Saturday 21 May 2022

Distance Theory Algebraically Supplemented Brane Simplified Model 3 S3 and Hoph Map

 Distance Theory Algebraically Supplemented

Brane Simplified Model <Continuation of Escalator Language Theory>
 
 
3
S3 and Hoph Map
 
 
1
From RS model, <distance> and <word value> are abstracted.
Refer to the next.
2
<Line segment>s from <Minus side> to <plus side> at <distance> and <word value> are both seemed as circle S1.
3
3-dimensional sphere S= { ( x1y1x2y2 ) | x1x22 + y1y22 = 1 }
Point of S3     x1y1x2y2 )
x1y1 ) ≠ ( 0, 0 )     π ( x1y1x2y2 ) = ( x2 + i y2 ) / ( xi y1 ) ∈ C
x1y1 ) = ( 0, 0 )    π ( x1y1x2y2 ) = ∞
Hopf map π : S→ Riemann Sphere, C ∪ {∞} 
Inverse image of a point p     π -1 (p) is S1.
Hoph map is fiber bundle that derived from fiber S1.
On fiber bundle, refer to the next.
4
Now <line segment > at <distance> is identificated as S1.
Two points a ( xaya ), b (xbyb ) at Gauss plane is objected to Riemann sphere.
At Riemann Sphere, two points ais marked by a’ ( x’ay’a ), b’ (x’by’b ) .
On Riemann sphere, refer to the next.
Point of Sis marked by ( x’ay’, x’by’b ) .
On S3, refer to the next.
5
<Line segment> at <word value> is also marked on S3.
6
<Distance> and <word value> are algebraically considered by S3.
 
Tokyo November 12, 2007
 


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