Multidimensional Generalization of Complex Numbers&Quaternions

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Multidimensional Generalization of Complex Numbers&Quaternions From: STALKER2002 Subject: General Date/Time 2008-07-07 01:08:49 Remote IP: 79.139.140.230 Message Now let us consider the multidimensional sphere in n-dimensional euclidean space R^n (n=2,3,4,5...N) and points A,B, C on its surface being three cities. The composition of travels along geodesical shortest way arcs from A to B and then from B to C is but MULTIPLICATION of GENERALIZED quternions: AC=AB@BC!!!!!! Geometrical Interpretation of Quaternions&MULTIPLICATION: (365) STALKER2002 (346) - - 2008-07-01 4:04 pm Re: Geometrical Interpretation of Complex Numbers&MULTIPLICATION: (277) STALKER2002 (346) - - 2008-07-07 00:06 am Multidimensional Generalization of Complex Numbers&Quaternions: (258) STALKER2002 (346) - - 2008-07-07 01:08 am Does anybody understand that this is a new DISCOVERY?: (314) STALKER2002 (346) - - 2008-07-07 2:59 pm NEW?: (272) Steven (680) - (Training Log) - 2008-07-08 4:20 pm Re: NEW? after quaternions and octanions there was NO other generalization of associative algebra: (262) STALKER2002 (346) - - 2008-07-08 4:39 pm Frobenius theorem (real division algebras): (279) STALKER2002 (346) - - 2008-07-08 5:13 pm Frobenius theorem (real division algebras): (256) Steven (680) - (Training Log) - 2008-07-08 5:18 pm Commutativity is not needed for quaternions and their generalization: (262) STALKER2002 (346) - - 2008-07-08 5:33 pm I never said that: (256) Steven (680) - (Training Log) - 2008-07-08 9:17 pm Not true: (264) Steven (680) - (Training Log) - 2008-07-08 5:14 pm The path AC is not the same as the pair of two arcs AB and BC: (242) STALKER2002 (346) - - 2008-07-08 5:22 pm Then how is your multiplication defined?: (249) Steven (680) - (Training Log) - 2008-07-08 9:31 pm Re: Then how is your multiplication defined? Look how!: (253) STALKER2002 (346) - - 2008-07-09 01:08 am Re: Then how is your multiplication defined? Look how!: (250) STALKER2002 (346) - - 2008-07-09 01:09 am Any composition of two rotations in R^4 is also a rotation in R^4 ,group SO(4)!: (256) STALKER2002 (346) - - 2008-07-09 01:28 am Any composition of two rotations in R^4 is also a rotation in R^4 ,group SO(4)!: (260) Steven (680) - (Training Log) - 2008-07-09 3:23 pm Not at All! The group of arcs deffers slightly from SO(4): (254) STALKER2002 (346) - - 2008-07-09 4:01 pm Not at All! The group of arcs differs slightly from SO(4): (264) STALKER2002 (346) - - 2008-07-09 4:05 pm However, some questions are still to be solved..: (262) STALKER2002 (346) - - 2008-07-09 5:00 pm Even in the AB@BC case, you have a well-defined problem (nt): (270) Steven (680) - (Training Log) - 2008-07-09 5:18 pm I know that; I wasn't talking about that (nt): (265) Steven (680) - (Training Log) - 2008-07-09 5:17 pm This is a problem: (271) Steven (680) - (Training Log) - 2008-07-09 3:13 pm The shortest way to Tokyo from Moscow is through China: (269) STALKER2002 (346) - - 2008-07-09 3:49 pm No, you can't force the continuity: (271) Steven (680) - (Training Log) - 2008-07-09 5:32 pm The second variation of functional length is zero on geodesic path: (274) STALKER2002 (346) - - 2008-07-10 01:56 am The FIRST, not the second variation of functional length is zero on geodesic path: (263) STALKER2002 (346) - - 2008-07-10 06:36 am That's my point!: (277) Steven (680) - (Training Log) - 2008-07-10 2:52 pm Of course Imust be more accurate and I care about geodesic preservation: (252) STALKER2002 (346) - - 2008-07-10 3:45 pm For any 3 points A, O, C there exists one and only one plane AOC,containing them,: (269) STALKER2002 (346) - - 2008-07-10 4:12 pm Yes: (256) Steven (680) - (Training Log) - 2008-07-10 11:15 pm Correct Idea! Because in 3D case I have defined multiplication of arcs via parallel translation of o: (266) STALKER2002 (346) - - 2008-07-11 00:17 am Antisymmetric 4-DYADS ARC[i,j]=N1(i)N2(j)-N2(i)N1(j). A New idea: (249) STALKER2002 (346) - - 2008-07-15 4:00 pm For simple 4-DYADS ARC1[i,j]=N1(i)N2(j) and ARC2[j,k]=N2(j)N3(k) ARC1@ARC2=N1(i)N3(k) : (234) STALKER2002 (346) - - 2008-07-15 4:51 pm This is why vector and tensor math is superior to and replaced quaternion interpretations (nt): (251) Steven (680) - (Training Log) - 2008-07-16 01:14 am By the WAY, in 2D case both of these arcs AC correspond to the same complex number!: (264) STALKER2002 (346) - - 2008-07-10 5:07 pm By the WAY, in 2D case both of these arcs AC correspond to the same complex number!: (260) Steven (680) - (Training Log) - 2008-07-10 10:49 pm [Top of List] [Previous Thread] [Next Thread] This Forum is for posting on topics of interest to the Taoist (Daoist) community. Tao is exceptionally broad. 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