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cubic quarternion group
*To*: mathgroup at smc.vnet.net
*Subject*: [mg53464] cubic quarternion group
*From*: "Roger L. Bagula" <rlbtftn at netscape.net>
*Date*: Thu, 13 Jan 2005 03:12:07 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
(* cubic quarternion group: *)
(* the fourth dimension without leaving the complex plane*)
(* a cyclotomic x^3+1=0 type of group *)
(* angles are {0,Pi,-2*Pi/3.Pi/6}*)
(* it is possible to map Hamilton quaternions to the complex plane using
this type of group*)
(* using a substitution conformal map: i->icube,j->jcube,k->kcube*)
(* this group owes it's origin to number theory quadratic fields and
Pell equations*)
(* when taken as polygon in the plane , it's "moment" as a center of
rotation is not zero *)
(* it is a boomerang type group for this reason*)
(* Roger L. Bagula 11 Jan 2005*)
i=-1
j=1+(-1)^(1/3)
k=-(-1)^(1/3)
Simplify[ExpandAll[(1+i+j+k)^3]]
N[j]
N[k]
(* group average/ center*)
N[(1+i+j+k)/4]
(* subgroup centers*)
N[(1+i)/2]
N[(j+k)/2]
Clear[i,j,k]
Solve[{ (1+i+j+k)^3-1==0,i^3+1==0,j^3+1==0,k^3+1==0},{i,j,k}]
Simplify[Expand[(z-1)*(z+1)*(z+(-1)^(1/3))*(z-(1+(-1)^(1/3)))]]
Solve[(-1+z^2)*(-I*Sqrt[3]-z-z^2)==0,z]
Arg[1]
Arg[-1]
Arg[1+(-1)^(1/3)]
Arg[-(-1)^(1/3)]
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