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)]