I think an appeal to the unit square in higher dimensions being spiky isn’t necessary, and is suspicious anyway. It’s easy to understand why the central ball pokes through the unit square, in high dimensions: the unit balls at the corners have vanishingly small volumes, so the central ball has plenty of room to grow.
Also, I wonder if the fact that most of the volume of a high-dimensional ball is near its boundary is a possible explanation for the holographic principle in string theory. If the universe is indeed 10-dimensional, then most of its volume would be squeezed to its boundary so most everything that happens, happens at the boundary.
I was always better at stories than equations, so I come at math the way a tourist comes at a foreign language: enthusiastic, largely incompetent, occasionally getting the gist. But the thing I keep finding is that the "underappreciated" formulas almost always turn out to be the ones where the mathematician was solving something that seemed purely abstract and then it turned out to describe the physical world almost embarrassingly well. It's the part of science history that feels most like a practical joke. The universe keeps doing the homework.
I'll be reading this one twice. Possibly three times. Still won't fully understand it, but that's sort of the point.
The fact that a sphere's volume peaks in a finite number of dimensions and then starts shrinking toward zero is one of those things that sounds wrong until you sit with it for a while. And then it still sounds wrong, but you've accepted that your spatial intuition was never built for this.
I spent my grad program at UC Santa Cruz, so seeing it filmed there is a nice bonus. Wonderful talk.
great talk, thanks. But im curious: did you move slides / animations yourself? I assume your animations had nome 'next slide like' feature, or your talk would be forced to follow the movie ;-) Expecially in response to audience reactions to questions (although sort of predictable) that seems challenging, were there different animations prepared? or would you have talked to the one you have if the audience responded differently?
Love it! One thing I think is interesting: You get an extra factor of pi every two dimensions, which kinda corresponds to the fact that 'rotation‘ is a two-dimensional idea, so there’s one power of pi for each independent axis you can rotate around.
Great talk!
I think an appeal to the unit square in higher dimensions being spiky isn’t necessary, and is suspicious anyway. It’s easy to understand why the central ball pokes through the unit square, in high dimensions: the unit balls at the corners have vanishingly small volumes, so the central ball has plenty of room to grow.
Also, I wonder if the fact that most of the volume of a high-dimensional ball is near its boundary is a possible explanation for the holographic principle in string theory. If the universe is indeed 10-dimensional, then most of its volume would be squeezed to its boundary so most everything that happens, happens at the boundary.
I was always better at stories than equations, so I come at math the way a tourist comes at a foreign language: enthusiastic, largely incompetent, occasionally getting the gist. But the thing I keep finding is that the "underappreciated" formulas almost always turn out to be the ones where the mathematician was solving something that seemed purely abstract and then it turned out to describe the physical world almost embarrassingly well. It's the part of science history that feels most like a practical joke. The universe keeps doing the homework.
I'll be reading this one twice. Possibly three times. Still won't fully understand it, but that's sort of the point.
The fact that a sphere's volume peaks in a finite number of dimensions and then starts shrinking toward zero is one of those things that sounds wrong until you sit with it for a while. And then it still sounds wrong, but you've accepted that your spatial intuition was never built for this.
I spent my grad program at UC Santa Cruz, so seeing it filmed there is a nice bonus. Wonderful talk.
And the founders of quantum mechanics noticed.
Schrödinger read the Upanishads. Heisenberg spoke with Indian philosophers. Oppenheimer quoted the Gita the day the first nuclear bomb exploded.
These weren't spiritual gestures. They were intellectual acknowledgements.
great talk, thanks. But im curious: did you move slides / animations yourself? I assume your animations had nome 'next slide like' feature, or your talk would be forced to follow the movie ;-) Expecially in response to audience reactions to questions (although sort of predictable) that seems challenging, were there different animations prepared? or would you have talked to the one you have if the audience responded differently?
It is really fun to derive this formula! Seeing what shows is mesmerizing.
Love it! One thing I think is interesting: You get an extra factor of pi every two dimensions, which kinda corresponds to the fact that 'rotation‘ is a two-dimensional idea, so there’s one power of pi for each independent axis you can rotate around.
Looking forward to it.
Coincidentally, I just watched this video about deriving the volume of a 4D sphere:
https://youtu.be/Zx0WC_R3-7w