I like "Z" with subscript (or superscript) ">=0" or Z with subscript ">0" depending what you want. "N" usually seems stupid by comparison. (How about "U" for the "unnatural numbers"?) For some reason I'm not too horrified by "Q" to denote the rationals instead of (say) "Z/Z" which seems less wasteful.
Interesting. I use Z with a positive integer subscript n for Z/nZ, and with an appended asterisk for the multiplicative group of integers mod n relatively prime to n. On 03-Mar-15 12:45, Warren D Smith wrote:
I like "Z" with subscript (or superscript) ">=0" or Z with subscript ">0" depending what you want.
"N" usually seems stupid by comparison. (How about "U" for the "unnatural numbers"?)
For some reason I'm not too horrified by "Q" to denote the rationals instead of (say) "Z/Z" which seems less wasteful.
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On Mar 3, 2015, at 10:04 AM, Mike Speciner <ms@alum.mit.edu> wrote:
Interesting. I use Z with a positive integer subscript n for Z/nZ, and with an appended asterisk for the multiplicative group of integers mod n relatively prime to n.
In non-number theory discussions, I prefer to do this, too. But many number theorists see Z_p (mainly for p prime) as the notation for the p-adic integers. So in those contexts I use Z/nZ (n prime or not) for the group or ring of integers mod n. When typography permits I prefer to denote the multiplicative group by adding an × superscript. (That's supposed to be a multiplication symbol.) PUZZLE: Find the n for which the invertible elements of Z/nZ are all square roots of unity. --Dan
On Tue, Mar 3, 2015 at 11:52 AM, Dan Asimov <dasimov@earthlink.net> wrote:
PUZZLE: Find the n for which the invertible elements of Z/nZ are all square roots of unity.
The factors of 24 (such a cool number!). The factor 1 works because 0=1 in the trivial ring. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Exactly -- nice solving! (This is one reason 24 is my favorite number.) --Dan
On Mar 3, 2015, at 12:17 PM, Mike Stay <metaweta@gmail.com> wrote:
On Tue, Mar 3, 2015 at 11:52 AM, Dan Asimov <dasimov@earthlink.net> wrote:
PUZZLE: Find the n for which the invertible elements of Z/nZ are all square roots of unity.
The factors of 24 (such a cool number!). The factor 1 works because 0=1 in the trivial ring.
Friends, mathematicians, awesome people; lend me your ears: I agree with using Z/nZ rather than Z_n for the integers mod n. Of course, if n is prime, one can write F_n instead (although admittedly F_n suggests extra structure of a field rather than a mere ring). Yes, 24 is my favourite natural number as well, for reasons including that one. In addition to the Leech lattice and its ilk, there is a completely unrelated (wait until someone finds a connection) reason: `24 is the least positive n such that there exists a cycle of length n in the integer lattice Z^3 such that its three projections (onto coordinate planes) are all trees.' Returning to the Leech lattice, it's bizarrely related to the fact that: 1^2 + 2^2 + 3^2 + 4^2 + ... + 24^2 = 70^2 Specifically, there's a beautiful lattice in Minkowski space called II_25,1, and (if I remember correctly) the Leech lattice arises as the set of vectors of a particular norm which are perpendicular to the vector: (1, 2, 3, 4, 5, ..., 21, 22, 23, 24; 70) which is lightlike (has norm zero). I think most of the other beautiful properties of 24 can be related to the Leech lattice. The Golay code obviously arises once one artificially imposes coordinate axes upon the lattice, and then of course you get the sporadic group M_24 completely for free. There's the string-theoretic fact that bosonic string theory only works in 25+1-dimensional spacetime, and I think that's also related to II_25,1. I was actually wondering whether one can define a `discrete string theory' on II_25,1 (I'm a combinatorialist, so I favour discrete objects), possibly as a cellular automaton (just to please Stephen Wolfram). Speaking of appeasing Wolfram, I recently launched a distributive search of random initial configurations in lifelike cellular automata. In order to contribute, it's as simple as running the Python script in Golly (preferably n instances if your computer has n cores). We're at 66 billion objects so far for the most popular search (Conway's Game of Life with no symmetry), which is satisfying given that it's only been running for 11 days: http://catagolue.appspot.com/census/b3s23/C1 (At the moment I'm running the script on 44 CPUs to attempt to reclaim my position as most prolific contributor, after it was stolen from me by my trans-Atlantic friend Dave Greene. He's only got 12 CPUs at his disposal, but they can run 24/7 rather than just overnight when no-one's looking!) Sincerely, Adam P. Goucher
Sent: Tuesday, March 03, 2015 at 8:19 PM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] natural numbers
Exactly -- nice solving!
(This is one reason 24 is my favorite number.)
--Dan
On Mar 3, 2015, at 12:17 PM, Mike Stay <metaweta@gmail.com> wrote:
On Tue, Mar 3, 2015 at 11:52 AM, Dan Asimov <dasimov@earthlink.net> wrote:
PUZZLE: Find the n for which the invertible elements of Z/nZ are all square roots of unity.
The factors of 24 (such a cool number!). The factor 1 works because 0=1 in the trivial ring.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I also like the fact that (the self-dual 4-dimensional regular polytope called) the 24-cell has no analogue among regular polytopes in any dimension. And its 24 vertices, if chosen well in the unit quaternions, form a group — the binary tetrahedral group. --Dan
On Mar 3, 2015, at 4:27 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Friends, mathematicians, awesome people; lend me your ears:
I agree with using Z/nZ rather than Z_n for the integers mod n. Of course, if n is prime, one can write F_n instead (although admittedly F_n suggests extra structure of a field rather than a mere ring).
Yes, 24 is my favourite natural number as well, for reasons including that one. In addition to the Leech lattice and its ilk, there is a completely unrelated (wait until someone finds a connection) reason:
`24 is the least positive n such that there exists a cycle of length n in the integer lattice Z^3 such that its three projections (onto coordinate planes) are all trees.'
Returning to the Leech lattice, it's bizarrely related to the fact that:
1^2 + 2^2 + 3^2 + 4^2 + ... + 24^2 = 70^2
Specifically, there's a beautiful lattice in Minkowski space called II_25,1, and (if I remember correctly) the Leech lattice arises as the set of vectors of a particular norm which are perpendicular to the vector:
(1, 2, 3, 4, 5, ..., 21, 22, 23, 24; 70)
which is lightlike (has norm zero).
I think most of the other beautiful properties of 24 can be related to the Leech lattice. The Golay code obviously arises once one artificially imposes coordinate axes upon the lattice, and then of course you get the sporadic group M_24 completely for free.
There's the string-theoretic fact that bosonic string theory only works in 25+1-dimensional spacetime, and I think that's also related to II_25,1. I was actually wondering whether one can define a `discrete string theory' on II_25,1 (I'm a combinatorialist, so I favour discrete objects), possibly as a cellular automaton (just to please Stephen Wolfram).
Speaking of appeasing Wolfram, I recently launched a distributive search of random initial configurations in lifelike cellular automata. In order to contribute, it's as simple as running the Python script in Golly (preferably n instances if your computer has n cores). We're at 66 billion objects so far for the most popular search (Conway's Game of Life with no symmetry), which is satisfying given that it's only been running for 11 days:
http://catagolue.appspot.com/census/b3s23/C1
(At the moment I'm running the script on 44 CPUs to attempt to reclaim my position as most prolific contributor, after it was stolen from me by my trans-Atlantic friend Dave Greene. He's only got 12 CPUs at his disposal, but they can run 24/7 rather than just overnight when no-one's looking!)
Sincerely,
Adam P. Goucher
Sent: Tuesday, March 03, 2015 at 8:19 PM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] natural numbers
Exactly -- nice solving!
(This is one reason 24 is my favorite number.)
--Dan
On Mar 3, 2015, at 12:17 PM, Mike Stay <metaweta@gmail.com> wrote:
On Tue, Mar 3, 2015 at 11:52 AM, Dan Asimov <dasimov@earthlink.net> wrote:
PUZZLE: Find the n for which the invertible elements of Z/nZ are all square roots of unity.
The factors of 24 (such a cool number!). The factor 1 works because 0=1 in the trivial ring.
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Here's a nice presentation of all those cool facts: http://math.ucr.edu/home/baez/numbers/24.pdf On Tue, Mar 3, 2015 at 5:39 PM, Dan Asimov <asimov@msri.org> wrote:
I also like the fact that (the self-dual 4-dimensional regular polytope called) the 24-cell has no analogue among regular polytopes in any dimension.
And its 24 vertices, if chosen well in the unit quaternions, form a group -- the binary tetrahedral group.
--Dan
On Mar 3, 2015, at 4:27 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Friends, mathematicians, awesome people; lend me your ears:
I agree with using Z/nZ rather than Z_n for the integers mod n. Of course, if n is prime, one can write F_n instead (although admittedly F_n suggests extra structure of a field rather than a mere ring).
Yes, 24 is my favourite natural number as well, for reasons including that one. In addition to the Leech lattice and its ilk, there is a completely unrelated (wait until someone finds a connection) reason:
`24 is the least positive n such that there exists a cycle of length n in the integer lattice Z^3 such that its three projections (onto coordinate planes) are all trees.'
Returning to the Leech lattice, it's bizarrely related to the fact that:
1^2 + 2^2 + 3^2 + 4^2 + ... + 24^2 = 70^2
Specifically, there's a beautiful lattice in Minkowski space called II_25,1, and (if I remember correctly) the Leech lattice arises as the set of vectors of a particular norm which are perpendicular to the vector:
(1, 2, 3, 4, 5, ..., 21, 22, 23, 24; 70)
which is lightlike (has norm zero).
I think most of the other beautiful properties of 24 can be related to the Leech lattice. The Golay code obviously arises once one artificially imposes coordinate axes upon the lattice, and then of course you get the sporadic group M_24 completely for free.
There's the string-theoretic fact that bosonic string theory only works in 25+1-dimensional spacetime, and I think that's also related to II_25,1. I was actually wondering whether one can define a `discrete string theory' on II_25,1 (I'm a combinatorialist, so I favour discrete objects), possibly as a cellular automaton (just to please Stephen Wolfram).
Speaking of appeasing Wolfram, I recently launched a distributive search of random initial configurations in lifelike cellular automata. In order to contribute, it's as simple as running the Python script in Golly (preferably n instances if your computer has n cores). We're at 66 billion objects so far for the most popular search (Conway's Game of Life with no symmetry), which is satisfying given that it's only been running for 11 days:
http://catagolue.appspot.com/census/b3s23/C1
(At the moment I'm running the script on 44 CPUs to attempt to reclaim my position as most prolific contributor, after it was stolen from me by my trans-Atlantic friend Dave Greene. He's only got 12 CPUs at his disposal, but they can run 24/7 rather than just overnight when no-one's looking!)
Sincerely,
Adam P. Goucher
Sent: Tuesday, March 03, 2015 at 8:19 PM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] natural numbers
Exactly -- nice solving!
(This is one reason 24 is my favorite number.)
--Dan
On Mar 3, 2015, at 12:17 PM, Mike Stay <metaweta@gmail.com> wrote:
On Tue, Mar 3, 2015 at 11:52 AM, Dan Asimov <dasimov@earthlink.net> wrote:
PUZZLE: Find the n for which the invertible elements of Z/nZ are all square roots of unity.
The factors of 24 (such a cool number!). The factor 1 works because 0=1 in the trivial ring.
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On 03/03/2015 17:45, Warren D Smith wrote:
I like "Z" with subscript (or superscript) ">=0" or Z with subscript ">0" depending what you want.
In view of our recent notational disagreements, it is a pleasure to say that I use the exact same notation for (I take it) the exact same reasons, and commend it to others' consideration. -- g
participants (7)
-
Adam P. Goucher -
Dan Asimov -
Dan Asimov -
Gareth McCaughan -
Mike Speciner -
Mike Stay -
Warren D Smith