( This post continues revisions to that of 2 November 2008 and before )
( It also substantially revises the posts of 2, 5, and 13 November 2010 )
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19 November 2010 :
Ever since i brought the hyperbolic sine and cosine curves into the conjecture
i have treated them as separate entities which might be able to explain separate effects .
This overlooked the way the (circular) sine and cosine worked together to build the core helix ;
and that thus, the hyperbolic sine and cosine would do likewise .
These figures express the hyperbolic sine and cosine together as a single curve or surface in what i call imaginary timespace ; or something else .
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In the top two figures and in the fourth row from the bottom,
linear Time is expressed as an axis perpendicular to the screen ; in the other figures, Time is vertical .
In all figures, imaginary time, (distinct from linear Time), is expressed as rotation around the Time axis .
Space, i believe to be compressed into the hyperbolic curves and the sheets they form .
( It can be helpful to think of a hidden dimension as comparable to color, tone or taste :
something which can be infinitely one way or another, or at a point in between,
without displacing itself in the dimension-system currently viewed. )
If, as the system seems to suggest, the limbs of the hyperbolic curves become two of the axes of the photon in spacetime,
it is interesting that these axes, on (very) close inspection, do not form a sharp corner .
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It is also part of the conjecture that the matter and antimatter universes are interleaved and alternating,
(at least in the proximity of a photon) .
That if one were to jump across the Time axis of a photon from the matter universe ~
instantly ~ so one landed on the other side at the same moment in time, one would be in the antimatter universe .
And that, similarly, if a person could disconnect themselves from the matter universe for the right length of time,
the antimatter universe would appear around them without their having to move .
It is to be considered that these universes, as represented in part by the sheets of the top figures,
are broader than 180° in imaginary timespace ...
that according to this model, the matter and antimatter realms overlap slightly .
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The linear curves in the bottom four rows each show a single instance of a three dimensional hyperbolic curve,
(together with a representation of the core helix) .
If the images in the fourth row from bottom were to be flattened, (movement on the axis perpendicular to the screen discarded),
their branching curves would become planar hyperbolas .
The branching curves of the third row, flattened, would become hyperbolic cosines ; and those of the second, hyperbolic sines .
Unlike the curves in the posts of this past 2nd, 5th and 13th, these are not modified for application to this space :
the bending of their form occurs naturally from the interaction between their h-sine and h-cosine components,
and from the acceleration of change near their origin, (the Time axis) .
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It remains worth considering that if these curves represent space from the perspective of a photon,
their slope : between the infinitesimal from horizontal which the hyperbolic limbs relax to at infinity,
and a slope of 1 where the curve meets the core helix, could be expressed as the photon's gravity .
The 'mechanism' of gravity would be the relative speed of time .
At that point where the curve touches the core, the slope of space would be equal to that of time ;
(as Time, when combined with the rotation of imaginary time, expresses itself as a helix with a slope of ≈ 45°) .
In those areas where the slope of space becomes equal to that of time, time (in the ordinary sense) stands still .
As one moved further out on the limbs, and their slope decreased, time would thaw and begin to flow .
Infinitely far away in an otherwise empty universe, where the slope of space was flat or all but, the passage of time would be fastest .
In much the same way that light can be contained within a medium where it travels more slowly, (a principle applied in optical fiber),
the paths of objects passing through a region of slower time would bend toward its center.
( I believe that similar reasoning is involved in Einstein's Theory of Relativity. )
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I also believe that the hyperbolic forms shown represent, (or are closely connected with),
the imaginary timespace face of the electrical and magnetic fields which accompany a photon ;
(which, i again believe, are sphere-bound hyperbolic extensions of a circular form which would fit within the core helix),
(this is gone into in more detail in other posts) .
The linear figures in the lower four rows imply that at their maximum, these fields can extend to infinity, (or to the local antipode of the universe) .
If so, the conjecture may need to be broadened to include elliptical hyperbolas to express the fields in standard spacetime .
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I believe a photon expresses the positive outcome of a probability,
(one which may hold a set of differing possibilities) .
Respecting this, the cyclical nature or the photon's fields might be modeled by considering the core helix of the event's possibility-set to be like a string of beads .
Starting from a given 'compass' bearing in imaginary time, (the field where i believe probability is resolved),
the helical curve governing the outcome's specific photon would wind around the string of beads .
The distance of a point on that wound helix, (where it is touched by the center of a hyperbolic span within the photon's imaginary timespace sheet),
from the Time axis ... will determine the scale applied to those hyperbolic limbs which extend from that point .
This is a late edit, and many of the figures already show a constant-radius core helix at center .
In that respect, i believe those figures are mistaken .
The bottom row shows the string-of-beads core of the possibility-set of a photon,
but does not show the photon's instance-specific helix winding around it .
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The upper eighteen figures represent in imaginary spacetime the sheets the hyperbolic curves might form for a +photon, (half of a ±photon pair),
as the curves' centerpoint rotates 180° around the position of its core helix .
As it does so, the helical radius cycles from zero, (at imaginary time : -90°), to one, (at imaginary time : 0°), and back to zero again, (at imaginary time : 90°) .
Because of this waveform, the scaled limit of the hyperbolic limbs which make up these sheets cycles from zero, (at imaginary time : -90°),
to about eleven and one half, (a substitute value for ∞, at imaginary time : 0°), and back to zero again, (at imaginary time : 90°) .
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( Though it is subtle, these surfaces are also shown cycling in their pitch, {between flat and an angle of 45°, and back},
as their member curves' point of contact with the beaded core helix adopts the core's timeslope
and expresses this upon its limbs as they contribute to the sheet. )
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The twin-lily-pad form the hyperbolic curve surface takes in this imaginary timespace
reflects the waxing and waning of the positive and negative limbs of the underlying (hyperbolic) curve,
while it is at the same time rotating, (in imaginary time) .
In real spacetime, this rotation of imaginary time would be 'unwound', making it invisible ;
and making the lily pads appear as perpendicular lines which advanced and retreated periodically .
An interesting aspect of this surface is that the pads pass through one another .