From Osher Doctorow
From the previous sections of this thread we have with * as the dot
product:
1) grad(f) * (1, 1, 1) = (x-->y), f(x, y, z) = z + y^2/2 - x^2/2
The function f(x, y, z) = 0 is a saddle (hyperbolic paraboloid) with
"center" at the origin, and the crossections f(x, y, z) = k for real
constant k move the center from the origin upward or downward with
respect to the z axis (up-down). So:
4) u = f(x, y, z) = z + y^2/2 - x^2/2
is an "infinite tower" of saddles with crossections u = k ordinary
saddles (hyperbolic paraboloids) with particular (different) centers.
Since for any unit vector v, grad(f) * v is the directional derivative
of f in the direction v or the rate of change of f in the direction v,
from (1) we have that:
5) (x-->y)/sqrt(3) = directional derivative of saddle tower in
direction of (1, 1, 1)
or correspondingly in 4 dimensions using (1, 1, 1, 1) and x, y, z, u.
With the interpretation of "time as duration" (T1) as t(1, 1, 1) or
t(1, 1, 1, 1), Probable Influence/Causation (represented above by
(x-->y) is the directional derivative of a saddle tower in the
direction of time. This relates both (1, 1, 1) and (x-->y) to time,
and makes more specific the definition of the second time dimension T2
as "time as (probable) causation", being represented by (x-->y) or
(x-->y)/sqrt(3).
Osher Doctorow
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