Conjecture A
2 Reflection of Word
Conjecture
Word has reflection.
[Explanation]
1
Point z, w
Function w = f(z)
2
Fractional transformation 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)
An-harmonic 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 points 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
= R2
7
(Theorem)
Affine transformation reserves symmetry.
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 symmetrical 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 occurred 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]
To be continued
Tokyo December 7, 2008
[Reference 2 / December 9, 2008]
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