[cschmid]

This is pretty obvious, no? If you choose a (uniformly) random point inside an n-sphere, the components of the vector will go to zero as n gets larger -- after all, the sum of their squares has to be less than 1.

A (uniformly) random point in an n-dimensional cube will have random coordinates from zero to 1, with no other constraint on their size.

[reikonomusha]

No, it is not obvious to people who haven't been trained in college-level mathematical thinking, or people who don't think about higher-dimensional objects using the handlebars of abstraction.

[jakear]

Pick N random numbers between -0.5 and 0.5. Square them. Then add them up. How does the probability that {sum of these N positive values exceeds 1} change as the N increases?

I think anyone with a basic understanding of arithmetic should be able to answer that.

[reikonomusha]

Is it obvious to anybody with "a basic understanding of arithmetic" that this has anything to do with higher dimensional spheres?

I think any great teacher can break down many complicated subjects (like the nature of the surface of a higher dimensional sphere) into a series of simpler subjects (if you have to choose a bunch of numbers so they add up to 100, will the numbers tend to be on the small side?), but that in no way implies a person capable of counting change will intuit this series of simpler subjects themselves.

I must ask, somewhat rhetorically, isn't that obvious?

[jakear]

Looks like this is just a debate of the semantics of the word "obvious". Feynman would likely find a great many things "obvious" that I could not hope to understand. This doesn't mean they aren't "obvious", just that I don't have the requisite mental frameworks.

The only frameworks required to reach this conclusion are the Pythagorean theorem (8th grade geometry), the definition of a circle (same), and the ability to generalize the Pythagorean theorem to higher dimensions. I take it this last bit is where you think people would struggle. I don't think so. Looking at the theorem, there are basically 2 ways one might attempt to generalize it, and the incorrect way (a^n + b^n + ... )^(1/n) can be demonstrated to fall on its face pretty easily by considering small values of higher dimensioned components.

I consider these a "basic understanding of arithmetic" as these frameworks have been around for thousands of years and are expected knowledge for the youngest of teenagers. Yes some hand holding may be required to generalize the idea of distance to higher dimensions, but I'd wager not as much as you seem to think.

Testing this is interesting. If a random person at a bar can be given a refresher on the above frameworks and reach the desired conclusion themselves in 5 minutes or less, would you grant the problem is "obvious to people who haven't been trained in college-level mathematical thinking, or people who don't think about higher-dimensional objects using the handlebars of abstraction"?

[sidlls]

I'm generally shocked when an average person I meet can add fractions without a calculator. I think you wildly overestimate just how "basic" the understanding of arithmetic is for the vast majority of people. This isn't to say these people are stupid--don't conflate knowledge of mechanical mathematics operations with intelligence.

[wardedVibe]

https://xkcd.com/2501/

[thomasahle]

Yes, this is a simple way to think about it:

If you choose n uniformly random values in [-1/2, 1/2] (the unit cube) then the sum of squares will be concentrated around n/12 (just take the variance).

This is way more than 1, which is what you would need to stay inside the unit ball.

[jstimpfle]

It's a nice explanation (maybe leave out the "obvious" part?). How do you explain that the volume goes up first, and peaks at N=5?

[j7ake]

This is nice way to think of the problem. Although I think for unit hyper sphere , I think of the sphere centred at the origin, so the distance from point to origin is always less than 0.5