Turn the World
Around
We are of the spirit
Truly of the spirit
Only can the spirit
Turn the world around
There is no discontinuity
in spirit!
A rebound, bounce, occurs at a physical discontinuity. The
motion approaching the discontinuity is rotated and reflected at the discontinuity.
When that motion is linear, then the rotated reflection will also be linear because a linear motion is symmetric
with respect to the discontinuity. If the motion approaching the discontinuity is
parabolic, which is also symmetric with respect to the discontinuity, then its rotated
reflection will be parabolic. But a hyperbolic motion approaching a physical discontinuity
is NOT reflected as a symmetrical hyperbolic motion in a rotated direction. It
is NOT symmetric in real physical dimensions, but it may be symmetric in the imaginary dimension.
A discontinuity need not be only physical. It can be an apparent,
unseen, discontinuity. When approaching this unseen discontinuity, the motion
can appear to be linear, but the rotated and reflected motion can appear to be parabolic. Rotation
by π/2 is identical to the inverse of a function, and the inverse of a parabolic
function, rotation by π/2, is
undefined with a change of signs. A change of signs is required at a discontinuity
from being observed to unobservable. This most probably indicates that the approaching
motion is probably hyperbolic, and the rotated and reflected motion is also hyperbolic.
Hyperbolic motion is NOT symmetrical in a real plane with respect to a discontinuity.
If the discontinuity is one of measurement, for example
traffic departure volume approaching the capacity of a road, the
departure volume may be hyperbolic with a discontinuity, but that is only
because you are observing departure volume. Departure volume is equal to
arrival volume before the discontinuity. If you could observe arrival volume,
which can never be expected to be negative, its motion might be expected to continue past
the discontinuity.
That discontinuity can be expected to be the separation between domains. For departure volumes
it can be volume as uncongested traffic in the laminar domain before the discontinuity,
which occurs at the transition speed and as congested traffic in the turbulent
domain which occurs below the transition speed. Again linear and parabolic
motion are symmetrical with respect to this transition speed, but hyperbolic
motion is NOT symmetrical with respect to this real transition speed. The
observations of traffic may appear linear before the discontinuity and may appear
parabolic after the discontinuity. This has been interpreted by Van Aerde (Van Aerde, 1995) and others that the
speed-volume curve is an asymmetrical parabola, which appears almost linear
above the transition speed and more parabolic below the transition speed. It is
instead proposed that the motion is hyperbolic which only appears linear in the
uncongested traffic above the transition speed and appears parabolic in the
congested traffic below the transition speed.
It is convenient to place the origin at the discontinuity
which creates four domains, quadrants, with respect to the axis of observability
( the y-axis) and the axis of transition
( the x- axis). These quadrants will be then:
1)
unobservable and laminar;
2)
observable and laminar;
3)
observable and turbulent; and
4)
unobservable and turbulent.
If an object approaches the discontinuity in quadrant 2),
then it will rebound, leave the discontinuity, in quadrant 3) remaining in an observable
quadrant. If an object approaches the discontinuity in quadrant 2) and passes THROUGH
the discontinuity ( travelling across the axis of observability) it will still
transition but it will leave in the unobservable quadrant 4).
Why would traffic, or any particle, have hyperbolic motion?
If the geodesic is on a flat surface, then the object will not leave that geodesic
unless it is acted upon by a force. If that particle is on a hyperbolic
surface, the geodesic will be hyperbolic and no force is required for an object
to change that hyperbolic, exponential, motion. Thus vehicles in traffic or any
particle, in the absence of force, might be expected to have hyperbolic motion
if the shape of the surface on which it is traveling is hyperbolic.
This has implications for traffic engineering. A speed-volume
curve can be expected to have a discontinuity at its capacity. In travel demand
modeling, the Volume Delay Function, VDF, appears to include both time and reliability
(expressed as a time). (Azizi
& Beagan, 2022) . The VDF arguably has three periods governed
by the discontinuity of time and when the reliability budget has been exhausted.
This means that from a volume to capacity ratio, v/c, from 0 to 1, the volume will be less than the
discontinuity at the capacity and the reliability budget has not been exhausted.
From a v/c of 1 to a v/c of 1.216, the volume
has passed through the discontinuity,
but the reliability time budget has not
been exhausted. For volumes above a v/c ratio of 1.216, the volume discontinuity
has been passed and the reliability time budget has been exhausted.
This also has implications for the overflow delay at an intersection
because delay, which is measurement of time, and it is the inverse of speed. This has traditionally
been solved as a rotation of a domain with a random departure queue and a constant
flat peak arrival volumes to a deterministic queuing departure. (Hurdle, 1984). Instead, if a transition at a discontinuity is
assumed, there need be no restriction on arrival volumes while preserving random
departures and deterministic departures.
This discontinuity is suggested to also appear in other situations
where an observable non‑physical discontinuity
exists, such as in fluid dynamics and its separation of regular laminar fluid
flow from white water, turbulent flow. If particles travel on geodesics on a
hyperbolic surface rather than geodesics on a flat surface, what is perceived
as the force of gravity on a flat surface might instead be motion without a force,
and there thus need be only three fundamental (the electrical, the strong
nuclear, and the weak nuclear) forces with gravity as only an apparent force, similar
to centrifugal force resulting from motion on the hyperbolic surface.
The solution to the Pythagoras’
Theorem for a triangle would be different on a curved hyperbolic surface than on a
flat Euclidean surface. This new solution would be c=ln(0±2*cosh(√(a2+b2))).
This would be virtually indistinguishable from the traditional solution for Pythagoras‘
Theorem when the distances involved are far less than 5/6 times the size of the
observable universe. In other words, the universe would be locally flat, but universally hyperbolic. The rth moment of a mean would not be 0 for odd
powers and highly negative for even powers, but would be 0 for all powers of r. The Standard Deviation
would not require Bessel’s Adjustment of n/(n-1). The relativistic
dilation factor, the Lorentz Transform, would be ln(0±2*cosh(√(1-v2/c2))).
The observable universe would not be flat but might be only one sheet of a two‑sheeted
hyperboloid where its two sheets intersect at the origin, the Big Bang. That
the observable universe is hyperbolic was proposed by Mabkhout (Mabkhout,
2012),
in which he also proposes that if Einstein’s tensors are solved for a
hyperbolic surface, Dark Energy and Dark Matter are not required.
It is suggested that Euler’s Formula is because a complex
number is being transformed from cylindrical coordinates to Cartesian coordinates,
and that reality is a surface intersecting the origin of that cylindrical volume
formed by the three dimensions of space, time and imagination where the coefficient
of imagination is zero, r*eix=r*cos(x)+r*i*sin
(x). This can be restated narratively as reality is the negative sheet of a two-sheeted
hyperboloid, having a negative coefficient, with the other sheet being positive,
and has a zero coefficient of the imaginary axis, i.e., r*eiπ=-r
+02i, because sin(π)=0 and cos(π)=-1.
It is proposed that the apparent discontinuity is only an illusion
which comes from the intersection of tanh(x-μ) and ‑tanh(x-x0-μ), both where x>x0,
and x0 is -CAP, capacity, for traffic.
The discontinuity appears at zero, but that is only because the x
coordinate axis has been translated. The
actual equations, if the translation is removed, are tanh(x), x>0 and ‑tanh(x-x0), x>0. If laminar real behavior continues THROUGH the discontinuity,
then it follows ‑tanh(x-x0) where x is always positive.
In each case, the amplitude, A, is 2. Given that σ2=½A2,this
means the variance, σ2,
is 1. This means that the Standard Deviation,
σ, the square root
of the variance, is √(2/2), 1, if the imaginary dimension is NOT considered. But since tanh repeats only in imaginary planes,
the standard deviation considering the imaginary axis should be ln(2*cosh(√(2/2)),
1.058
Assuming that the mean is only zero leads to making the
mistake that luck must be destiny. Assuming
that the absolute has no error and thus its Standard Deviation, SD, and
variance must be zero is confusing the limit of error, Standard Error, SE=SD/√n
where n→∞, with deviation. The discontinuity shows that variance is not equal
to zero AND the applicable mean is not equal to zero.
References
Azizi, L., & Beagan, D. (2022, January).
Inclusion of Reliability in the Volume Delay Function. Poster Presented at
Annual TRB Meeting.
Hurdle, V. F. (1984). Signalized intersection delay
models–a primer for the uninitiated. Transportation Research Record,
971(112), 89.
Mabkhout, S. (2012). The infinite distance horizon
and the hyperbolic inflation in the hyperbolic universe. Phys. Essays,
25(1), p.112.
Van Aerde, M. (1995). A single regime
speed-flow-density relationship for freeways and arterials. Washington D.
C.,: Presented at the 74th TRB Annual Meeting,.
