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00:05:04 apparently mkPolar 1 is also known as cis
00:05:21 yes
00:05:27 "cos + i sin"
00:05:51 im pissed off about this cyclotomics
00:06:01 yeah i know, i meant it was a defined haskell function
00:07:07 hm
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00:08:04 z^5 = 1 and not z = 1 means z^4 + z^3 + z^2 + z + 1 = 0 fwiw
00:08:25 i suppose that's one of the results, but it also is simply polynomial division
00:08:49 well
00:08:52 z^4 + z^3 + z^2 + z = -1
00:09:29 but we should be able to extract two sequences, x1 = z^4 + z^2, x2 = z^3 + z (say) such that x1+x2=-1 & x1*x2 =
00:09:35 -!- BeholdMyGlory has quit (Remote host closed the connection).
00:10:13 ah
00:10:25 it doesn't seem to be true though :(
00:10:49 well z and z^4 are also conjugate, so z+z^2 and z^4+z^3 are conjugate
00:11:06 and multiplying conjugates at least gives a positive real
00:11:20 2.6180339887498945
00:11:36 (that is 1+phi, which is not rational)
00:11:58 so I'm a bit lost now as for what I should do
00:12:03 another option would be z+z^3 and z^4+z^2
00:12:16 well that gives 0.38196601125010515
00:12:30 which is 1/2.6180339887498945
00:13:29 -!- pikhq has joined.
00:13:49 well i don't know this stuff, anyway
00:16:04 cyclotomic polys are easy
00:16:05 jeez
00:17:51 !hs let w = exp (2*pi*sqrt(-1)*(1/17)) :: Complex Double in (sum $ map (w^) [3,10,5,11,14,7,12,6])*(sum $ map (w^) [9,13,15,16,8,4,2,1])
00:17:55 !haskell let w = exp (2*pi*sqrt(-1)*(1/17)) :: Complex Double in (sum $ map (w^) [3,10,5,11,14,7,12,6])*(sum $ map (w^) [9,13,15,16,8,4,2,1])
00:17:59 `haskell let w = exp (2*pi*sqrt(-1)*(1/17)) :: Complex Double in (sum $ map (w^) [3,10,5,11,14,7,12,6])*(sum $ map (w^) [9,13,15,16,8,4,2,1])
00:18:02 egobot is not here
00:18:42 (-4.0) :+ (-5.06434096082401e-016) :: Complex Double
00:20:10 you may try privmsg'ing lambdabot. prepend with "> "
00:20:23 *Galois Data.Complex> multiplicativeGroupGenerator 52
00:20:23 *Galois Data.Complex> map(\i->2^i`mod`5)[1..4][2,4,3,1]
00:20:23 *Galois Data.Complex> let w = exp (2*pi*sqrt(-1)*(1/5)) :: Complex Double in (sum $ map (w^) [2,3])*(sum $ map (w^) [4,1])
00:20:26 (-0.9999999999999999) :+ 6.861555643110582e-17
00:20:38 i guess that won't show up here
00:20:53 multiplicativeGroupGenerator 5 = 2
00:21:13 orbit of 2 in multiplicative Z/5Z = [2,4,3,1]
00:21:22 hm
00:21:33 so the periods (w^2+w^3)(w^4+w) = -1
00:21:34 3 is also a generator, i should think
00:21:37 but I thought I did this......
00:21:45 I just pick the first generator
00:22:33 as i said, exp (2*pi*sqrt(-1)*(1/5)) = cis (2*pi/5)
00:22:56 I don\t understand what I had wrong before
00:23:17 This mouse still feels too weird
00:23:33 well we didn't try that splitting
00:23:34 Also, changing the speed of the thing also changes the speed of the trackpad
00:23:39 (w+w^4)*(w^3+w^4) <------- augh!!!
00:23:41 im so stupid
00:25:31 * Sgeo switches back to his wired mouse
00:26:28 fax: i was trying to suggest things that were conjugates of each other, if the product was supposed to be negative no wonder that didn't work :D
00:26:51 oerjan, but you are right -- they are conjugates
00:27:12 um no
00:27:23 if they were conjugates their product would be positive
00:27:42 (w^2+w^3) and (w^4+w), that is
00:27:49 oh right I see
00:28:23 so there's a 'fast' way to calculate the value of (w^2+w^3)*(w^4+w)
00:28:31 but it's really confusind and complicaetd ;D
00:29:03 exactly the right kind of thing for a computer program, you'd think
00:29:46 I am trying to program this algorithm but it is really a lot harder than anything I have worked on before
00:29:53 well I guess that is the wrong way to put it
00:29:59 What if you tried it imperatively?
00:30:01 >.>
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00:40:17 this... appears to only work for fermat primes
00:40:27 but that is not good because there are other primes
00:42:28 oh? what goes wrong for 7, say?
00:42:48 well give me 10 mins I will try p = 7 on paper
00:43:17 {1,2,3,4,5,6} lessee
00:43:37 {2,4,1} nope
00:44:00 [3,2,6,4,5,1]
00:44:05 that's the orbit of 3
00:44:07 {3,2,6,
00:44:13 i was _getting_ there :D
00:44:15 :P
00:44:16 sorry
00:45:10 so do you take (w^3+z^2+w^6)*(w^4+w^5+w^1) ?
00:45:19 or would, if it worked
00:46:11 how did i get that z there
00:46:23 http://www.pasteit4me.com/312001
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00:47:52 huh, so [3,6,5] and [2,4,1], or [3,4] and [6,1]
00:48:36 [3,4], [2,5] and [6,1]
00:48:45 so the thing splits into 2*3 or 3*x
00:48:47 so the thing splits into 2*3 or 3*2 ***
00:48:52 oh any two?
00:49:12 er wait
00:49:17 you multiplied all three
00:49:27 yeah
00:49:37 in general if we have an orbit that covers the whole group [a,b,c,d...]
00:50:07 it seems like you can share it out (one for me, one for you, one for him, one for me, ...) into sets which (obviously sum to -1) and multiply to make a rational number
00:50:21 but why they multiply to make a rational..... I have no idea
00:50:28 oh you are taking all the cosets of a subgroup?
00:50:38 hmm
00:50:43 yes I guess they are cosets aren't they
00:50:54 the group is Z/pZ by the way (multiplicative)
00:51:10 (I don'tknow if I neglected to say that earlier)
00:51:31 i understood that much
00:51:42 although i wouldn't call that Z/pZ...
00:52:04 wikipedia used a superscript X
00:53:03 hm since the multiplicative group is cyclic, i guess all subgroups are cyclic too so given by a single element
00:53:26 oerjan, ooh that explains why you can keep doing it!
00:54:34 it's a bit usesless to know that y1+y2+y3 = -1 & y1*y2*y3 = 1
00:54:36 you can't solve this equation
00:54:47 in fact if this is the case you don't need to find a primitive root at all, unless you want to classify everything
00:54:56 hmmm
00:55:00 p-1 is always even
00:55:11 (p > 2)
00:57:08 OMG BLUE BUBBLEGUM JONES
00:57:21 also, fax! :D
00:57:24 * augur glomps fax
00:57:28 hey augur :)))
00:57:40 i had an idea for a fun little challenge
00:57:46 the universe thing?
00:57:51 yeah did i mention it to you?
00:58:00 ive been modelling simple universes
00:58:08 what is a universe ?
00:59:11 ok so in this context, a universe is some topology, some number of particles that inhabit that topology (or some properties over that topology, take your pick), and some number of rules that govern the physics of those particles in that topology
00:59:59 augur WOLFRAM??
01:00:10 no.
01:00:19 this sounds like cellular automata
01:00:24 im sure you could model (some) of these universes as CAs
01:00:28 except on arbitrary topology
01:00:31 but the physics dont have to be defined CAly
01:00:36 oh there are non-CA rule sets?
01:00:38 like how
01:00:44 well you could do so if you wanted.
01:01:21 just as an example, imagine you had an infinite discrete universe with a particle occupying one position in the universe
01:01:46 you could say that it has a constant velocity in some direction and so forth
01:01:53 im sure you could also define a CA for this tho
01:06:12 so do you know the setup of the game?
01:07:39 I guess so
01:07:42 it sounds very difficult
01:08:09 it is rather, i think
01:08:15 just to reiterate, the game goes like this
01:08:44 i secretly design a (simple) universe. then i provide to you the causal graph of the universe
01:08:52 oerjan, btw -- x1 = z^3+z^6+z^5 ; x2 = z^2+z^4+z^1 gives x1 = -1/2-sqrt(-7/4) ; x2 = -1/2+sqrt(-7/4)
01:08:53 and you have to determine the design i used.
01:09:11 augur, the causal graph is a picture ?
01:09:25 augur I want to try this but I am not likely to win
01:10:08 the causal graph is basically just a directed graph, where each node is one of the possible configurations of the universe, and each directed arc connects a configuration to a "successor" confiuguration
01:10:17 so the problem is, I know that z^3+z^6+z^5 = -1/2-sqrt(-7/4) -- but that doesn't help me find any expression for z^3 or whatever
01:10:22 ie the configuration the universe would be in after running the physics one time unit forward
01:11:27 you could imagine a universe with a single point in it, and that point has a single property Energy that is one of the integers, and the "physics" is such that Energy increases by one every time step
01:11:51 well z^3 is a 7th root of unity, i don't know whether they can be expressed with real roots
01:12:13 so the causal graph of this universe is basically 0 -> 1 -> 2 -> 3 -> ... where each state corresponds to one of the configurations {(point, energy)} of the universe
01:12:24 does this make sense, fax?
01:12:46 vaugely :)
01:13:00 or imagine you have a universe that is a string of text
01:13:04 of length 3
01:13:23 with say 1 of 26 "particles" (letters) in each position
01:13:34 "aaa", "aab", ...
01:13:35 oerjan, well there is a theorem that every root of unity can be expressed in radicals (from Gauss)
01:13:39 these are the configurations of this universe
01:13:53 fax: oh
01:13:55 oerjan, but I am completely failing to see how it works for non-fermat primes
01:14:02 then theres some physics that says each "configuration" is followed by, lets say, the "next" string
01:14:13 or god only knows what the universe is, right
01:14:22 augur yeah I get that it just seems if you can choose any rule you want that it's impossible to figure it out
01:14:33 well let me give you a very simple one :)
01:14:37 ok
01:14:42 here is a simple causal graph: 1 -> 1
01:16:05 this is the simplest universe i can think of
01:16:30 that sucks
01:16:31 :/
01:16:39 idiot universe
01:16:41 cmon, think of what kinds of universes this could be :)
01:16:49 what is the universe and the physics
01:16:59 actually its a whole class of universes, but
01:17:01 the universe is {1} and the physics is I
01:17:35 basically, yeah. the universe is a 1x1 universe with no particles (or 1 particle) and no physical laws that change anything.
01:17:51 alternatively it could be seen as any universe with no particles an no physical laws that change anything
01:18:05 heres the next simplest one i can think of
01:18:08 1 -> 1, 2 -> 2
01:18:24 the universe is {1,2} and the physics is I
01:18:28 ?
01:18:37 well, what do you mean bye {1,2}?
01:18:40 idk
01:18:47 the 1 and 2 that i used are just names for configurations
01:18:47 What do you mean, BYE! {1,2}
01:18:51 it's just the nodes of a graph
01:18:55 there are no edges yet
01:18:56 (Sorry, couldn't help myself)
01:19:10 but the nodes in the graphs are not the universe
01:19:25 the universe is a topology plus a physics that this graph describes
01:20:14 the universe envisioned is as follows:
01:20:30 a 1x2 universe with one particle (which can be in either location) and no physics
01:20:54 or, a 1x1 universe with one particle (which can be pointing in either one of two directions) and no physics
01:21:12 no, it's a particle which can be either red or blue!
01:21:17 the configurations for the former are: [o| ] and [ |o]
01:21:23 oerjan, :)
01:21:33 and for the latter: [<] and [>] lets say.
01:22:43 heres another universe thats similar: 1 -> 1, 2 -> 2, 3 -> 3, 4 -> 4
01:24:34 im sure you can imagine what this is
01:24:58 yes, it's a 1x2 universe with two particles *ducks*
01:25:04 nope :)
01:25:14 unless you hve some physics in there
01:25:24 well no physics
01:25:29 nothing changes
01:25:30 then definitely not
01:25:38 because that universe would be [o|o]
01:25:44 and would have only one state
01:25:55 making its causal graph the same as the first universe: 1 -> 1
01:25:56 you didn't say particles couldn't be in the same position
01:26:07 ahhaaa good thinking oerjan :)
01:26:12 then yes, you're right
01:26:15 it could be that universe
01:26:42 anyway i predict the number of possibilities will become hopelessly large, fast
01:26:49 or it could be a 1x1 universe with a particle that faces one of four directions, or two particles that face one of two directions, or, ...
01:26:58 well, the number of possibilities might
01:27:08 but im not looking for _my_ solution, just _a_ solution
01:27:28 so heres another universe thats kind of interesting
01:27:33 "faces directions" sounds similar to momentum.
01:27:47 :)
01:28:09 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 1; 7 -> 1; 8 -> 4
01:28:15 * Sgeo wants to use Haskell for everything
01:29:07 ill even be generous and tell you the physical makeup of the universe
01:29:13 1x2, one particle.
01:30:07 well we have 2^3 states, so one bit could correspond to position of particle
01:30:30 ok :)
01:30:42 thats a truism, ofcourse
01:30:52 because, modulo anything else, the particle can be in exactly two positions
01:31:03 so the universe has at least 2*n configurations
01:31:06 well it means the particle can have 4 inner states in each position
01:31:13 ok
01:31:27 you're veeeery close
01:31:33 -!- Oranjer has left (?).
01:31:41 now just figure out what the physics is :)
01:31:46 it could point in each of four directions
01:31:54 true.
01:32:44 hm...
01:33:01 i make it a rule that the particle is _always_ changing position, each step
01:33:06 ok
01:33:50 so 1,3,5 and 8 correspond to the same position, 2,4,6,7 to the other
01:34:02 ok
01:34:30 now we can say that 7 and 8 are both pointing down
01:34:35 ok
01:36:20 btw, when you interpreted 1x2, did you interpret that to be 1 high or 1 across?
01:36:38 the particle always rotates its pointing direction each step, but skipping the down direction
01:37:02 i believe this gives the required graph
01:37:15 1 high
01:37:19 ok
01:37:25 thats what i intended :p
01:37:42 er what's what you intended
01:37:59 1 high
01:38:37 1 and 4 point left, 2 and 5 up, and 3 and 6 right
01:38:45 and rotation is clockwise
01:39:04 ok
01:40:27 thatll do it :)
01:41:26 now let me do one thing to my universe: expand the size to 2x2 now. the causal graph is as follows:
01:41:41 this analysis was made somewhat easier by the fact it contained two aspects that did not interfere with each other
01:42:00 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8 -> 1; 9 -> 10 -> 1; 11 -> 12 -> 3; 13 -> 14 -> 5; 15 -> 16 -> 7
01:42:17 identical physics, one particle, just in 2x2 not 1x2
01:42:34 what do you mean by identical physics?
01:42:39 that is, not the physics oyu just devised, but the physics that i had in mind.
01:42:44 ah
01:42:49 as before.
01:42:57 so this universe and the previous one are different _only in size_
01:43:02 now we have 4 positions rather than 2
01:43:07 we do.
01:43:32