The Zero Delusion v0.99x

d H (z 2 , z 3 ) = ln (z 1 , z 2 ; z 3 , z 4 )
Physically, we can suppose that the disk is a body, and the superposition
of the configurations ±1 and ±ı describes its quantum state. Since circles in Ǎ
tend to be straight lines when the radius of a geodesic diverges, we can take
−1, 1, s, and ∞ as consecutive concyclic points of the real-algebraic projective
line. Then, an object S outside the unit disk at Euclidean distance s from its
boundary is at hyperbolic distance d H (1, s) = ln (−1, 1; s, ∞) = ln (s+1) /(s−1).
Circle inversion z → 1 /z leaves the conformal distance invariant, i.e., 0, d = 1 /s, 1,
and −1 are consecutive within the disk along a diameter satisfying (s+1) /(s−1) =
(−1, 1; d, 0) = (1+d) /(1−d). S is now within the disk infinitely far away from the
origin and at Euclidean distance d from the boundary, and hence at conformal
distance d C (d) = ½d H (1, d) = ½ (−1, 1; d, 0) = ½ ln ( (1+d) /(1−d)) = artanh (d),
which is in the range (0, ∞). While the sum of distances u and v is linear within
this coding space, namely artanh (u) + artanh (v) [90] (addition of two areas),
it corresponds to Einstein’s addition and subtraction formula (on the collinear
form) of velocities tanh (artanh (u) + artanh (v)) = (u±v) /(1±uv) in the external
world.
The Poincaré distance is a measure of relativistic speed, i.e., the hyperbolic
area that separates two frames of reference in relative motion, ergo the punctured
Poincaré disk is a rapidity space that encodes what happens in the Möbius
plane. Alternatively, one sheet of a two-dimensional hyperboloid of revolution
embedded in the Minkowskian three-space serves as another model of H 2 . The
Möbius, Laguerre, and Minkowski planes are flat, and so is the Minkowski spacetime.
Minkowskian spaces handle hyperbolic angles, but the distance between
two vectors is the norm of their difference, i.e., Euclidean, while the embedded
hyperboloids measure hyperbolic distances along geodesics. In other words, only
"the non-Euclidean style of Minkowskian relativity" [171] is doubly conformal,
not the algebraic ambient spaces. The geometries of relativity are de Sitter space,
Minkowski spacetime, and anti-de Sitter space corresponding to elliptic (positive
curvature), Euclidean (no curvature), and hyperbolic (negative curvature)
geometries. All three can give place to doubly conformal and coding spaces.
According to Liouville’s theorem, a conformal map in n ≥ 3 dimensions between
two open regions of Euclidean space is equivalent to a composition of
n-dimensional Möbius transformations, namely homotheties, translations, rotations,
and inversions in the (n − 1)-sphere [67]. A conformal transformation is
informally a diffeomorphic mapping between two manifolds implemented as a
Möbius map in an open neighborhood of each point. If f is a bijective mapping
of an open set S in E n onto f (S) with vanishing differential df x nowhere in
x ∈ S, then f is conformal if and only if 〈df x u, df x v〉 = е 2ω(x) 〈u, v〉 for all
u, v ∈ E n , where the brackets denote the inner product (Theorem 3.8 in [18]);
ω (x) is a rational-valued function that is nonnull if the angles ∠ v u and ∠ dfxv
df xu
differ. Conformal transformations of E n map m-spheres to m-spheres (m < n)
without exception because degenerate flat subspaces do not exist.