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European Journal of Applied Sciences – Vol. 11, No. 5
Publication Date: October 25, 2023
DOI:10.14738/aivp.115.15905
Groot, C. T. (2023). The Possible Existence of Parallel Universes Within the Total Number of Hypercubic Lattices of a
Multidirectional Discrete Space. European Journal of Applied Sciences, Vol - 11(5). 56-67.
Services for Science and Education – United Kingdom
The Possible Existence of Parallel Universes Within the Total
Number of Hypercubic Lattices of a Multidirectional Discrete
Space
Christiaan T. Groot
ABSTRACT
The very many subspaces of a multidirectional hypercubic lattice provide a space
structure not present in conventional space demonstrating the possibility of
superstructures of self-contained universes of nuniverse subspaces. To show that, first
the total number of subspaces nsubspaces of the expanded lattice is determined from
the discrete gravitational constant, that is, expressed in the units of a lattice. The
total number of subspaces nsubspaces is so large that further structuring in universes
of lattices is possible. Starting from the plausible assumption that interactions are
limited to one subspace, it is argued that only particles present in one of the
universes will interact. Since nsubspaces/nuniverse >1, different autonomous universes
are possible.
Keywords: Multidirectional discrete space, hypercubic lattice, gravitational constant,
black matter, universe.
INTRODUCTION
When opting for a discrete space, the hypercubic lattice is an obvious choice due to its very
discrete basic structure. However, such a lattice is not isotropic in the spatial directions. The
multidirectional hypercubic lattice of Ref [1] offers possible solution to the isotropy problem.
This lattice consists of a plurality of nsubspaces perfectly equal hypercubic lattices, called
subspaces, arranged in time series of npoint coupled subspaces. Here, each subspace represents
a spatial direction.
For the multidirectional hypercubic lattice to be a discrete space, the number of different
subspaces nsubspaces, though very large, must be limited. Purpose of this article is to determine
nsubspaces from the gravitational constant.
The set of multiple hypercubic lattices offers a discrete alternative Euclidian space having
features that do not exist in such as grouping the subspaces into sets with a common spatial
characteristic Ref [1]. Each set contains a number of subspaces represented by the
characteristic natural number ngroup. In Ref [2] it is shown that some groups can be associated
with a physical constant. There it has also been shown that there are various relationships
between the characteristic numbers. Some of these relations are used in Chapter 2 to express
the discrete gravitational constant in the only two units Δx and S of the multidirectional lattice.
To find the expression for the discrete gravitational constant, a result of general relativity is
used i.e., the formulation of time dilation in a gravitational field. This is because time dilation is
quite easy to describe in a lattice.
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Groot, C. T. (2023). The Possible Existence of Parallel Universes Within the Total Number of Hypercubic Lattices of a Multidirectional Discrete Space.
European Journal of Applied Sciences, Vol - 11(5). 56-67.
URL: http://dx.doi.org/10.14738/aivp.115.15905
In Chapter 3 it is shown that the time dilation in a multidirectional lattice can be used to relate
the discrete gravitational constant to the total number of subspaces nsubspaces. No attempt is
made to explain gravity from the perspective of a discrete space. Using the enormous numerical
value of the nsubspaces found, Chapter 4 explores the possibility of a further grouping of subspaces
by defining superclusters of self-contained universes. Every universe is supposed to have
enough subspaces to incorporate all known particles and their motions. It is also shown that
there is no particle interaction between universes, making each universe a place to situate
matter or energy that is dark to other universes. The multidirectional discrete space offers the
possibility to explain dark energy or matter from the perspective of spaces.
THE GRAVITATIONAL CONSTANT IN DISCRETE UNITS
The multidirectional discrete space consists of very many completely identical hypercubic
lattices differing only in spatial direction, the characteristic of the lattice Ref [1]. This makes it
possible to distinguish several structural elements of the lattice, such as the group of subspaces
consisting of the smallest set of all the different spatial directions with as characteristic number
ndirections.
To get mutually comparable expressions for the characteristic numbers, the usual SI-units are
converted into the two units Δx and S of the multidirectional hypercubic lattice. One of the
physical constants, the fine structure constant, turns out to be a quotient of two characteristic
numbers. Interestingly, this quotient eliminates the conversion constant ρconv of metric in
discrete units (see relation (3)), making it possible to express some of the characteristic
numbers in a sequence of natural numbers Ref [2], as shown in Table 1.
Only, for the total number of subspaces nsubspaces, no relation to a physical constant has been
established. The gigantic number of subspaces nsubspaces that make up the multidirectional space
is probably be much greater than any other characteristic number. Because of the smallness of
the gravitational constant, this chapter examines the possibility whether nsubspaces is inversely
proportional to this constant.
Some Quantities Expressed in Discrete Units
To obtain the gravitational constant in the discrete units Δx and S, some quantities need to be
expressed in these units. The unit Δx stands for the smallest distance of a lattice and the unit S
for the subspaces in which the phenomenon is present. The following relations of Ref [2] are
used in the next section to convert the expression for the gravitational constant in discrete
units:
xd = xcontinu/ρconv Δx. (1)
td = tcontinu c/ρconv Δt. (2)
In a hypercubic lattice holds Δt = Δx. The conversion constant ρconv is given by
ρconv = md electron/mec
2 e
2/ed
2 = re/Rd ecm/Δx, (3)
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where Rd e is the discrete electron radius and ed the discrete elementary charge, whose numerical
value(s) are determined in Ref [2]. md electron is the discrete electron mass given by
md electron = ed
2/Rd eS/Δx.
The relation discrete mass md to the usual mass m is given by
md = m md electron/meS/Δx. (4)
The Gravitational Constant in Discrete Units
Time dilation relation of the general theory of relativity describes gravitational action of one
mass field. The mathematical results will be used to describe the discrete gravitational constant
in a discrete space.
According to the general theory of relativity, the time interval dτ within a gravitational field
with regards to a time dt at great distance to the center of mass is given by (see for example Ref
[4]):
dτ = g001/2 dt. (5)
In the case of a spherically symmetric field, the relation proper time τ and Cartesian time t is
given in good order approximation by (see Ref [4]):
τ/t = (1 - G m/c2r). (6)
By using (1), (2), equation (6) is converted into discrete quantities resulting in the following
relation between proper time τ and Cartesian time
τd/td = τ/t = 1 - G md/rd me/md e 1/c2ρconv.
Take as the discrete gravitational constant
Gd = G me/md e 1/c2ρconv.
Then (6) becomes after eliminating ρconv with (3)
τd/td = τ/t = (1 - Gd md/rd), (7)
with
Gd = G (me/md e)
2 (e2/ed
2)
-1 Δ2x/S.
Note: Since the variables of τ/t are both expressed in the discrete time unit, the time quotient
is dimensionless. This makes the dimensions of 1/Gd those of md/rd, which is S/Δ2x (see (4) for
the dimension of md).
After eliminating the mass quotient in Gd with (3), the expression for Gd can be simplified to
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Groot, C. T. (2023). The Possible Existence of Parallel Universes Within the Total Number of Hypercubic Lattices of a Multidirectional Discrete Space.
European Journal of Applied Sciences, Vol - 11(5). 56-67.
URL: http://dx.doi.org/10.14738/aivp.115.15905
Gd = G (e2/ed
2) (Rd e/c2re)
2 Δ2x/S. (8)
All continuous parameters in above equation are known. The discrete ones have been
calculated in Ref [2], repeated here in Table 1. This allows the numerical value of Gd to be
calculated, which will be done in Section 3.5.
Because all discrete variables are natural numbers, Ref [2] yielded relations as
ed
2 = npoint S.
Where npoint is the number of subspaces coupled along the time-axis as defined in (10). It has
also been found that the discrete electron radius Rd e satisfies
Rd e = npoint S.
With the above expressions, equation (8) after eliminating of ed
2 and Rd e becomes
Gd = G npoint e
2 c
-4re
-2 x
2/S. (9)
TIME DILATION IN A MULTIDIRECTIONAL HYPERCUBIC LATTICE
In ref [3], to describe motion within a lattice, physical phenomena have been assumed to arise
from deviations from the regular structure of the multidirectional hypercubic lattice, i.e., the
structure of empty discrete space.
With this assumption it has been found that there are deviations of the space structure that
form a corridor bounded in space. The space structure in this enclosed space can be such that
there is movement of the entire space of the corridor relative to the surroundings. The deviation
in space-time structure at the boundaries is the cause of both time dilation and length
contraction within the corridor.
In a gravitational field, time dilation is present as very tiny but experimentally confirmed effect.
In the hypercubic lattice, being a completely autonomous space without reference to another
space (e.g., Euclidean space), the axes are straight lines by definition. This makes it difficult to
apply the concept of general relativity of curved geodesics in a hypercubic lattice when one of
the axes are the geodesic.
Regardless of the origin of the effect, the time dilation due to gravitation will be described in
the same way as for motion. The result, the mathematical description of time dilation, is
assumed to be equivalent to that of general relativity.
It will be assumed that there is a deviation from the regular space structure that causes the
dilation of time. The possible reason for the deviation from the structure will not be given here.
It must be found in the mass of discrete fields because of Einstein’s principle of equivalence of
inertial and gravitational mass. The solution is probably part of a particle model in a hypercubic
lattice, i.e., a description in terms of deviations of the spatial structure of the particle field.
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Time Dilation
A lattice consists solely of vertices interconnected in pairs by edges. The hypercubic lattice is
structured so that each vertex of the lattice is part of eight axes, which are by definition
perpendicular to each other.
The distance in a lattice between two vertices A and B is defined as the number of intermediate
vertices along the axis through A and B, which is also the case in a multidirectional hypercubic
lattice. This extended lattice is a chain of npoint hypercubic lattices connected along the time-axis
that repeats in time after Δt. The thus defined time-axis is called a space point. In a
multidirectional hypercubic lattice there is a difference between the minimum length of the
lattice Δx= Δt and the minimum time difference δt between two consecutive vertices (i.e.,
hypercubic lattices) of a space point with
Δt = npoint δt. (10)
This makes that there are two-time units, Δt within one of the lattices and δt along the time axis
of a space-point. This is of importance for the describing of the time deletion.
For some reason, let there be a gap in time on the time axis of a space point relative to the space
points in the direct surroundings. The anomaly in the structure of the lattice causes the local
distance in time in one region of space to differ from the distance in time in another region.
Figure 1 illustrates the possibility of a periodic time gap of size δt, which is a missing vertex in
the time series of vertices. If there are many adjacent space points with a time gap, the missing
vertices form a missing spatial axis. It is assumed that the time gap is regularly present within
one of the lattices of the multidirectional discrete space, i.e., at ngap time distances Δt. Seen from
the surrounding space, the repetition time Δtgap is
Δtgap = tsurround = ngapΔt. (11)
Viewed locally within the space point there is one less time interval Δt belonging to a time
segment of the lattice
tlocal= (ngap – 1) Δt.
The ratio of the time intervals of the surrounding space to the local space is
tsurround/tlocal = (1-1/ngap).
In Ref [2], it has been found that a mass field in a discrete space have its repetition frequency
Tparticle with
mdiscrete = ndirections/Tparticle S/Δt. (12)
With the time dependence of the particle mass, an object with frequency ΔT measured locally
has a fractional longer repeating time ΔT ́ as measured from surrounding space, with
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Groot, C. T. (2023). The Possible Existence of Parallel Universes Within the Total Number of Hypercubic Lattices of a Multidirectional Discrete Space.
European Journal of Applied Sciences, Vol - 11(5). 56-67.
URL: http://dx.doi.org/10.14738/aivp.115.15905
ΔT ́ = ΔT(1-1/ngap) Δt. (13)
What makes that seen from the surrounding space the particle mass is fractionally larger.
Gap in
time
Gap in
time
ngap
t-axis
Surrounding
space
Local space
ngap - 1
x,y,z-axis
x
spacepoint
t
Figure 1: Time differences in a lattice-like discrete space
Assume that for some reason every ngap there is a time gap in the time series of vertices as seen
from the surrounding space. In the local space of the space point, this time gap occurs every
tlocal = ngap-1. Since the surrounding space and the local space are one and the same lattice, the
time period of tsurround time steps corresponds exactly to tlocal steps in local space. This means
that from the surrounding space the local time period ΔT is seen as ΔT ́ = ΔT(1-1/ngap). Objects
in local space experience a time dilation of the factor 1(1-1/ngap).
The Frequency of Time Gaps from Special Relativity
Equating the time dilatation as given by general relativity (7) with the comparable time
dilatation of the discrete space (13) one gets
τ/t = ΔT ́/ΔT. (14)
Uniting of the above equation with (13) gives
τ/t = ΔT ́/ΔT = 1/(1-1/ngap).
And uniting above equation with (7) and (14) makes that
ΔT ́/ΔT = (1 - Gd md/rd).
The combination of the two above equations for ΔT ́/ΔT gives the following relation between
ngap and Gd.
1/ngap= Gd md/rd. (15)
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The Number of Time Gaps Per Unit of Time
In Ref [3] it is shown that, by assuming that particle fields consist of an agglomeration of field
granules, movement of particle fields in a hypercubic lattice can be described.
Each granule consists of a piece of the hypercubic lattice enclosed by a spatial cube where each
cube is repeated in time, depending on the size of the cube. The properties of a cube-like granule
are determined by the boundaries, the transition of the inner space with its surrounding space.
With the assumption that there are three different types of field granules, the velocity of a
particle field can be described Ref [3] whereby the particle field consists of a multitude of
concentric granules of various sizes. In Ref [2] it has been found that that each granule is
distributed in time over ndirections subspaces where in each of these subspaces, the mean
repetition period of the granule is Tparticle, averaged over all various sizes of the particle's field
granules.
To relate the time gaps to the gravitational constant, the frequency of the gaps in time is first
determined. It is assumed that the time gaps arise from the granules of the particle's mass field.
• First, it should be noted that a particle field is present in only ndirections subspaces of the
nsubspaces subspaces of the multidirectional space. The number of time gaps is therefore
proportional to the fraction of subspaces in which a particle field is present.
• Secondly, it turns out that there is a gap in time at each Tparticle of the particle field, i.e., at
the moment a field granule is present.
• Thirdly, in ref [[3] it is determined that, from a spatial point of view, the active part of a
granule is found in the boundary surfaces of the granule. That is in a layer in one of the
lattices of the thickness of the unit length Δx (=1). Since the gaps can only be found in
boundaries of the granule, this means that each instant in time the presence of the gaps
is inversely proportional to the average size rd of the particle granules.
Taken together is the number of the time-gaps per time unit at the boundary of the granule
ρgap = ndirections/nsubspaces 1/Tparticle Δx/rd1/Δt.
The above equation in combination with the expression of the discrete particle mass (12)
makes that
ρgap = md particle/nsubspaces Δx/rd1/Δt. (16)
The Number of Subspaces of the Multidirectional Lattice
In one lattice, the time difference along the time axis can be simply defined as the number of
intermediate vertices. In the multidirectional lattice, the time difference along the space point
is also related to the number of subspaces due to the definition of the multiple lattices as a
concatenation in time of npoint different lattices Ref [1]. Each lattice in turn connects a multitude
of other time series of fully different lattices. If the number of lattices thus created is limited,
the multidirectional space is a discrete space. Therein all the lattices have a minimum time
spacing δt. Due to the successive presence of lattices there is a time difference between the
occurrence of various lattices.
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Groot, C. T. (2023). The Possible Existence of Parallel Universes Within the Total Number of Hypercubic Lattices of a Multidirectional Discrete Space.
European Journal of Applied Sciences, Vol - 11(5). 56-67.
URL: http://dx.doi.org/10.14738/aivp.115.15905
With ρgap as the average number of gaps per Δt caused by the granules of the mass field over all
the subspaces in which the field is present, the mean time-difference between two consecutive
gaps is
Δtgap= 1/ρgap Δt. (17)
Which is with (16)
1/Δtgap= ρgap = md particle /nsubspaces 1/rd 1/Δt. (18)
Union of equations (13) and (17) gives ngap = 1/ρgap. After elimination of both rd and the particle
mass md with (18) and (15), the relation between the discrete gravitational constant Gd and the
total number of subspaces nsubspaces of the multidirectional hypercubic lattice is obtained:
nsubspaces = 1/Gd S. (19)
Expressing all quantities in discrete units yields the strikingly simple relation (19). Due to the
equation (18) and (15), the dimensions of both md particle and rd have also disappeared.
After combining equation (19) with (9), the number of subspaces becomes
nsubspaces = G-1 npoint-1 e
-2 c
4re
2 S. (20)
Calculation of the Total Number of Subspaces
Take the equation (20). All variables herein are known:
• Newtonian constant of gravitation: G-1= 6.674 28(67)-1 x 1011 m-3 kg s2 {NIST};
• Elementary charge: e
-2 = 1.602176487(40)-21038 coulomb-2 {NIST};
• Velocity of light:c
4= 8,987552 1032 m4 s
-4 {NIST};
• Classical electron radius: re
2 = 2.817 940 2894(58) 210-30 m2 {NIST};
• npoint = 3.3827 1019.
This value is one of the several parameters of the multi-directional hypercubic lattice. In Ref [2]
these are determined from the fine structure constant. In Table 1 they can be found in the form
of a sequence of incrementally increasing natural numbers.
Table 1
nc2 npivot npoint ndirections 1/α {Δ}
1 5.8160 109 3.3827 1019 2.9126 1022 137.035 999 706{ 27}
1 5.8163 109 3.3829 1019 2.9128 1022 137.035 999 658{-20}
1 5.8164 109 3.3831 1019 2.9129 1022 137.035 999 611{-68}
2 2.3264 1010 5.4122 1020 4.6600 1023 137.035 999 729{ 50}
2 2.3265 1010 5.4125 1020 4.6602 1023 137.035 999 706{ 27}
2 2.3265 1010 5.4125 1020 4.6603 1023 137.035 999 682{ 3}
2 .. ..
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Using the minimum value of the above parameters, a straightforward calculation of nsubspaces
shows that its (maximum) value is
nsubspaces = G-1 e
-2 c
4 re
2 npoint-1 = 1.10677(11) 1033S. (20)
UNIVERSES WITHIN THE TOTAL NUMBER OF SUBSPACES
In ref [2] a subdivision of the subspaces of multi directional space is made into subspaces with
a certain characteristic. It turns out that the number of some of these subspaces are related to
a physical constant. In this chapter, the subspaces will be further structured, made possible by
the enormous size of nsubspaces (20). The value of the total number of distinguishable subspaces
nsubspaces is much greater than the number of subspaces with different directions ndirections
needed to contain a particle field; 1.10677(11) 1033 (20) versus 2.9126 1022 from Table 1. The
resulting question is how a subdivision of total number of subspaces can be made into groups
of ndirections subspaces.
Self-Contained Universes of Subspaces
It is assumed that each particle field is present in the ndirections subspaces with a different
direction. With nsubspaces very much larger than ndirections, there is a substantial number of
completely parallel subspaces, i.e., completely different subspaces with the same axis- directions. This finding leads to the following consequence.
The conglomerate of subspaces in which different types of particle fields are present will
consist of many more than ndirections subspaces. In Ref [1], the subgroup of npivot subspaces has
been introduced to give the multidimensional hypercubic lattice its semi-Euclidean character.
The subgroup consists of subspaces, which have one spatial axis in a particular direction and
the two perpendicular axes in all possible spatial directions. The group of ndirections npivot
subspaces is therefore more suitable to contain all types of all particles field having ndirections
directions.
Define as a universe the group of subspaces in which all pairs of subspaces have a common
space axis, allowing interaction between fields within the subspaces along the common space
axis. The number of subspaces of a universe is, with the values obtained from Table1
nuniverse = ndirections npivot
= 2.91 1022 5.81 109 = 1.69 1032S. (21)
Completeness in Diversity of Subspaces of a Universe
The question is whether a universe as defined in (21) is capable of containing the most diverse
phenomena in all their manifestations.
Completeness in Directions of Polarization:
Each universe is so rich in subspaces that for each space axis there are npivot subspaces
consisting of a parallel space axis combined with two axes perpendicular to it Ref [1]. For a
particle field moving in certain direction, this means that there are npivot subspaces for the
particle field to reside in, enough to accommodate a field with different albeit limited number
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Groot, C. T. (2023). The Possible Existence of Parallel Universes Within the Total Number of Hypercubic Lattices of a Multidirectional Discrete Space.
European Journal of Applied Sciences, Vol - 11(5). 56-67.
URL: http://dx.doi.org/10.14738/aivp.115.15905
of polarization directions. This makes a universe as define in (21) complete in possible
polarization directions for each spatial direction.
Completeness in Possible Directions of Movement:
Ref [3] describes the movement of a field in a hypercubic lattice. It shows that a particle field,
itself present in ndirections subspaces, can move as a total field in any desired direction when each
of the ndirections granules of the particle field is united with an external step-wise moving field.
This external field in turn is present in subspaces with an axis in the direction of motion. The
external stepwise moving field can be interpreted as the de Broglie matter waves.
The motion model provides an indication of the required number of subspaces. For a moving
particle field, itself present in ndirections subspaces with all possible directions, only one of the
ndirections subspaces has the direction of the motion, leaving npivot subspaces per direction to
support the motion.
As a result, in a universe such as (21), each of the npivot subspaces needs to supports
ndirections/npivot granules of the particle field. This indicates that the motion model of Ref [3]
needs to be improved.
The Number of Universes
On the assumption that the number of subspaces of a universe indeed is given by (21), then the
fraction of universes within a multidirectional lattice can easily be calculated. Using the
smallest values of npivot, npoint and ndirections from Table 1, the fraction of universes is with (20)
and (21)
ρuniverses = nsubspaces/nuniverse = 1,10 1033/1,69 1032 = 6,53. (22)
When universes overlap, meaning there are subspaces belonging to multiple universes, then
the number of universes can be the natural number 7 or higher. Overlapping universes offers
the possibility of avoiding a blockade for the further development of the multidirectional
discrete space, namely the inherent problem of the representation of groups of subspaces by
natural numbers. The danger of such a representation is that the various natural numbers will
become inconsistent at some point. Such an inconsistency arises when the total of nuniverse must
be equal to nsubspaces. If there were an exact match, the gravitational constant would have very
precisely resulted in nuniverse times a natural number. With no agreement this would mean the
end of the multidirectional lattice as a possible discrete space. With overlap of universes, the
proposed multidirectional hypercubic lattice offers a good possibility to describe a discrete
space. Of course, the number of different universes can be chosen and with that the degree of
overlap.
Using the next substantially larger values for npivot, npoint and ndirections (2.32 1010, 5.41 1020 and
4.66 1023) from Table1, the fraction of universes becomes ρuniverses= 0,25 10-1, meaning
nsubspaces<nuniverse, ie. there is only one universe. This is impossible given the need for
overlapping universes, making (20) the only possibility. Which leads to the conclusion that the
values of Table 1 are limited to the top three, belonging to nc2 =1. This means that the numerical
values of the discrete physical constants are known to a high degree of accuracy. From this
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follow the numerical values of the discrete electron radius ed, the discrete electron radius Rd e
and the discrete Planck's constant hd and from these the exact continuous values. Exact values
are verifiable, which means that the multidirectional discrete space from which they originate
can also be verified.
Interaction of Matter Limited to One Universe
Regarding phenomena in a multidirectional lattice, the central premise is that fields can interact
only if parts of the fields are in the same subspace. An indication for this is given in Ref [3].
A universe that is supposed to consist of ndirections.npivot subspaces can contain multiple particles
present in ndirections subspaces. These particles only interact if the particle fields are
simultaneously present at the same location within one common subspace.
Because only a fraction of the nsubspaces subspaces of a hypercubic lattice belongs to one universe,
makes that the most particle fields do not interact, only particle fields in one universe can
interact. As a result, phenomena belonging to one universe are completely separate from
phenomena within other universes.
Note: spatial distances can be very large, but universes are extremely close, namely within a
unit cube Δx0
3 of each lattice. This is because all the hypercubic lattices have the same unit cube
Δx0
3 as their smallest size ensures that all subspaces are present within a spatial unit cube Δx0
3
with Δx0 = 1 as measured in discrete units Δx, = 8,33 10-35 m, expressed in the usual metric
units.
CONCLUDING REMARKS
The lattice concept is used here to describe a discrete space. The immediate advantage is that
within a lattice the electron radius problem does not exist. The main drawbacks are the
complete lack of anisotropy in spatial directions and the lack of clarity in which manner
movement occurs in such a rigid frame.
The anisotropy problem can be solved by assuming an extended hypercubic lattice with an
almost unlimited number of mutually equal subspaces. The proposed multidirectional space is
semi-Euclidean, i.e., Euclidean with a small uncertainty in distances along the hypotenuse. The
advantage is also that physical constants can be expressed in terms of number of subspaces of
a group with a given characteristic. With these expressions, the characteristic numbers can be
calculated as a sequence of natural numbers, as shown in Table 1.
New in this article is the calculation of the still missing total number of subspaces from the
gravitational constant. The existence of such a total number implies that all natural numbers
representing number of subspaces must form a complete set. If not, the characteristic numbers
are not coherent.
One way around this problem is to structure the subspaces into a number of parallel universes
containing a portion of the total number of subspaces. Parallel universes may be the
explanation of the so-called dark matter, whose existence has been suspected for almost a
century [5]. If parallel universes are an adequate description of reality, there appears to be
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Groot, C. T. (2023). The Possible Existence of Parallel Universes Within the Total Number of Hypercubic Lattices of a Multidirectional Discrete Space.
European Journal of Applied Sciences, Vol - 11(5). 56-67.
URL: http://dx.doi.org/10.14738/aivp.115.15905
several universes between which no signal can be exchanged. The fraction one universe to the
total number of universes is 1/7, making that matter in one universe is 14 % of the total matter
given a homogeneous distribution of all matter over all universes. Compared to the results of
Ref [6], matter does not seem to be homogeneously distributed over the different universes.
The multidimensional discrete space based on myriad lattices offers explanatory possibilities
unprecedented within a continuous Euclidean space, allowing the reinterpretation of many
hitherto not understood physical phenomena such as the physical constants. Dark matter can
also be understood as arising from universes of lattices filled with ordinary matter, without
having to assume exotic particles.
References
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