Hyper-Spacetime and Intelligent dimensions

a) The first characteristic is that it is projective (i.e. it is considered by Projective Geometry) whereupon it holds a duality symmetry (something alike it exists in String theory but also in other physical theories). In other words, it constitutes a mixture of two different substances. The first substance (Ordinary Spacetime) is the usual Reality that relates with Electromagnetism and Nuclear (strong, weak) Forces and conceives immediately by man. The other substance (Anti-Spacetime) is somehow vague (something as shade) and is related with the Inertial reference frames and Gravity and is, indirectly, perceptible by man.

b) The second characteristic is that it is trinitarian i.e. it has three orientations (or axes), the known x, y, z. However, in each orientation we may correspond two senses. Consequently, it has six directions (x, y, z and u, v, w)

We rimind that: direction = orientation + sense.

c) The third characteristic is that its dimensions are dynamical (contrarily to Classical Geometry where dimensions are static and play an auxiliary role (active interpretation)). It means that they contain all of information and exert forces on geometrical and material objects that dwell in Spacetime and they, thus, define their form, location and any properties (physical, chemical and even biological) ( passive intepretation). Thus, Spacetime (according to IDT) is omnipotent and omniscient!

- In Classic Geometry (and in Classic Physics), Space is represented by position vector (r) which defines the location of a point (or a paricle, respectively). This vector is related (in its contravariant form) by a 3x1-column matrix, which has as elements the known dimensions x, y, z. The Relativity incorporated the time and changed the position vector in a 4x1-column matrix.

- In Theory of Intelligent Dimensions, instead, the imposing dimensional building of Extended Spacetime requires the 3x1 position vector (r) is changed in a 8x8-matrix (R) that should, under regular conditions, contained 64 dimensions but because of dual symmetry their number is limited in 32.

This matrix is related with the SO(32) matrix of String Theories (of Type I and Heterotic).

This matrix is actually a generalisation of Poincare’s matrix.
But the matrix of Poincare contains parameters of transformations while the matrix of IDT, which we will analyze, contains intelligent dimensions that incorporate the transformations, and represents the Spacetime itself.
These dimensions are not scattered but are distributed in 12 subspaces (or else, packets of dimensions).

These distributed in three categories:

a) Pointed Spacetime

The four sub-spaces of first category contain pointed dimensions.

These dimensions substitute discrete transformations (i.e inversions).

Their matrix-representatives take up the four corners of the 8x8-matrix (R).

Two of them, namely, Pointed Space and Pointed Time, take up the first and the last 1x1-element of the 8x8-matrix (R), on main diagonal, while the two others, namely, the Pointed Anti-Space and Pointed Anti- Time, lie on the two ends of secondary diagonal.

b) Linear Spacetime

The four sub-spaces of second category contain linear dimensions.

These dimensions substitute continuous linear transformations (i.e. translations).

Their matrix-representatives take up the four sides, namely, exterior columns and rows
of 8x8-matrix (R).

Two of them, namely, Linear Space and Linear Time, take up the first 8x1-column (i.e. left) of the 8x8-matrix (R) while the two others, namely, the Anti-linear Space and Anti-inlinear Time, take up the last (i.e. down) 1x8-row of the 8x8-matrix (R).
(Their duals go respectively, namely, to right 8x1-column and to up 1x8-row).

c) Bilinear Spacetime

The four sub-spaces of third category contain bilinear dimensions.

These dimensions substitute continuous bilinear transformations (i.e. rotations).

Their matrix-representatives take up the 6x6-interior of 8x8-matrix (R).
Two of them, namely, Βilinear Space and the Βilinear Time, include the main diagonal while the two others, namely, the Bilinear Anti-Space and Bilinear Anti-Time, include the secondary diagonal.

(At the end, very fleetingly, I will report that the number “8” of 8x8-matrix (R) and the number “8” of the bits that constitute the “byte” are not coincidental but are a result of consequence of endogenous incorporation of information by dimensions of this Extended Spacetime).

In the following, we will examine these sub-spaces analytically.

Part II –Pointed Spacetime

As we already quoted, this category includes four subspaces which contain six pointed degenerated dimensions. These are : (1), (i) and its duals (-1), (-i):

1) Pointed ( or Imagine Conoid) Space

This is moulded by three (degenerated to one) real dimensions of zero degree.
This tree-fold dimension is the real negative unit (-1).

It is represented by a 1x1 single-element matrix.

- It constitutes a source. It creates charged strings as copies of dimensions.

Perhaps, it is related with a “white hole”.

- It substitutes the transformation of “space inversion”.

- It is related with the generator of symmetry that is represented by the physical quantity:
“potential (or field) energy”.

- They are related with symmetry breaker that is represented by the physical quantity:

“potential (or field) power”.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1

2) Pointed ( or Imagine Ellipsoid) Time

This is moulded by three (degenerated to one) imagine dimensions of zero degree.

This tree-fold dimension is the positive imagine unit (i).

-It is represented by a 1x1 single-element matrix.

It constitutes the sink-hole (in other words, it is a dimension of annihilation of massive strings).
Perhaps, it is related with a black hole.
(in other words, if you enter it then you leave our Cosmos and you enter other Cosmos of Universe).

This dimension has the the following attributes:

- It substitutes the transformation of “combination of charge conjugation, space inversion and time reversal) (CPT)”
- It is related with the generator of symmetry that is represented by the physical quantity:
“kinetic (or inertial) energy or mass”.

- They are related with symmetry breaker that is represented by the physical quantity:

“kinetic (or inertial) power”.

- They erect forces on string and oblige it to become elliminated.
So, string is, perpetually, moving on surface of a imagine ellipsoid (lied at -i∞).

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1






i

3) Pointed ( or Imagine Conoid) Anti-Space

This is moulded by three (degenerated to one) real dimensions of zero degree.
This three-fold dimension is the real positive unit (1).

It is represented by a 1x1 single-element matrix.
- It constitutes a source. It creates maassive strings as copies of dimensions.

Perhaps, it is related with a “white hole”.

This dimension has the the following attributes:

- It substitutes also the transformation of “identity (I)”

- It is related with the generator of symmetry that is represented by the physical quantity:
(mass).

- They are related with symmetry breaker that is represented by the physical quantity:

“?”.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1






i							1

4) Pointed ( or Imagine Ellipsoid) Anti-Time

This is moulded by one imagine dimension of degree zero.
This dimension is the known negative imagine unit (- i).

It is represented by a 1x1 single-element matrix.

It constitutes the sink-hole (in other words, it is a dimension of annihilation of charged strings).
Perhaps, it is related with a black hole.

This dimension has the the following attributes:

- It substitutes the transformation of “combination of time reversal and charge conjugation) (TC)”.
- It is related with the generator of syymmetry that is represented by the physical quantity:
“electric charge”.

- They are related with symmetry breaker that is represented by the physical quantity:
“electric dipole moment”.

It is the dual dimension of the dimension of Pointed Anti-Space.

- They erect forces on a string and oblige it to become elliminated.
So, string is, perpetually, moving on surface of a imagine ellipsoid (lied at -i∞).

- It is the dual dimension that of Pointed Time.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1							-i






i							1

Part III – Linear Spacetime

As we already quoted, this category includes four subspaces which contain twelve linear dimensions: (s_i) , (t_i) , (s_m) and (t_m) : i Î (x,y,z) and m Î (u,v,w)

1) Linear Space

This is moulded by three linear real dimensions of degree one.
They are ordinary space distances (x, y, z) or else, in notation of IDT, (s_x, s_y, s_z).

It is represented by a 3x1-column (and its corresponding dual 1x3-row, besides).

s_x

s_y

s_z

These dimensions have the the following attributes:

- They incorporate the symmetry of “space homogeneity”
- They substitute the transformation of< compination of “inversional rotation and translation”.
- They are related with the generator off symmetry that is represented by the physical quantity “magnetic angular momentum”.

- They are related with symmetry breaker that is represented by the physical quantity:

“magnetic moment of force”.

- They erect attractive forces on a string and oblige it to intersect them at one point of its positive hemi-axe.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	-i
s_x
s_y
s_z



i				1

2) Linear Time

This is moulded by three linear imagine dimensions of degree one.
They are the time distances (t_x, t_y, t_z). These dimensions are the usual time distance (t) and other two extra dimensions of the denomitated Hyper-time or 3D-time.
It is represented by a 3x1-column-matrix (and corresponding dual 1x3-matrix, besides).

it_x

it_y

it_z

These dimensions have the following attributes:

- They incorporate the symmetry of “time homogeneity”.

- They substitute the transformation of “pure translation”.
- They are related with the generator off symmetry that is represented by the physical quantity:
“electric momentum” .

- They are related symmetry breaker that is represented by the physical quantity:
“electric force”.

- They exert repulsive forces to a string and obliges it to intersect them at no points ( i.e. to lie parallel to them).

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	it_x	it_y	it_z	-i
s_x
s_y
s_z
it_x
it_y
it_z
i							1

3) Linear Anti-Space

This is moulded by three anti-linear real dimensions of degree one.
They are the inertial space distances (s_u, s_v, s_w).

It is represented by 1x3-row-matrix (and its dual corresponding 3x1-matrix, besides).

s_u

s_v

s_w

These dimensions have the following attributes:

- They incorporate the symmetry of “anti-space homogeneity”.

-They substitute the transformation “combination of rotation and reversal translation”
- They are related with the generator off symmetry that is represented by the physical quantity “vorticity (or inertial) angular momentum”.

- They are related with the symmetry breaker that is represented by the physical quantity
“mechanical (or inertial) torque”.

- They erect attractive forces on a string and oblige it to intersect them at one point of its negative hemi-axe.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	it_x	it_y	it_z	-i
s_x
s_y
s_z
it_x							s_u
it_y							s_v
it_z							s_w
i				s_u	s_u	s_w	1

4) Linear Anti-Time

This is moulded by three anti-linear imagine dimensions of degree one.
They are the inertial
time distances (t_u, t_v, t_w).

It is represented by 1x3-row-matrix (and its corresponding dual 3x1-matrix, besides).

it_u

it_v

it_w

These dimensions have the following attributes:
- They incorporate the symmetry of “anti-time homogeneity”.
- They substitute the transformation of “pure reversal translation”
- They are related with the generator of symmetry that is represented by the physical quantity
“inertial (or mechanical) momentum”.
- They are related with the with symmetrry breaker that is represented by the physical quantity
“inertial (or mechanical) force”.

- They exert torques to strings and force these to intersect on

- They erect attractive forces on medium of a string and oblige it to intersect them at zero point origin (0) and they, simultaneously, erect repulsive torque on its ends.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	it_x	it_y	it_z	-i
s_x							-it_u
s_y							-it_v
s_z							-it_w
it_x							s_u
it_y							s_v
it_z							s_w
i	it_u	it_v	it_w	s_u	s_u	s_w	1

Part IV – Bilinear Spacetime

As we already quoted, this category includes four subspaces which contain twenty eight bilinear dimensions: (φ_i) , (θ_i), (ψ_i ), (η_i ), (ζ_mm) and (ξ_mn) : i Î (x,y,z) and m,n Î (u,v,w)

1) Bilinear Ellipsoid Space

This is moulded by three bilinear real dimensions of degree two.

They are the elliptic angles (φ_{x ,} φ_y, φ_z).

It is represented by a screw-symmetric 3x3-matrix (which contains these dimensions and its corresponding duals, besides).

0	-φ_z	φ_y
φ_z	0	-φ_x
-φ_y	φ_x	0

These dimensions have the following attributes:
- They incorporate the symmetry of “space isotropy”
- They substitutte the transformation of “elliptic rotation”).
Perhaps, this the transformation is related with color-SU(3) one.
- They are related with the generator off symmetry that is represented by the physical quantity
“color angular momentum” .

- They with symmetry breaker that is represented by the physical quantity
“color torque”.

- They erect attractive forces on the two ends of a string and oblige it to intersect them at two points (one at its positive hemi-axe and one at its negative hemi-axe).
So, string is, perpetually, moving on surface of a ellipsoid.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	it_x	it_y	it_z	-i
s_x	0	-φ_z	φ_y				-it_u
s_y	φ_z	0	-φ_x				-it_v
s_z	-φ_y	φ_x	0				-it_w
it_x							s_u
it_y							s_v
it_z							s_w
i	it_u	it_v	it_w	s_u	s_u	s_w	1

2) Bilinear Mono-chone Hyperboloid Space

This is moulded by three anti-bilinear imagine dimensions of degree two.
They are the hyperbolic angles (θ_{x ,} θ_y, θ_z).

It is represented by 4x4 hermitean matrix (It contains these three dimensions and their three corresponding duals, besides).

		iθ_x	0
		-iθ_y	iθ_z
-iθ_z	iθ_y
0	-iθ_x

These dimensions have the following attributes:

- They incorporate the symmetry of “isotropic coupling” . It related with the dubbed “center of mass” - They substitute the transformation of “monochone boost” or “monochone hyperbolic rotation”).
Perhaps, this the transformation is related with isospin-SU(2) one.
- They are related with the generator off symmetry that is represented by the physical quantity:
“dia-magnetic magnetization twist hyper-strain angular momentum”.

- They are related with a symmetry breaker that is represented by the physical quantity:
“dia-magnetic magnetization torsion hyper-stress”.

- They erect repulsive forces on the two ends of a string and oblige it to intersect at no points.

So, string is, perpetually, moving on surface of a monochone hyperboloid.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	it_x	it_y	it_z	-i
s_x	0	-φ_z	φ_y				-it_u
s_y	φ_z	0	-φ_x	iθ_x			-it_v
s_z	-φ_y	φ_x	0	-iθ_y	iθ_z		-it_w
it_x		-iθ_z	iθ_y				s_u
it_y			-iθ_x				s_v
it_z							s_w
i	it_u	it_v	it_w	s_u	s_u	s_w	1

3) Bilinear Conoid Time

This is moulded by three bilinear real dimensions of degree two.
These dimensions are the conic distances (ψ_x, ψ_y, ψ_z).
These dimensions are self-adjoint (i.e. they are coincide (not, simply, equal) to its duals)

It is represented by 6x6 hermitean matrix (It contains these three dimensions and their three corresponding duals, besides).

			-iψ_u
				-iψ_v
					-iψ_w
iψ_u
	iψ_v
		iψ_w

These dimensions have the following attributes:
- They incorporate the symmetry of “homootropic bisection”.
- They substitute the transformation of “conic translation”.
- They are related with a generator of ssymmetry that is represented by the physical quantity “polarization dilation hyper-strain angular momentum”.
- They related with a symmetry breaker that is represented by the physical quantity
“polarization tension hyper-stress couple”.

- They erect tensions on the two ends of a string and oblige it to be stretched.

So, string is,, perpetually, moving on surface of a hyperboloid (real) cone.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	it_x	it_y	it_z	-i
s_x	0	-φ_z	φ_y	-iψ_x			-it_u
s_y	φ_z	0	-φ_x	iθ_x	-iψ_y		-it_v
s_z	-φ_y	φ_x	0	-iθ_y	iθ_z	-iψ_z	-it_w
it_x	iψ_x	-iθ_z	iθ_y				s_u
it_y		iψ_y	-iθ_x				s_v
it_z			iψ_z				s_w
i	it_u	it_v	it_w	s_u	s_u	s_w	1

4) Bilinear Di-chone Hyperboloid Time

This is moulded by three anti-bilinear imagine dimensions of degree two.
They are the hyperbolic angles (η_{u ,} η_v, η_w).
It is represented by 6x6 Hermitean matrix (It contains these three dimensions and their three corresponding duals, besides).

		-iη_u	iη_v
			-iη_w


iη_w
-iη_v	iη_u

These dimensions have the following attributes:
- They incorporate the symmetry of “isottropic bisection”. It related with the dubbed “center of mass” - They substitute the transformation of “dichone boost” or “dichone hyperbolic rotation”).
Perhaps, this the transformation is related with isospin-SU(2) one, too.
- They are related with the generator off symmetry that is represented by the physical quantity:
“para-magnetic magnetization flexion hyper-strain angular momentum”.
- They are related with a symmetry breaker that is represented by the physical quantity:

“para-magnetic magnetization bending hyper-stress torque”.

- They erect repulsive forces on the two ends of a string and oblige it to intersect at no points.

So, string is, perpetually, moving on surface of a dichone hyperboloid.

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	it_x	it_y	it_z	-i
s_x	0	-φ_z	φ_y	-iψ_x	-iη_u	iη_v	-it_u
s_y	φ_z	0	-φ_x	iθ_x	-iψ_y	-iη_w	-it_v
s_z	-φ_y	φ_x	0	-iθ_y	iθ_z	-iψ_z	-it_w
it_x	iψ_x	-iθ_z	iθ_y				s_u
it_y	iη_w	iψ_y	-iθ_x				s_v
it_z	-iη_v	iη_u	iψ_z				s_w
i	it_u	it_v	it_w	s_u	s_u	s_w	1

5) Bilinear Spiroid Space

This is moulded by three bilinear real dimensions of degree two.
These dimensions are the “compactificated” angles (ζ_xy, ζ_yz, ζ_zx)

It is represented by a symmetric off-diagonal 3x3 matrix (which contains these three dimensions and three duals, besides).

0	ζ_xy	ζ_yz
ζ_yx	0	ζ_yz
ζ_zx	ζ_zy	0

These dimensions have the following attributes:
- They incorporate the symmetry of “isotteny”
- They substitute the transformation of “distortional deformation”.

- They are related with a generator of symmetry that is represented by the physical quantity “gravitational slipping strain angular momentum”.
- They are related with a symmetry breaker that is represented by the physical quantity
“gravitational shear (or tangentional) stress torque”.

- They erect shear on the two ends of a string and oblige it to be twisted.

So, string is, perpetually, residing in the interior of a ellipsoid.

Especially, they are, spirally, lied on its three planes of symmetry .

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	it_x	it_y	it_z	-i
s_x	0	-φ_z	φ_y	-iψ_x	-iη_u	iη_v	-it_u
s_y	φ_z	0	-φ_x	iθ_x	-iψ_y	-iη_w	-it_v
s_z	-φ_y	φ_x	0	-iθ_y	iθ_z	-iψ_z	-it_w
it_x	iψ_x	-iθ_z	iθ_y		ζ_uv	ζ_wu	s_u
it_y	iη_w	iψ_y	-iθ_x	ζ_vu		ζ_vw	s_v
it_z	-iη_v	iη_u	iψ_z	ζ_wu	ζ_wv		s_w
i	it_u	it_v	it_w	s_u	s_u	s_w	1

6) Bilinear Helicoid Space

This is moulded by three bilinear real dimensions of degree two.
These dimensions are the “compactificated” distances (ξ_uu, ξ_vv, ξ_ww).
These dimensions are self-adjoint (i.e. they are coincide (not, simply, equal) to its duals)

ξ_uu
	ξ_vv
		ξ_ww

These dimensions have the following attributes:
- They incorporate the symmetry of “homooteneity”.
- They substitute the transformation of “dilatational deformation”.
- They are related with a generator of ssymmetry that is represented by the physical quantity “gravitational dilation strain angular momentum”.
- They related with a symmetry breaker that is represented by the physical quantity
“gravitational tension ( or transverse stress) couple”.

- They erect tensions on the two ends of a string and oblige it to be stretched.

So, string is, perpetually, residing in the interior of a ellipsoid.

Especially, they are, helically, entwined on its three axes of symmetry .

Now, we put the dimensions of this subspace at its locations,
in the position 8x8 matrix (R).

-1	-s_x	-s_y	-s_z	it_x	it_y	it_z	-i
s_x	0	-φ_z	φ_y	-iψ_x	-iη_u	iη_v	-it_u
s_y	φ_z	0	-φ_x	iθ_x	-iψ_y	-iη_w	-it_v
s_z	-φ_y	φ_x	0	-iθ_y	iθ_z	-iψ_z	-it_w
it_x	iψ_x	-iθ_z	iθ_y	ξ_uu	ζ_uv	ζ_wu	s_u
it_y	iη_w	iψ_y	-iθ_x	ζ_vu	ξ_vv	ζ_vw	s_v
it_z	-iη_v	iη_u	iψ_z	ζ_wu	ζ_wv	ξ_ww	s_w
i	it_u	it_v	it_w	s_u	s_u	s_w	1

Part V – Epilogue

Thus, we have 1-1 correspondence between Geometry and Physics (String theory sees this correspondence as 2-1).

Here, let we comment, that the last six dimensions that are contained in the Spiroid Space (paragraph 5 ) and in Helicoid Space (paragraph 6) are, very probably, related with the Calabi-Yau Space of String Theory.

At the end, let us remark that the physical quantities (symmety generators or breakers, field intensities etc), which are used by IDT, are extended tensors (which contain, innately, the spinors).

Especially, we must hightlight that the physical quantities, which emerge from original Spacetime, interpretate mathematically by k-differential forms and demand covariant indices.

Instead, the physical quantities, which emerge from inertial Spacetime, interpretate mathematically by k-vector fields and demand contravariant indices.

Everything is writed down cursorly. Maybe, there are some, non-essential technical faults but, in general lines, the report is right and gives the general idea of formulation of the theory.

I ΄ll return with the relation between Quantum Mechanics and IDT where it become entirely clear why the quantum cloak is nessesary for comprehension of Physical World by Man (here is involved the anthropic principle) but doesn’t constitute main feature of Universe!