Some terms I have come to use in
this process:
real-element = a perceivable and measurable element
in 3d space and time.
hyper-element = a non perceivable or measurable element in 3d+
space and time.
hyper-cord = a string or (line) that is non real too our senses yet
real in an equation. representing a line between a real coordinate and a
hyper dimensional coordinate.
hyper-sheet = as its name implies a plotted two(2) hyper-element
plane
hyper-solution surface = a plotted three(3) hyper-element surface
which results between the real and hyper solutions
real-volume = a plotted real three(3) element volume
hyper-volume = a plotted three(3) hyper-element volume
hyper-origin = the start point for the (3 plus or hyper) plotting.
Note: the origin is always carried by the 3d plotted point in the
equation.
hyper-real intersect = a point where hyper elements intersect a
real 3d point.
hyper-real swap = a point where a real and hyper element exchange,
this can be viewed as a point in the equation where a rotation in space
occurs and forces a real-element to take up a hyper state and a
hyper-element takes up a real state.
{X0, Y0, Z0} =
static 3 dimensional point solution, composed or 3 real-elements or
variables
X-scale, Y-scale, Z-scale = Note; the scale is arbitrary and selected for
ease of plotting and the convenience of the visualization
{X0, Y0, Z0} tn =
dynamic 3 dimensional point, composed or 3 real-elements in relationship
to t - (time).
this first image represents a 3 dimensional point solution
in time solved and plotted at t1 to tn
a real-volume is described from the origin too t1
and this volume changes in time.
described as a displacement in X, Y, Z. Note: a cord from
the origin to {X, Y, Z} can at times represent a magnitude or scalar
value, the vector in this instance being the vectors of time and the
direction of path of the solution in time.
This is easy enough to describe, plot and represent on any
computer. and can be rotated to better view the path described in time.
Some will say that time represents a dimension also and
thus what we see is a 4 dimensional solution. Yet I will go with the
assumption that time for my purpose will be the discriminating scale of
sequential events and thus representing as it were a snapshot of a state
or condition. Also for clarity I must assume time is constant and non
deformed and of equal units or increments. At least for this simplified
example. Note: (For some equations sometimes the scale of time must be
deformed to make the plotting intelligible.)
{X'1, Y'1} = hyper 3+ dimensional
point, composed or 2 hyper-elements or variables
In this simplified example {X'1, Y'1}
are plotted considering the hyper origin as always being carried by the
three dimensional/element solution. in this instance t1
This gives us a hyper plane or sheet, which contains the
cord {X'1, Y'1} to t1.
Again this cord can be viewed as a magnitude or scalar
value at time t1. The red line in this
instance.
Using this method I will hope to show that any
multi-element/dimensional equation can be plotted to give an intelligible
view of its components.
also in this instance and example (X and X' are parallel)
and (Y and Y' are parallel). Rotation of the hyper coordinate frame of
reference can occur in some instances but more commonly it remains
oriented to the 3d reference frame.
Each subsequent hyper sheet is calculated with the origin
being the 3d solution point in this instance t2
Thus the hyper origin is dynamic as opposed to the real
origin
Scaling becomes the important in keeping the image intelligible.
In the end we find we can see a real solution path, a
hyper solution path, and a hyper solution surface between them. Note this
has been a simple 5d example, this processes can be applied to equations
of 12 dimensions or elements and still remain intelligible.
