Distance Theory Algebraically Supplemented Brane Simplified Model 3 S3 and Hoph Map / November 12, 2007

Distance Theory Algebraically Supplemented

Brane Simplified Model <Continuation of Escalator Language Theory>

3

S³ and Hoph Map

TANAKA Akio

1

From RS model, <distance> and <word value> are abstracted.

Refer to the next.

Distance Theory Algebraically Supplemented / 2 Distance / Tokyo October 26, 2007

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.

Symmetry Flow Language / Leaf of Language / Tokyo May 11, 2007

4

Now <line segment > at <distance> is identificated as S¹.

Two points a ( x_a, y_a ), b (x_b, y_b ) 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.

Symmetry Flow Language / Simplex, Simplicial Complex and polyhedron / Tokyo May 17, 2007

Symmetry Flow Language / Meaning Variation and Time Shift in word as Homotogy / Tokyo May 17, 2007

Point of S³ is marked by ( x’_a, y’_a , x’_b, y’_b ) .

On S³, refer to the next.

Symmetry Flow Language 2 / Time Shift of Meaning in Moduli Space / Tokyo May 23, 2007

5

<Line segment> at <word value> is also marked on S^3.

6

<Distance> and <word value> are algebraically considered by S³.

Tokyo November 12, 2007

Sekinan Research Field of Language