Complex Manifold Deformation Theory 2 Reflection of Word

Complex Manifold Deformation Theory

2 Reflection of Word

TANAKA Akio

  

Conjecture

Word has reflection.

[Explanation]
1
Point     z, w
Function     w = f(z)
2
Fractional trasformation     w = S(z) = 
S(z) is written by Sz.
z = z₁/z₂,   w = w₁/ w₂
Affine transformation by homogeneous coordinate     w₁ = az₁ + bz₂, w2 = cz₁ + dz₂
Also transformed by matrix   =
3
(Theorem)
Anharmonic ratio is invariant by affine transformation.
Different points on plane     z₁, z₂, z₃, z₄
Affine transformation     T
(Tz₁, Tz₂, Tz₃,Tz₄) = ( z₁, z₂, z₃, z₄ )
4
(Theorem)
Circle is mapped to circle by affine transformation.
5
(Definition)
What pints z and z* is symmetric about circle C is what 3 different points z₁, z₂, z₃ on C satisfies the next.
(z*, z₁, z₂, z₃) =
6
Circle      C
Center     a
Radius     R
Symmetric point of z     z*
(z*-a)(-) = R²
7
(Theorem)
Affine transformation rezerves symmetricity.
8
Affine transformation     w = k・
Transformation     w = Tz
a, b is transformed to a', b'.
 = k・
When k>0, transformation is hyperbolic.

[Comment]
1 If the real world is expressed by circle C, map from z to z* is reflection.
2 On affine transformation, circle and line are not distinguished.
3 At the simplest C, line of the real world has symmetricsl points z and z*.
4 When z is a meaning minimum of language, z* is a meaning minimum of mirror language.
5 If a certain "degeneration" is occured in C, meaning minimum is expressed by one point. The situation is called "stop" in language.
6 Mirror Theory and Mirror Language are proto-theory of reflection on language.

[References]
Distance Theory / Tokyo May 5, 2005 / Sekian Linguistic Field
Mirror Theory / Tokyo June 5, 2004
Mirror Language / Tokyo June 10, 2004
Reversion Theory / Tokyo September 27, 2004


To be continued
Tokyo December 7, 2008

[Reference 2 / December 9, 2008]
Mirror Theory Group / Tokyo December 9, 2008 / Sekinan Linguistic Field

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