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Re: Then how is your multiplication defined? Look how!From: STALKER2002
Subject: General
Date/Time 2008-07-09 01:09:43
Remote IP: 79.139.140.114
MessageWhen you move continually from B to C, being at time t in the point X[t] on the arc BC, the resulting product AB@BX[t] changes also continually from AB to AC. This is a usual normal well-definition of the path AC. The more algebraic definition of my multiplication @ requires definition of the generalization of i,j,k quaternions. By the analogy with 3D unit sphere geometrical interpretation of quaternions i,j,k, let us consider 4D unit sphere in euclidean R^4 and four points A=(1,0,0,0),
B=(0,1,0,0),C=(0,0,1,0) and D=(0,0,0,1)on it. The shortest arc from any two of them is an analog to quaternions i,j,k with muliplication rule being
AB@BC=AC, BC@CD=BD, etc,6 equations for 6 pairs among 4 points ABCD. This 6 pairs are the analogs of 3 quaternions i,j,k in 3D spere case. Hence the implied generalization of quaternions numbers should be
S=a+b(AB)+c(BC)+d(CA)+e(AD)+f(DC)+g(BD)! 7D S-quaternions!!
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