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 S3 = { ( x1, y1, x2, y2 ) | x12 + x22 + y12 + y22 = 1 }
Point of S3 ( x1, y1, x2, y2 )
( x1, y1 ) ≠ ( 0, 0 ) π ( x1, y1, x2, y2 ) = ( x2 + i y2 ) / ( x1 + i y1 ) ∈ C
( x1, y1 ) = ( 0, 0 ) π ( x1, y1, x2, y2 ) = ∞
Hopf map π : S3 → 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 ( xa, ya ), b (xb, yb ) at Gauss plane is objected to Riemann sphere.
At Riemann Sphere, two points a, b is marked by a’ ( x’a, y’a ), b’ (x’b, y’b ) .
On Riemann sphere, refer to the next.
Point of S3 is marked by ( x’a, y’a , x’b, y’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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