III
Geometries
Curved Space, the angle step 4 →
3 in terms of pure geometries:
1. A perpendicular geometry:
It is a fact that a perpendicular geometry appears in several
physical relations:
It's pointed out that the gravitational
(centripetal) force acts strongest along the rotation axis,
while the centrifugal force as its opposite acts along the
equatorial plane.
Other examples: In old physics the orbitals
of planets are illustrated as the combination of two vectors,
one pointing inwards towards center, one the tangent to this.
And we have the orthogonal relation between electric and magnetic
components in an EMwave, as between the amplitude and radial
jumps in electron shells.
Most elementary: equality should reign around a first
center. Equal distances form a circle, perpendicular to the
radius. If divided, it seems natural to think it should be
divided in equal parts. Halvings seems to be (or has been
given ?) the principle for spin. Quarters for something else
 as from a second polarization.
Hence, starting with a center as first prerequisite,
we could see the perpendicular geometry, assumed in ddegree
3, as an inherent, inevitable development from a principally
antiparallel one?
Another aspect: According to our model we have one ddegree
of motion in a 4dimensional vector field, a 1dimensional,
i.e. linear motion, ("to and from each other"),
which means a longitudinal one, as variations in density,
defining spherical layers in the geometry. The motion moment
acts as a polarizing force. (Note that we have proposed Density
to be the first "physical quantity" in step 5 →
4, when identifying the usual physical concepts in the dimension
chain.) So from a 4dimensional, antiparallel structure plus
a 1dimensional motion we get a 3dimensional geometry defined.
2. NonEuclidean geometry:
The views above seem much too simpleminded and bound to an
Euclidean geometry to explain geometrical realities? What
if we look at cosmos in terms of non Euclidean geometries?
Scientists say that big masses curve the
space around them, but do they tell us why? Masses have positive
radius of curvature, the space around big masses have negative
curvature. But as far out as one have been able to measure
the cosmic space in itself, (the angular sum of a triangle),
it seems to have an Euclidean geometry.
Departing from our model we could rather presume that curvatures
in the geometries precede the creation of masses or at least
are an intrinsic part in that creation and in a polarization
Mass  Vacant Space (or E= +/ mc^{2}).
The surface of a globe has an elliptic geometry;
the angular sum of a triangle is more than 180°. The area
between three adjacent circles has a negative curvature; the
angular sum of the triangle is less than 180°.
An hyperbolic geometry, a combination of positive
and negative curvature, is of course the most consistent with
the model here, characterized by polarizations. It's often
compared to a horse saddle when it concerns surfaces. With
this metaphor, there are simultaneously 2 polarizations in
the geometry: a step to a perpendicular relation between coordinate
axes and a curving of these in negative / positive directions.
Such opposite curvatures may in fact be
identified as transformations of Direction inwards  outwards:
the positive one defining an enclosed center, a transformed
inward direction, in ddegree 3 a globe as volumes for masses.
Whereas the negative one describes an excluded center, as
a transformation of outward direction  defining an anticenter
as Vacant Space in ddegree 3.
3. Constant positive or negative curvature:
A constant positive curvature, along both coordinate axes
of a surface, gives the globular, elliptic form of volumes
for masses in cosmos. The hyperbolic geometry rather describes
the relation between two such globes with intermediate space:
perhaps a reason to interpret what is called "gravitational"
centers as responsible for both the attraction and separation
between celestial bodies? A manifestation of ddegree 3 of
the unpolarized kind of dimension degrees that we haven't
been able to find in physical terms?
Where could we find negative curvature along
both axes? Such a form has been compared with two counterdirected
trumpets, what is called a "pseudosphere". (Introducing
a distance between their border circles and positive curvature
joining them, we should get a form similar to a spiral galaxy.)
The doubletrumpet could be described as if each arc of a
quadrant in a circle was inverted.
One example of this geometry could be the
magnetic field between 2 magnetic Npoles, forced to meet,
repelling each other.
4. Why curvature?
The geometrical aspects depart from 1dimensional lines. That's
a view from lower ddegrees towards higher, inwards in our
dimension chain.
A curved line implies an "intrusion"
of ddegree 1 into a 2dimensional world or a start of defining
such a world, a beginning of an orbit plane (a step 2 <—
1). A curved surface implies an intrusion into a 3dimensional
world.
With the assumptions in our model of debranched
degrees meeting "the other way around", Step 4 →3
corresponds to the step 2 <—1.
The "intrusions" inwards in the
chain along the main axis could in very general words be regarded
as transforming the geometry between the principally antiparallel
vectors in ddegree 4 into a 3dimensional geometry with polarized
volumes through curved surfaces.
It seems easier to interpret the curving
as builtin motional structures from the end of a dimension
chain, with the chain as doubledirected, giving us an observable
world with 3 and 2dimensional forms.
5. Centers in the geometry as shrunk or increased:
Another aspect on the nonEuclidean geometries with positive
and negative curvatures departs from the description that
a surface which grows faster than proportional to the radius
squared give a negative curvature, a surface growing slower
than proportional to the squared radius gets the positive
curvature.
Now, taking a piece of cloth and pursing
it up in the middle (representing a shrinking center or origin),
one gets a wavy surface outwards on the cloth as representing
a form with negative curvature.
If adding a piece of cloth into a hole in
the center, increasing it, the surface will curve in a positive
manner, more adjustable to the football principle.
Hence, the curvature is depending on the
size of the center  or the unity with the role of a center
pole (the 0pole) in our model.
The negative curvature could derive from
an underlying level, a smaller origin or 0pole from which
we have the outward Direction in ddegree 4. With growing
complexity of the realities after Big Bang the center will
grow more complex and may be thought of as increasing in size.
The positive curvature and gravitational
formation of Direction inwards would follow purely out of
this increased center! That is, if we start from an Euclidean
geometry.
The relation between Vacant Space as divergent
and Mass volumes as convergent would then simply be interpreted
as a relation between center and anticenter, as suggested
in our model  with the addition of a Time factor.
A main concept used in the background texts here is "center
displacement". Neglecting the assumption of a gradual
growth, a center displacement implies that the circumference
(as a 00pole) on one stage becomes the center for the next
stage.
In a description of the elliptic geometry it's stated that
a) a line in Euclidean geometry may be represented
by a point in elliptic geometry,
b) a plane in Euclidean geometry may be represented
by a line in the elliptic one,
c) a solid angle between planes in Euclidean geometry
may be represented by plane angle (as 2dimensional) in elliptic
geometry.
This points towards the interpretation of
the elliptic curvatures as of a lower ddegree. The opposite
should apply to a the complementary geometry with negative
curvature, lines representing (or growing to) surfaces etc.
 and points representing lines...
Such descriptions indicate that we should
see the elliptic geometry and the geometry of negative curvature
as of different ddegrees. (Or potentially pointing towards
lower and higher ddegrees respectively?) Elementary, as there
exist an infinity of surfaces in a volume, a higher ddegree
represents unity in relation to a lower as a multitude. We
have the unity of Vacant Space and the multitude of celestial
bodies.
6. A Time factor again  and the curvature of sine waves:
With (another) Time (or "phase") displacement, we
have the form of usual sine curves  as projected outwards
from a vector rotation in a unity circle.
The curvature is positive to 180°, then
becomes negative in relation to the first part of the curve,
a relation between a and b around the inflection point. Curve
b represents the concept of an "excluded center".
It could be regarded as the curve a mirrored two times in
two axes, both horizontally and vertically.
It's the form of surface waves, positive and negative curvature
in a mutual relation, as a circle of rotation broken up and
polarized in time: We have the opposites convex/concave (representing
opposite signs), suggested as one description of the complementary
poles of ddegree 2, but following one successively in time.
The inflection points in a 2dimensional wave make up or
define a 1dimensional line. This is perhaps one answer to
the question how on earth the polarization of a 2dimensional
surface in "inside/outside", "convex/concave"
may define an 1dimensional line according to our model !?
Two notes:
a) Hyperbolic geometry in the atom?
In an atom mass volumes with positive curvature is concentrated
in the center. Could we eventually find some expression for
a negative curvature, if connected with empty space, at the
electrons?
Could an hyperbolic or negative curvature
described as excluding a center, be connected with Pauli's
"exclusion principle" between electron pairs  or
something else in the electron shells? In our views on protons
versus electrons we have seen them as "grandchildren"
of the opposite vector fields in ddegree 4, children of Mass
and Vacant Space respectively (see file Forces).
Besides this question: In the tentative
interpretation of quarks in nuclei as p and n, we have suggested
a parallel to the gastrulation process in embryos, including
both elliptic and negative curvature in the motional structure
of growth.
b) Olber's paradox:
It has been stated that if the Universe eventually had a hyperbolic
geometry, then we should be able to find more and more galaxies
the further out in cosmos the telescopes reach. However, with
the assumptions in our model this conclusion must be wrong?
If at a certain stage a hyperbolic geometry
is polarized in positive curvature of mass volumes and negative
curvature of "Vacant Space", then it's only empty
space that give the widening lines of sight, and proportionally
fewer and fewer galaxies would be found further out. One possible
explanation to Olber's paradox?
*
Multiplicity of Mass and its Distribution
The manifold of masses but apparently unity
of "vacant space" should in one sense follow from
pure geometries and relations between ddegrees as said above.
But how explain the fragmentation of Mass or the multitude
of centers for gravitational concentration?
And how explain the distribution of masses
from what is supposed to be a uniform development of geometries
from a Big Bang center?
It has been said that the observed, nearly
homogeneous micro wave background radiation in cosmos  taken
as supporting the Big Bang theory  has not been able to unite
with the "unequal" distribution of masses. It's
unclear in which sense it is regarded as unequal.
1. Gravitation as a polarizing force!
Gravitation, FG, as an inward directed force with its foundation
in the 00pole is also, according to first postulates or hypotheses
in our model a polarizing, splitting force. How can this apparently
wrong hypothesis be justified and maintained?
In fact, there is such a polarizing force acting in cosmos,
according to the scientists' observations:
Contraction through gravitation in cosmic
clouds lead to a fragmentation, a splitting up of the clouds
in smaller clouds and the birth of a great number of stars
in groups or crowds.
It's stated that a cloud under certain conditions
(of total mass, density and temperature) begins to contract,
and then gets fragmented "in several steps" to smaller
and smaller "clumps".
Secondly, one has also observed a "spontaneous"
polarization between hotter and colder regions in celestial
clouds, which seems to contradict usual thermodynamic laws.
The contraction means too that the H2molecules
are split into Hatoms (probably explainable simply through
increased temperature?).
Hence, we seem to have first a polarizing force, which could
be attributed to the 00pole of ddegree 4 in our model, then
or apparently simultaneous a contracting force, as Direction
inwards (pole 4a) in ddegree 3 in our model. A third appearance
of the gravitational force in ddegree 3→
to 3b (one pole in ddegree 2) should give the birth of stars
and rotation, probably also with a factor of polarization.
(Rotation may be interpreted as a "haploid" motion,
pointing to another half with opposite rotational direction.)
Should we talk about Gravitation in all
three ddegrees or perhaps give the first polarizing force
(00) another name?
2. The polarization principle not recognized as such?
The fragmentation of these celestial clouds is not explained
in the sources used here. There is only talk about "small
disturbances". The problem seems to be the same as with
turbulence.
A general polarization principle  in several
steps, of several kinds or properties and not called attention
to as such, partly unknown or unexplained, could be the answer.
From the viewpoint of polarizing motions
(the end of the 5dimensional chain in our model) and the
assumed 1dimensional longitudinal waves attributed to vector
fields in ddegree 4, they should create rings or shells of
denser regions.
It's difficult to avoid the assumption that
a secondary polarization through motion of a transversal kind
is needed too in explaining the distribution of mass centers
in cosmos; some kind of inherent waves in step 2 1 as results
of ddegree step 4 →3 in the
structure, along the circumferences (compare our hypothesis
about side waves).
Such waves are never mentioned in the used
sources. However, the sun for instance is said to be divided
in sectors, with opposite directions or signs for the magnetic
field  and cells of convection streams.
Examples, where a general polarization principle
intuitively is applied: Pauli's "invention" of the
"exclusion principle" between electron pairs in
the atom, given the explanation of opposite "spins".
The still chiefly theoretical "up" and "down"
quarks.
3. Why this "unequal", " not homogeneous"
distribution of mass in Macrocosm?
Scientists mean that the nearly homogeneous
background microwave radiation found in cosmos should imply
an equally even distribution of Mass in cosmos. This contradicts
the irregular or not "homogeneous" distribution
of Mass, as they see it. There is no good explanation found.
In one source used here it's proposed that
a rapid increase in size from Big Bang led to the result that
different parts of Space "lost contact with each other"
which should imply that "local fluctuations" in
density could be permanent. Such a description doesn't feels
satisfactory, at least not with our model here in mind. "Local
fluctuations" and "small disturbances" and
such references sound without contact with any scientific
principles?
To approach the problem, we could test to
look at dimension chains as a genetic code: We could ask:
How many (crossingout) principles of differentiation
are needed to explain the individualization of mass in cosmos?
(Eventually starting with 4 forces, FA,
FG, FE,
FM, as there are 4 bases A, G,
C and U in the genetic code!)
For instance: a) gradients of densities,
b) gradients of forces' strengths, c) gradients of velocities,
d) gradients of radii of curvature, and with Time: e) generations
of masses. (Cf. Hoyle's "Cfields".) Perhaps it
is enough, or do we have to add chance, fluctuations within
the borders of uncertainty in microcosm?
There is of course no answer here. We could
just make the supplementary note that gradients may have the
character of discontinuous steps, changing the "quality*
of energy" when a certain amount of energy or a certain
interval is reached?
* (Said to be the view of Sarfatti 1975.)
4. What creates the many centers? Or how are they identified
as such?
Before differentiating processes:
In our simple geometrical terms the starting
point of a vector field inwards has a spread out position
with the word from quantum mechanics. The same holds for the
first indefinable target of the outward directed vector field.
Somewhere in the literature it's stated
that convergent vector fields (vconv)
give an undefined center, as if not pointing to a common,
singular one. Why? No explanation in that text. Because the
divergent vector field from the primary center already has
given birth to a multiplicity of secondary centers, a process
preceding the convergent vector fields? Because convergent
vectors are pointing backwards in Time  ?  referring to
a center already on its way? The convergent field meeting
the divergent one "halfway"?
Such a view could be connected with the
idea of an eventual inflationary stage of development after
Big Bang.
There exists something of a similar relation between "vectors
outwards and inwards in the nervous system: In the inward
directed parasympathetic nervous system the nerves
depart from the peripheral ends of the vertebral column, from
head and tail vertebras (at least in human beings), while
nerves from the sympathetic system, outward directed towards
brain and muscles, start from the middle region of the spine.
Ganglions as centers (?) for the sympathetic
system are situated near the spine, with many connections
with one another, while the ganglions for the parasympathetic
system are situated far away, distributed and localized as
separate to the neighborhood of the individualized intestines.
These are organs developed from within, roughly said from
the vegetative pole of the first embryo.
Another question: Should we think of vectors outwards as
branched? Compare "bifurcations" at
certain stages in chaos research.
It could perhaps be appropriate to apply aspects from quantum
mechanics on the question about the multitude of centers.
Identifying a higher ddegree is undefined in the lower ddegree,
the higher one representing a "superposition", this
could imply that the outcome when it "collapses"
(here through a ddegree step) may show up as yes or no (
~ mass or vacant space)  and anywhere? Compare what is said
about quanta of forces, that they may have any mass whatsoever,
that's undefined. It sounds like one possible answer to the
not homogeneous distribution too.
Where do we find the first ovum in a developed human body?
An idea about divisions (equal to polarizations?) becomes
implicit in such a question, if a parallel to Universe. And
copying of a code, which as a suggestion here should be represented
by dimension chains as the general pattern within the frame
of surrounding conditions and actual Time.
After all, the cosmic multitude of centers resembles
turbulence, smaller whirls or bubbles born from bigger ones
etc. The long (or protracted) distances between celestial
mass concentrations seem perhaps to contradict this similarity
but may be a chimera. Expansion of Vacant Space creates the
distances, with more or less of negative curvature.
Very simpleminded: Why are gathered masses
so small in relation to empty Space between them? One equally
simpleminded answer: The "lengths" of vectors pointing
outwards are principally unlimited, while the "lengths"
of inward directed vectors gets principally limited by definition
at their meeting points (ultimately the black holes?):
5. Bubbles:
Looking at macrocosm as 3dimensional, as from a 3dimensional
point of view, we may perceive the accumulations or "bubbles"
of masses, stars and galaxies, as elevations from a more highdimensional
world. As unavoidable irregularities in a "degraded"
space.
Aggregation of masses becomes " intrusions"
from a 54dimensional world into this lower degree. (Is that
why mountains are regarded as homes for Gods?)
We have mentioned such "intrusions" from
the other direction, lower ddegrees as 1dimensional lines
when curved making "intrusions" in ddegree 2, curved
surfaces in ddegree 3. Extra(or inter)polation gives "curved"
3dimensional volumes implying intrusions in ddegree 4: What
should such "new" curvature of volumes involve?
In which form should it appear? Perhaps just that which Einstein
said, that big masses curve the space around them! I.e. another
aspect on Space: not only a simple, allpenetrating coordinate
system x, y, z, applicable in the same way in masses and empty
space, but with another, separate more highdimensional manifestation
in these "bubbles" of masses. (Another possibility
is to associate it with negative curvature but inwards, the
principle of life!)
Viewing the development in the other direction:
Do the "intrusions" of higher ddegrees into lower
ones, of a pole of ddegree 5 and ddegree 4 into ddegree
3, imply something else than intrusions seen in the opposite
direction?
It seems so in one sense: We get the highdimensional
world inside, within the aggregated masses, not as external
as with the opposite view.
If a 4dimensional and partially 5dimensional reality shall
show up and find room in a 3dimensional world, there unavoidably
will be "bubbles" in the xyzspace! Bubbles as
the result and a solution of he problem.
Vector fields outwards/inwards have to curl,
volumes transform to surfaces, as surfaces into lines, a way
of transformations from a hyperbolic geometry with negative
curvatures* towards an elliptic one in its adjustment to an
Euclidean surrounding. Ddegree 0 →
4 →3 →(2)
(Mass is a very effective way to stow energy.
A shirt for volumes of chests gets flat as a surface when
packed up. Diagonals reduced to points!)
*Is there a possibility that
the meeting of 2 negatively curved surfaces
(or volumes?)  which would presuppose several centers or
"0poles" 
may imply formations of enclosed volumes and the start of
the "gravitational force" and the elliptic geometry
for masses?
Einstein is said to have imagined the property Mass as one
dimension added to the 4dimensional spacetime. It sounds
curious and doesn't agree with the views in the model here,
but it could eventually be understood in the sense above.
(Compare the suggested interpretation of formula E = mc^{2}
(file...): mass as 3dimensional and the velocity c
squared as representing two steps from ddegree 5 to 3.)
How is such a view compatible with the fundamental assumption
in our model that higher ddegrees in masses, when interpreted
as 3dimensional, are transformed into external motions as
rotation? Is it only a question about viewpoints? Is there
perhaps no rotation as an "absolute" motion when
taking the view "from inside" a mass, the rotation
only a relative motion seen from outside in a 3dimensionally
interpreted cosmos? Perhaps only a question about the level
 or ddegree  of analysis? Or inner 4dimensional vector
fields as binding forces only partly transformed into rotation
in ddegree 3?
The two views on "intrusions" could be illustrated
by the perpendicular aspect on our dimension chain  and connected
with the development of higher levels versus the process of
reproduction, two directions which can be regarded as orthogonal
in a dimension chain:
Motions builtin as structural elements
When debranched degrees in first steps of a dimension chain
meet from the end of the chain (0/00) inwards in step 32,
a vertical axes through this step can illustrate a development
direction towards higher levels as bubbles of celestial bodies
 or atoms.
The one way direction of the chain,
5 → 4 →
3 → 2 →1
→0/00 (~5)
can illustrate a reproduction way on the ground level. What
is left of these ddegrees 2 →1
→0/00 may be identified with
the external environment in which the bodies move and from
which they get their nourishment (as Hatoms or He for the
galaxies).
Figure illustrating the thought about
Level Development:
At first these views may seem incompatible with our descriptions
of Mass and Space as poles of ddegree 3, but the a and bpoles
of a ddegree in our model inherit the characteristics of
00 and 0poles.
We may also look at a dimension chain as
"haploid" between 0 and 00poles: (not 50/00)
and assume that only the 0pole develops towards intrinsic
complexity and "bubbles" in the middle step 32,
defining enclosed centers, while the indefinite 00pole as
anticenter only is defined through this process, only defined
as the environment on each stage: the designing of mass bubbles
given from "outside", ensuing from lower ddegrees
through the condition: a 32dimensional Space.
With such views it seems possible to unite
the aspect of more highdimensional bubbles of Mass in space
with the suggested elementary definitions of Mass and Vacant
Space as complementary "poles" of ddegree 3.
6. About Mass as an effective way to store energy:
Compare with transformations between numberbase systems.
Suppose ddegrees are connected with such
different systems. A number in base10, if first divided in
parts, most often give a smaller sum when transformed to base8,
than if transformed as a whole. Yet, a development from higher
numberbase systems toward lower ones implies growing numbers,
mirroring the growth of cosmos.
In the opposite direction, a cosmos of binary
digits may be packed up with higher numberbase systems until
we reach number 1 (as billions of cells in one head) and beyond
that.
To From 4th
to 3rd dimension degree  problematic issues,VVI
