The Zero Delusion v0.99x

(l, m; j, −j) =
=
=
j−l
j−m
−j−l
−j−m
(l−j)(m+j)
(l+j)(m−j)
(l−j) 2 (m+j) 2
(l 2 −1)(m 2 −1)
= (1−lm)2 +(m−l) 2 +2j(1−lm)(m−l)
(1−lm) 2 −(m−l) 2
(∗)
= 1+tanh2 ∠ m l
1−tanh 2 ∠ m l
+ j 2 tanh ∠m l
1−tanh 2 ∠ m l
= cosh (2∠ m l
) + j sinh (2∠ m l
)
= exp (2j∠ m l
)
where we use the hyperbolic tangent subtraction formula tanh (∠ m l
) = (m−l) /(1−lm)
in (∗). Likewise, ±ı and ±ɛ represent the diverging slopes of two pairs of lines
meeting infinity in the Möbius and Laguerre planes, respectively. The reader
can check (l, m; ı, −ı) = exp ( 2ı∠ l m)
and (l, m; ɛ, −ɛ) = 1 + 2ɛ ( 1/m − 1 /l) =
exp ( 2ɛ∠ l m)
using the tangent subtraction formulas tan (∠
m
l
) = (m−l) /(1+lm) and
tanp (∠ m l
) = (m−l) /(lm), where tanp is the parabolic tangent [37].
These results confirm that we can extend the cross-ratio to rings and calculate
angles using cross-ratios. The (principal value of the) natural logarithm’s
imaginary part of an algebraic, dual-algebraic, or split-algebraic number of modulus
one is its (double) argument. In other words, a generalized angle is half the
area swept by the rotation about the origin on a segment of the unit cycle. Indeed,
0 is the area swept by the "straight" unit cycle, i.e., along the unit line
segment (1-ball), like P, е, and π are the "angles subtended" by the parabolic,
elliptic, and hyperbolic unit cycles.
This view expands the concept of the Exponential Map, which, applied to
the imaginary axis and parametrized by the generalized angle y, generates cycles
of elliptic (standard) (exp (yı) = cos y + ı sin y, ı 2 = −1), hyperbolic (exp (yj) =
cosh y+j sinh y, j 2 = +1), and parabolic (exp (yɛ) = 1+ɛy, ɛ 2 = ı 2 +j 2 ) geometry.
Any conic, e.g., a rotated ellipse, admits this interpretation of angle via the crossratio,
in agreement with the statement "there is scarcely any part of conics to
which the theory of cross-ratio is not applicable" (preface of [122]).
Moreover, we can express an LFT as a composition of cross-ratios of one variable
z taking values in an extended ring, namely a translation 5 (− d /c, z − 1; z, ∞),
a conjugation 6 (z − 1, 0; z, ∞), a rotation 7 ( 0, z − c2 /(bc−ad); z, ∞ ) , and another
translation 8 (− a /c, z − 1; z, ∞). Thus, an LFT is a cross-ratio of cross-ratios, and
we can express the cross-ratio projective invariance that generalizes 12 just in
cross-ratio terms as
(∞, t; t + 1, z) ≡ z − t
(− a /c, z − 1; z, ∞)◦ ( 0, z − c2 /(bc−ad); z, ∞ ) ◦(z − 1, 0; z, ∞)◦(− d /c, z − 1; z, ∞) ≡ f C (z)
(z 1 , z 2 ; z 3 , z 4 ) = (f C (z 1 ) , f C (z 2 ) ; f C (z 3 ) , f C (z 4 ))