[math-fun] More examples of serious weirdness for higher integers?
I'm interested in very natural examples in math where something goes askew in a big way after some low integers. In topology, it's known that the n-dimensional sphere (the unit sphere in R^(n+1) cannot have more than one smooth structure if n = 1, 2, 3, 5, 6. Almost every sphere of dimension >= 7 does have more than 1. (The case n = 4 is unresolved.) As Adam pointed out, the symmetric and alternating groups S_n, A_n, have some strange things happening for low n. Like A_n is simple, except not for n = 4. The outer automorphism group Out(S_n) is trivial for all n *except* for n = 6 (there is an outer automorphism). What are the best examples of this kind of thing in various branches of math? I'm not so interested in things like "Every integer greater than 127401 is the sum of 17 9th powers." I just made that up, but you get the idea. —Dan
Here's a repository of results of this form: https://math.stackexchange.com/questions/186103
Sent: Wednesday, January 16, 2019 at 3:21 AM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] More examples of serious weirdness for higher integers?
I'm interested in very natural examples in math where something goes askew in a big way after some low integers.
In topology, it's known that the n-dimensional sphere (the unit sphere in R^(n+1) cannot have more than one smooth structure if n = 1, 2, 3, 5, 6. Almost every sphere of dimension >= 7 does have more than 1. (The case n = 4 is unresolved.)
As Adam pointed out, the symmetric and alternating groups S_n, A_n, have some strange things happening for low n. Like A_n is simple, except not for n = 4. The outer automorphism group Out(S_n) is trivial for all n *except* for n = 6 (there is an outer automorphism).
What are the best examples of this kind of thing in various branches of math?
I'm not so interested in things like "Every integer greater than 127401 is the sum of 17 9th powers." I just made that up, but you get the idea.
—Dan
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not in that list: Frobenius Theorem. The only n > 0 for which there is an n-dimensional real associative division algebra are n = 1,2 and 4. ---and they are the usual suspects. Theorem. The only spheres which are Lie groups are S^0, S^1 and S^3. Theorem. The only finite dimensional real alternative (or normed) division algebras are the real numbers, the complex numbers, the quaternions and the octonion. On Tue, Jan 15, 2019 at 10:39 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
Here's a repository of results of this form:
https://math.stackexchange.com/questions/186103
Sent: Wednesday, January 16, 2019 at 3:21 AM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] More examples of serious weirdness for higher integers?
I'm interested in very natural examples in math where something goes askew in a big way after some low integers.
In topology, it's known that the n-dimensional sphere (the unit sphere in R^(n+1) cannot have more than one smooth structure if n = 1, 2, 3, 5, 6. Almost every sphere of dimension >= 7 does have more than 1. (The case n = 4 is unresolved.)
As Adam pointed out, the symmetric and alternating groups S_n, A_n, have some strange things happening for low n. Like A_n is simple, except not for n = 4. The outer automorphism group Out(S_n) is trivial for all n *except* for n = 6 (there is an outer automorphism).
What are the best examples of this kind of thing in various branches of math?
I'm not so interested in things like "Every integer greater than 127401 is the sum of 17 9th powers." I just made that up, but you get the idea.
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Not quite what Dan asks for, but it seems worth mentioning that all regular 4-dimensional regular polytopes correspond to 3-dimensional ones (the regular polyhedra) except one, the 24-cell discovered in 1852 by Ludwig Schäfli. On Tue, Jan 15, 2019 at 9:23 PM Dan Asimov <dasimov@earthlink.net> wrote:
I'm interested in very natural examples in math where something goes askew in a big way after some low integers.
In topology, it's known that the n-dimensional sphere (the unit sphere in R^(n+1) cannot have more than one smooth structure if n = 1, 2, 3, 5, 6. Almost every sphere of dimension >= 7 does have more than 1. (The case n = 4 is unresolved.)
As Adam pointed out, the symmetric and alternating groups S_n, A_n, have some strange things happening for low n. Like A_n is simple, except not for n = 4. The outer automorphism group Out(S_n) is trivial for all n *except* for n = 6 (there is an outer automorphism).
What are the best examples of this kind of thing in various branches of math?
I'm not so interested in things like "Every integer greater than 127401 is the sum of 17 9th powers." I just made that up, but you get the idea.
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The way to make this precise is to note that the 24-cell is the only regular 4-dimensional polytope which *doesn't* have a degree-2 vertex with two unannotated edges in its Coxeter-Dynkin diagram: *----*----* which, when contracted to give: *---------* yields the 'corresponding' 3-dimensional regular polytope. The correspondence works in both directions, because every 3-dimensional regular polytope contains a single (up to isomorphism) unannotated edge in its Coxeter-Dynkin diagram which can be subdivided to yield the 'corresponding' 4-polytope. More generally, Coxeter-Dynkin (and indeed ordinary Dynkin) diagrams tend to explain the lion's share of low-dimensional coincidences. Best wishes, Adam P. Goucher
Sent: Wednesday, January 16, 2019 at 5:36 PM From: "James Buddenhagen" <jbuddenh@gmail.com> To: "Dan Asimov" <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] More examples of serious weirdness for higher integers?
Not quite what Dan asks for, but it seems worth mentioning that all regular 4-dimensional regular polytopes correspond to 3-dimensional ones (the regular polyhedra) except one, the 24-cell discovered in 1852 by Ludwig Schäfli.
On Tue, Jan 15, 2019 at 9:23 PM Dan Asimov <dasimov@earthlink.net> wrote:
I'm interested in very natural examples in math where something goes askew in a big way after some low integers.
In topology, it's known that the n-dimensional sphere (the unit sphere in R^(n+1) cannot have more than one smooth structure if n = 1, 2, 3, 5, 6. Almost every sphere of dimension >= 7 does have more than 1. (The case n = 4 is unresolved.)
As Adam pointed out, the symmetric and alternating groups S_n, A_n, have some strange things happening for low n. Like A_n is simple, except not for n = 4. The outer automorphism group Out(S_n) is trivial for all n *except* for n = 6 (there is an outer automorphism).
What are the best examples of this kind of thing in various branches of math?
I'm not so interested in things like "Every integer greater than 127401 is the sum of 17 9th powers." I just made that up, but you get the idea.
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Adam P. Goucher -
Dan Asimov -
James Buddenhagen -
W. Edwin Clark