Geometrical Interpretation of Quaternions&MULTIPLICATION

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Geometrical Interpretation of Quaternions&MULTIPLICATION From: STALKER2002 Subject: General Date/Time 2008-07-01 16:04:52 Remote IP: 79.139.140.28 Message Let's imagine that the Earth is an ideal Sphere of unit Radius. Points A , B and C are some Cities on it: Moscow(A), New York(B) and Tokyo(C) for example. When you travel by the shortest way from Moscow to New York (AB) and then from New York to Tokyo(BC) also by the shortest way, the resulting geodesical arc on the Earth from Moscow to Tokyo (AC)being the shortest way from Moscow to Tokyo(AC)which IS BUT MULTIPLICATION product of Quaternions (AB) & (BC): (AC)=(AB)@(BC), with all needed requirements for multiplication @ being satisfied. Every geodesical arc (AB)on the Unit Sphere correspondes to a Quaternion. If A=B then quaternion (ABB) is equal to 1. If A is diametrally opposite to B, then (AB) is equal to -1.The i j k quaternions being arcs along equator, Grinvich ang 90 degree after Grenvich meridians with longitude 90 degrees. Twisely taken i or j or k quternion gives rise to -1. etc. http://www.tao.nm.ru 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: (257) 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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