U tweetove putem weba ili aplikacija drugih proizvođača možete dodati podatke o lokaciji, kao što su grad ili točna lokacija. Povijest lokacija tweetova uvijek možete izbrisati. Saznajte više
One great thing about math is that you can still learn new things about the sphere - yes, the good old 2-sphere, S². There's more than one way to wrap an n-sphere around the S² except for n=1! More precisely: (1/n)pic.twitter.com/6BSDzf5E7z
It's a hard calculation, and the hardest case is when n = 1 mod 8: https://arxiv.org/abs/1506.00952 Sergei O. Ivanov, Roman Mikhailov and Jie Wu wrote this in 2015 - and then realized that Brayton Gray could have known this result back in 1984. (2/n)
It instantly follows that πₙ(S³) has more than one element except for n=0,1,2. (Oh, duh - πₙ(S²) has just one element when n = 0. I ignored that case in my first tweet.) Curtis showed earlier that πₙ(S⁴) has more than one element for all n > 3. (3/n)
Mahowald and M. Mori showed that πₙ(S⁵) has more than one element for all n > 5. But that's where this business stops! If k > 5, πₙ of the k-sphere has just one element when n=k+4. I don't understand these computations: I merely admire them. (4/n, n = 4)
A thing I've always tried to wrap my head around: Higher homotopy groups vanish for the circle S^1 as, naively speaking, there are not higher order holes to be grasped by them. Why are higher homotopy groups not vanishing for higher S^n, Why is S^1 different?pic.twitter.com/ht3I2ZoeIZ
Because S^1 has a contractible universal cover.
I remember back in the 60’s I had a professor who said he wanted to write a book about the 2-sphere. I remember because I was totally baffled - how could you fill a book talking about such a simple thing!?
Simple things get complicated if you start talking about their interactions with all other things!
Yes. Or consider this beautiful paper by Igor Turkanov https://arxiv.org/abs/1603.02914 given a non recursive formula the function counting primes! Perhaps one of the oldest mathematical problems, coming from Euclid!
Not nearly as basic, but today I got to lookup the list of known functions of great circle distance that define valid covariance matrices on a sphere. It's a much shorter list than in flat space.
Twitter je možda preopterećen ili ima kratkotrajnih poteškoća u radu. Pokušajte ponovno ili potražite dodatne informacije u odjeljku Status Twittera.
