Friday, June 30, 2006

Group Theory

Now someone with a pet peeve might decry my use of "number theory" in the linked/archived post (link), as I'm speaking of brightly colored polyhedra, twisting about under various camera lights (like in POV-Ray), with permutations to match. Matrices get into the act. But where's the "number theory" here? "Isn't this all group theory then?" (quivering chin, indignation) .

Ah, but there's a hidden bridge, like in Uru. Twixt group theory and number theory we have these little groups and fields modulo N. Extract the totients of a number (if prime, it'll be everyone lower -- real prima donnas those primes), and intermultiply them modulo that number. Woah hoah: those little doggies got Closure, Associative, Inverts, and Neutrality (CAIN -- some Biblical allusion I didn't invent (got it from a colleague)). And if modulus N is a prima donna (a prime), then throw in addition, and you've got CAIN in two operators, plus a distributive principle, a little Galois field (I hear he was a chick magnet -- unconfirmed), suitable for study by elementary school students.

So that was coming back over the bridge, from number theory (totatives are the positive integers relatively prime to N, but smaller i.e. no in-common factors but 1 -- a number's totient, an attribute, is "how many" of said totatives it has (if p is prime, phi(p) == p-1 (this phi, which I've seen attributed to Euler, is a name collision with the golden mean, another meaning for the same lowercase greek letter))).

There's another pun lurking here, again not my invention: CAIN reminds us of Abel in the Bible, and a pioneer of group theory is Abel, such that if you not only have Associativity (a(bc)=(ab)c), but Commutativity as well (ab == ba), then your group is described as Abelian. The maths are full of such jokes, which is why the hackers' Gnu Math is such a good fit. We get one anothers' jokes, sometimes at least.