very vague understanding if you are aiming for a general audience. Not clear it is worth it for the reader. For that more targeted audience it could be a lot shorter and give a little more detail.
Probably too small a market, but I would definitely enjoy that type of content a lot.
To be fair, manuals are often pretty opaque without the requisite background knowledge.
ne supra crepidam
What a nice 65th birthday present to finally close off the last dangling piece of your almost-complete research agenda!
Cheeky.
P.s. I have to assume the rules forbid shapes with surfaces of zero thickness. Otherwise I can just smash a ball into an inner-tube. If the shapes have thickness mandated, what is it? Are the thickness of the surfaces a consideration when morphing from one shape to another? Is the surface thickness negative or positive from zero? All of these questions stem from my experience in 3D modeling where these parameters must be defined.
In topology, a doughnut and a coffee mug are equivalent (a mug has exactly one hole, in the handle where your fingers grab). Because the mathematician doesn't care about how hard, thick, or breakable it is; they only care about how complex the shape is. So throw out thickness, size, elasticity, etc.
The study is about the properties of (higher dimensional) shapes rather than concrete objects. It’s like asking what’s the thickness of a circle.
[1] https://en.wikipedia.org/wiki/Homotopy#Homotopy_equivalence
[2] https://en.wikipedia.org/wiki/Retraction_(topology)
Deleted Comment
is it actually infinitely - or just a lot?
Clearly IANAM.
A torus is like an inner tube - an inner void and a big hole in the middle.
A solid torus just has a big hole in the middle, like a donut.
https://en.wikipedia.org/wiki/Solid_torus
Usually people just use donut/bagel for an illustration of a torus. Not sure why they used inner tube here - maybe to make it clear it is just the surface?