J. Modern Physics, 2010, 1, 147-149
doi:10.4236/jmp.2010.12021 Published Online June 2010 (http://www.SciRP.org/journal/jmp)
Empirical Relations about the Number of Dimensions
in Theoretical Physics with the Concept of
Common and Unshared Dimensions
Tomofumi Miyashita
Miyashita Clinic Mitsuya-Kita, Osaka, Japan
Received April 21st, 2010; revised May 16th, accepted May 21st, 2010.
Abstract
How many dimensions are there in the universe? Currently, there is confusion about the number of dimensions in the
universe. Empirical rela tions about the numb er of dimensions in th eoretical physics with the con cept of common sp ace-
time 4 dimensions and unshared dimensions are described in this report.
Keywords: Kaluza-Klein, Superstring, Super gravity, N a m b u St ri n g, Unshared Dimensions
1. Introduction
Einstein discovered space-time 4 dimensions. In order to
complete the theory of everything (TOE), the numbers of
dimensions have been increased. Currently, there is con-
fusion about the number of dimensions in the universe.
There is a simple question, “How many dimensions are
there in the universe?”
The concept of “unshared dimensions” is very suitable
solution. Unshared dimensions belong to each particle
and the numbers of unshared dimensions are different
between the diffe rent ki n ds o f part i cl es.
This paper presents empirical relations about the
number of dimensions in theoretical physics with the
concept of common space-time 4 dimensions and un-
shared dimensions.
2. Empirical Relations about the Number of
Dimensions in Theoretical Physics
2.1 The Concept of Unshared Dimension
There are two kinds of dimensions. One is common space-
time 4 dimensions and the other is unshared dimensions
which belong to each particle. The common space-time 4
dimensions are entirely same dimensions which Einstein
discovered.
Unshared dimensions are something internal dimen-
sions in each particle. The concept of unshared dimen-
sions is discovered empirically. So, the mathematical
explanation of “unshared dimensions” is the next stage
argument. It is important that even the space has one
unshared dimension. So, the definition of the space is
different from the common space-time 4 dimensions.
2.2 Empirical Relations about the Number of
Dimensions in Theoretical Physics
The number of unshared dimensions can be expressed
empirically as:
Unshared dimension = 4 × (4 – N + 1)/N (1)
N is the number related with the symmetry. Calculated
unshared dimensions are shown in Table 1. Empirical
relations about the number of dimensions in theoretical
physics are shown in Table 2.
When N = 3, then unshared dimensions are 8/3. In this
case, explaining for quarks is difficult. 8 and 1/3 is re-
lated with a quark, because the number of gluon is 8 and
a quark must exist with three particles.
Simple evidences supported unshared dimensions can
be shown in the next section.
3. The Evidence of Unshared Dimensions
3.1 Empirical Relations between the Masses of
Leptons
If leptons have 6 unshared dimensions, the mass of lep-
tons are expected like this.
76
7x
k
dxxkM   (2)
Here, M, k and x are the mass of lepton, constant co-
efficient and the value of unshared dimensions.

Empirical Relations about the Number of Dimensions in Theoretical Physics
with the Concept of Common and Unshared Dimensions
148
7
1






m
M
y (3)
Here, m and y are the mass of electron and the seventh
power root of ratio with electron. The calculation results
are shown in Table 3.The square of correlation coeffi-
cient between the generation and the seventh power root
of mass ratio is 0.9996 shown in Figure 1. It is impossi-
ble that this result is only coincidence. The expected
mass of the forth lepton and the fifth lep ton are 14.3 GeV
and 70.5 GeV respectively. The peak of 14.3 eV was
reported [1] and th ere remains the possibility of the forth
lepton.
The expected leptons which have larger generation
number should be very unstable, since they cannot be
discovered still yet. But there are many experimental
reports about the reactions around the expected energy.
The reports published in the internet are much more than
those around the unexpected energy.
3.2 The Mass of Weak Boson
Weak boson is related with electrons and neutrino. Then,
the number of unshared dimension is 26 (= 10 + 20 – 4).
The 27th power root of ratio between Weak boson and
electron are shown in Table 4. The result is near the
generation number 2. If the generation number is 3, the
mass of Weak boson is 4 × 106 TeV and it is impossible
to observe.
Figure 1. Empirical Relation about the mass of lepton
Table 1. Calculated unshared dimensions
N 4 3 2 1 (0)
particle The space quark lepton neutrino consciousness
Unshared dim ension 1 8/3 6 16 infinite
Table 2. Empirical relations about the number of dimensions in theoretical physics
Theory common The space quark lepton neutrino total
Kaluza-Klein 4 1 - - - 5
Superstring
(~1980s) 4 - - 6 - 10
Super gravity 4 1 - 6 - 11
Nambu string 4 - - 6 16 26
Superstring (~2000s ) 4 1 - 6 16 27
Table 3. The seventh power root
Generation mass (MeV) Mass ratio The seventh power root of mass ratio
electron 1 0.51099906 1 1
Muon 2 105.6 206.6540005 2.141653053
Tauon 3 1776.99 3477.481935 3.20549411
4th 4 14371.76031 28124.82729 4.321
5th 5 70548.8140 138060.5555 5.424
6th 6 257713 504331 6.526
7th 7 768658 1504226 7.629
8th 8 1977637 3870138 8.732

Empirical Relations about the Number of Dimensions in Theoretical Physics
with the Concept of Common and Unshared Dimensions
149
Table 4. The 27th power root
Generation mass (MeV) Mass ratio The 27th power root of mass ratio
electron 1 0.51099906 1 1
W-boson 2 80398 157334.9274 2.021601712
Z-boson 2 91187.6 178449.6433 2.036632582
4. Summary
The concept of common space-time 4 dimensions and
unshared dimensions are discovered empirically. Simple
evidences supported unshared dimensions can be shown
from the mass of particles. It is very useful concept to
answer for the question, “How many dimensions are
there in the universe?”
REFERENCES
[1] C. Friberg, E. Norrbin, T. Sjöstrand, “QCD Aspects of
Leptoquark Production at HERA,” Physics Letter B, Vol.
403, No. 3-4, June 1997, pp. 329-334.