Noticeable length dilatation is the remarkable result of a simple numerical experiment.
Given is an Euclidian space E of vectors P(x,y,z) with -11 and a mapping E--> U to the space U of unit Quaternions Q
with Q=(re(Q),im(Q)=(i,j,k)) . We use q for the mapping E-->U and p for U-->E.
q: Q=q(P): Q=(cos(alpha/2),sin(alpha/2)*Pu) and alpha=PI*|P| and Pu=P/|P|.
(i=P.x,…)
p: P=p(Q) : P=alpha/PI*im(Q)/|im(Q)| with cos(alpha/2)=re(Q).
Also given is a displacement operation d: E-->E
d: P’=d(P,D); P’=P(x,y,z) +D(x,y,z). with P’=p(q(P + O) * q(D)) – O with O=-P-D/2;
In “words”, the displacement begins with an offset and the quaternion mapping of the shifted position and the displacement followed by quaternion multiplication and back-mapping of the resulted quaternion to the position. Finally the inverse offset is applied. The offset ensures the isotropic and homogenous nature of the displacement operation.
If we apply the displacement d to the corners C of a cube around P we get
C’= p(q(C + O) * q(D)) – O with O=-P-D/2;
If C(i),C(j) is perpendicular to D, we get
|C’(i)-C’(j)| =al*|C(i)-C(j)| with lateral dilatation al>1.
In accordance to the rotation symmetric geometry, defined by the quaternion mapping, the transport over the Euclidian distance D corresponds to a rotation angle alpha=|D|*PI. This means that at the target position the transported cube arrives rotated by alpha. The rotation axe is perpendicular to the transport direction. Because we have zero dilatation in transport direction and dilatation a in lateral direction, we get a resulting length dilatation LD in transport direction at the target position, LD=al*sin(alpha).
Numerical results (with "large" displacements executed as a sequence of about 1000 small displacements) are : LD=1.52% with alpha=54°, LD=5.3% with alpha=90°.
With the description above it should be easy to reproduce the results.
Can these results alternatively explain the red shift, which we currently think is caused by accelerated expansion of the universe?