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
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.
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?
Sure, no problem, author.
An n-manifold would just mean any n-dimensional manifold. These are very particular n-dimensional manifolds, namely, spheres.
Now of course, this is topology, so our equivalences are broad; but the thing these are all equivalent (homeomorphic) to is a sphere. Sure, you can take a more complicated shape that's equivalent to a sphere, but that complexity is incidental; the broad equivalences of topology let us ignore them. (Although, alternatively, they also let us turn the sphere into, say, a cube, if that's easier to think about, which often it is.)
also a hypercube is not a cube--it's an n-cube. otherwise this is just lazy pop science rhetoric to get the kids excited about their field (and eventually suppress wages in mathematics with their newly-supplied labor, degree in hand). except not even science, so even less important
I understand that these objects are topologically equivalent to n-spheres, but that doesn't make them n-spheres, let alone spheres proper. In fact, you point out that cubes and spheres are topologically equivalent despite zero spheres being cubes and zero cubes being spheres.
please no knee-jerk 'this is pure mathematics, it doesn't need applicability' answers.
There's an applications section in the Wikipedia article, but it's all to other parts of pure math. It's hard to summarize, but they've got to do with obstructions to untangling, unwrapping, or otherwise solving things to do with spaces.
