Applied Mathematics, 2013, 4, 22-29
Published Online November 2013 (
Open Access AM
Chaotic Fractal Tiling for the Missing Dark Energy and
Veneziano Model
L. Marek-Crnjac1, M. S. El Naschie2
1Technical School Center, Maribor, Slovenia
2Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt
Received September 5, 2013; revised October 5, 2013; accepted October 12, 2013
Copyright © 2013 L. Marek-Crnjac, M. S. El Naschie. This is an open access article distributed under the Creative Commons Attri-
bution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
The formula for the quantum amplitude of the Veneziano dual resonance model is shown to be formally analogous to
the dimensionality of a K-theoretical fractal quotient manifold of the non-commutative geometrical type. Subsequently
this analogy is used to deduce the ordinary energy of the quantum particle and the dark energy of the quantum wave.
The results agree completely with cosmological measurements. Even more surprisingly the sum of both energy expres-
sions turned out to be exactly equal to Einstein’s iconic formula . Consequently Einstein’s formula makes no
distinction between ordinary and dark energy.
Keywords: Hausdorff Dimension; Cantor Set; Dark Energy; Kähler Manifold; Quantum Entanglement; Veneziano
1. Introduction
We utilize a formal analogy between Veneziano’s for-
mula for the amplitude of the dual resonance model and
that for the Hausdorff fractal dimensionality of quotient
manifold to develop an energy-mass quantum relativity
equation which turned to be of the form QR
where is the famous formula of Einstein’s
special relativity and
is a scaling quasi exponent
equal to 52122
where 5
is Hardy’s quantum
entanglement and
. We calculate that
way the deficit energy density balance of the universe
and find that this 95.4915028% deficit compared with
Einstein’s maximal total energy density is the
hypothetical dark energy which is in turn a confirmation
not only of the real existence of dark energy but also of
the exactness of the cosmological measurements of
WMAP and certain supernova analysis by the three 2011
Nobel Laureates. More importantly the analysis clearly
shows the inadequacy of the traditional interpretation of
the energy-mass relationship of special relativity when
applied to the problem of immensely large distance
scales of the order of magnitude of Hubble radius. The
correct way to understand is to realize that
222EO mc is ordinary energy, which can be
measured and corresponds to the energy of the quantum
particle. The rest, i.e.
21 22ED mc is the dark
energy of the quantum wave and cannot be measured in a
direct way because of quantum wave collapse. It is
gratifying to note a new result not realized in our earlier
paper, namely that the sum is E(O) + E(D) = E(Einstein)
and that chaotic dynamics and fractal geometry lead to an
elegant unified picture for nature [1-12].
2. Background Information and Preliminary
There is a serious well documented huge discrepancy
between cosmological measurements and theory with
regard to the total energy content of the cosmos [1-4].
This was brought about when cosmologists discovered
around 1998 that the universe is not only expanding but
that its expansion is accelerating [1,2]. This is in clear
contradiction to our classical understanding of gravity
which says that attraction between mass in the universe
should cause the expansion to slow down [1-4]. The de-
viation of theory from reality is not two or even ten per-
cent but rather a staggering 95.5% of the energy of the
universe is presumed to be missing [1]. This totally un-
expected result of the most profound problem in physics
and cosmology lead to postulating a hypothetical repul-
sive force called dark energy to explain the accelerated
expansion of the universe [1-6]. We trace this discrep-
ancy back to wrongly interpreting special relativity
[3-6]. Thus in this paper we will be concerned
with revising the interpretation of where E is
the energy, m is the mass and c is the velocity of light by
including quantum particle physics [7-11] as well as
quantum entanglement [12], topological quantum field
theory [13], effective quantum gravity [14] and super-
strings [15,16] in our deterministic chaos-fractal analysis
There are many ways to show that the correct quantum
relativity [27,28] must be a simple scaling [4,5,23]
of E so that QR
is the scaling expo-
nent [4-6]. It turns out that Veneziano’s model of dual
resonance is one of numerous other ways to handle the
problem but has the advantage of being relatively con-
servative and familiar way [7,8]. At the end we find that
where the 22 are Veneziano’s 26 dimensions
needed for getting rid of negative norms [7-11] from his
theory after substituting the four dimensionality of
space-time so that 26 4 = 22 and 222Emc [27].
Explaining how to do this and distinguishing between
ordinary energy and dark energy is our main concern in
what follows [22-28].
3. The Veneziano Amplitude and the
Dimensionality of Quotient Manifolds
Based on previous work by Regge and others, Veneziano
[7-10] was able to write down an amplitude for the dis-
persion scattering of four particles according to his dual
resonance model. In conventional notation this is [7,9]
  
,1d . (1)
Here a and b are parameters depending upon the mo-
mentum of colliding particles, the function xa is real line
multiplicative character and the gamma function in-
volved is labeled [9]. Suppose we can set
,ab ab then there
will be a formal analogy between A(a,b) and the inverse
of the dimension of a quantum manifold of the X space
type used in non-commutative geometry [10]
where a and b here are the dimensions of two sub-mani-
folds forming an X quotient manifold such as Penrose
fractal tiling [11]. Since D is either a Hausdorff or topo-
logical dimension, the 1/D is a normed probability and
can be compared directly with A(a,b) of Veneziano [7,9].
That means
 
,1 ,
abD ab.
The above is naturally part and parcel of the Bosonic
Nambu-Goto [16] string which will not be explained here
but we will touch upon this point again later on.
4. Calculating the Topological and
Hausdorff Dimensions as Well as
Let us use D to derive first the topological dimension and
second the Hausdorff dimension of a Hilbert 4D hyper-
cube [17,18]. In the first case we just need to set a = b =
1/2 and find
1212 114 4.
12 12
In the second case we have ,a
where 2
and one finds [10,17,18]
Hilbert .
In terms of a continued fraction expansion of 3
44 4.23606799.
 
Needless to mention that the same result is found from
the inverse value of Veneziano’s amplitude formula [7-
9]. It was not long before trouble came into the dual reso-
nance model in the form of ghosts or negative norms, i.e.
negative quasi probabilities [8,9,15] which made this
tangible simple model unreal. To get rid of negative
norms, the price was to assume that the model is embed-
ded in 26 dimensions [18]. Thus we either accept ghosts
or reconcile the anomalies via 26 dimensions. It seems
that Veneziano reluctantly accepted the 26 dimensional
space which was the beginning of modern string theory
[15] while convinced that it will turn out to be just a
mathematical neat trick and no one will ever worry about
seeing the invisible 26 4 = 22 dimensions. Now with
hindsight and in retrospect, if dark energy really exists
and we believe that it exists because of its qualitative and
quantitative consistency with the increased rate of accel-
eration of the expansion of the cosmos, then it is hidden
in the 22 extra dimensions of Veneziano-Nambu-Goto
Bosonic string theory [7-9,16] as we will reason shortly.
To show that we consider first a hyper Hilbert cube
[17,18] not in 4 but in 11 dimensions.
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5. An Eleven Dimensional M-Theory Hyper
Hilbert-Fractal Cube
In analogy with 44D we now construct D
11 11. This basically gives us a Hausdorff dimension
11 1111
11 1
11 11
11.09016994393 112
 
where and
1 0.18033989k
5 120.618033989
 . This is in effect an 11
dimensional cube inside another 11 dimensional cube
and so on like an infinite Russian doll [17,18]. Now the
probability corresponding to D(11) which we call a frac-
tal M-theory space-time for obvious reasons, is simply
the inverse value which due to nice number theoretical
properties of
112 0.090169945.
Actually this 52k
corresponds exactly to
Hardy’s generic quantum entanglement probability [12].
Now we seek to determine the probability A(a,b) from
either A(a,b) or D(a,b). Taking first A(a,b) and setting
 as a sub-manifold probability, the joint quo-
tient space probability is given by
 
55 5
,2Aa b
 
 
From D(a,b) we have on the other hand the inverse
 
2 1122.18033989.
Dab Aab
 (9)
Consequently we see a clear indication of the in-
volvement of and role played by the extra 22 “dark” di-
mensions which initially troubled Veneziano and his col-
leagues at the time. There is thus a strong and direct link
between the 22 “dark” dimensions, Hardy’s quantum
entanglement which was obtained using orthodox quan-
tum mechanics and which was firmly confirmed experi-
mentally and dark matter which is strictly speaking still
hypothetical but also firmly established in real measure-
ment of accelerated expansion consistent with repulsive
antigravity nature of this dark energy [1,2,11]. However,
the situation changes completely when we realize that
dark energy is nothing else but the energy of the quantum
wave as we will reason later on.
6. Revising Einstein’s E = mc2 to the Sum of
E1 = mc2/22 and E2 = mc2 (21/22)
From the above it is clear that special relativity is not
quantized in any suitable form, says the Bosonic Gupta-
Bleuler quantization and thus did not take the needed
extra 22 dimensions into account nor did it have any pro-
vision for quantum entanglement. To improve special
relativity its equations must intersect the equations of
quantum mechanics in one way or another and a Hardy
or Immirzi-type of probabilistic quantum entanglement
must be planted in it [11]. Assuming the validity of Weyl
scale relativity [4] and having faith in the perfect sim-
plicity of capricious but not malicious mother nature and
father time, we could reduce the task to intersecting
with Hardy’s probability
[12] in the form
of 5
2 of the Veneziano model [7,16]. This ex-
pectation is readily fulfilled in a remarkable almost sur-
real way. Setting 52A
as a scaling
, we see that
so that
22222.18033989 22
QR mcmc mc
Emc k
 
Remember that special relativity is based on only 1
degree of freedom elementary particle which is the pho-
ton [11] while the standard model has 12 messenger par-
ticles [10,11,14], i.e. the dimension of
3SU 8
2SU 3
where SU(3), SU(2) and
U(1) denote the Lie symmetry groups of the standard
model [4]. Subtracting the photon then special relativity
has 11 particles less. These are equal to the degrees of
freedom of the standard model [11] minus one (12 1 =
11). Not only that but also the eleven super symmetric
partners are neither known nor considered, not even hy-
pothetically. Thus in all 2(11) = 22 needed degrees of
freedom needed to minimize energy were not considered.
This is how the reduction scaling 1/22 could be explained.
It is one explanation of many others. The end result must
however be real because these are not idle theoretical
discussions of academic value but rather serious meas-
urement data [1] suggesting serious shortcomings of
something which was shown here to be interpreting spe-
cial relativity wrongly. Of course we have not yet found
any super symmetric particles but this would not invalid-
date our argument because we have many others, all
leading to the same conclusions. For instance the second
Betti number b2 [13] is equal 1 for flat connected spaces
just like that of special relativity. However, b2 is exactly
22 for a K3 Kähler manifold [19,20] with 22 more 3 di-
mensional holes in it than in a flat connected manifold or
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a de Sitter space-time [11]. This again leads to
 
3K hler122bSRbK
ä and a reduction in
the predicted energy by 95.49% [1]. The truly surprising
part of our analysis is that this shortfall in the energy
density household exists in the form of dark energy
which is the energy of the quantum wave [22-26]. Fur-
ther careful analysis conducted over the last three years
using K-theory in conjunction with non-commutative
geometry, E-infinity theory and Wooden’s ultimate
L-theory lead to the realization that the quantum wave is
an empty set in a quintic Kaluza-Klein space and is the
source of non-measurable dark energy density equal
EEDmc mc
 2
By contrast the meas-
ured ordinary energy
EEO mc
 
found to be the energy of the zero set also in quintic 5D
7. Kolmogorov and the 26 Strings
In Ref [21] Kolmogorov, who of course never worked in
string theory or high energy physics is quoted on a
“question of topology” where he wrote: “It seems to me
that it is not difficult to prove that an n dimensional
closed set may be embedded in a space of sufficiently
large dimension and in only one way. I know how to
prove this for polyhedra which is embedded in Euclidean
space of dimension [21]”
42Dn. (13)
Here is a faint connection to Nash’s powerful embed-
ding theorem, however it is the same formula given in
Vol. II, page 371 of Green, Schwarz and Witten’s classi-
cal book on string theory [15]. Thus n could be consid-
ered not just a number but a dimension of the object to be
embedded. For n = 0 one finds the world sheet D = 2. For
n = 1 we find (4)(1) + 2 = 6, i.e. D = 6 of the mass sector.
For n = 6 one finds (4)(6) + 2 = 26 of Bosonic strings.
Interestingly for a five dimensional object, say a quantum
object with three space dimensions, one Klein-Kaluza
dimension and one spin 1/2 degree of freedom, we find
(4)(5) + 2 = 22 dimensions. Adding the 3 + 1 space and
time dimension to that we retrieve the 26 dimensions
again. We discussed this point only because it highlights
the important role played by Euclidean embedding in
high energy space-time topology and its interesting rela-
tion to the vital 22 “dark” dimensions behind the so
called missing dark energy of the cosmos [1-4].
8. Discussion
In this section we would like to contemplate the reason
why our present analysis at the simple answers which
agrees with measurements for a problem which seemed
to require far more complex reasoning and analysis [27-
63]. In addition we discuss some recent advances in our
understanding of basic problems connected to funda-
mental questions in quantum mechanics and cosmology
Let us start with a general observation documenting
two basic shortcomings in the mathematical formulation
of physical processes and the geometrical shape of the
space-time in which these processes are taking place or
are part of. Considering the vast mathematical literature
on the continuum hypothesis and set theory, it is quite
surprising that physicists do not make a proper and sharp
distinction between three notions namely zero, being
empty or not being there at all [31,56,57]. Philosophers
make a distinction between being and nothingness and
devoted entire monumental books to this subject, for in-
stance “Being and Nothingness” by J. P. Sartre or even
“Being and Time” by M. Heidegger [64,65] which in-
fluenced Sartre. On the other hand, at least since the
founding father of set theory, G. Cantor, pure mathema-
ticians make a definite distinction between a zero set
which has only zeros in it as elements and the empty set
which has no elements what so ever in it despite being a
set and pure insubstantial nothingness which is not there
at all as a set [31,32]. In physics on the other hand, ex-
cept when it is handled by mathematicians of the calibre
of Grothendieck, Attaya and Connes [10,57,58], zero
means empty as well as nothing and here lies the origin
of the difficulties which crept into physics and mani-
fested itself in our understanding or rather our misunder-
standing of the wave function of quantum mechanics and
dark energy in cosmology to mention only two of the
major problems in theoretical physics [53,54]. The theory
presented here is more connected to reality mainly be-
cause we started from Cantor sets and fractals and went
further all the way to the ultimate logic of H. Wooden
[32] and defined the wave as the empty set which covers
the particle modelled by the zero set. That way the nega-
tive energy of the wave reveals itself as being nothing
else but the dark energy we are searching for while the
measurement problem disappears altogether as a problem
and becomes a natural consequent of interfering with an
empty set rendering it non-empty [55-58]. Conventional
particle-wave duality is seen as a kind of Hamlet to be or
not to be situation. It is a particle or it is a wave depend-
ing upon the set up. Our theory presented here proves it
to be slightly but importantly different because the wave
is the surface (i.e. cobordism) of the particle in a five
dimensional space-time. Since a particle is a zero set, its
topological dimension is zero and therefore by Menger-
Urysohn deductive dimension theory, the surface, i.e. the
wave topological dimension is minus one. That means it
is the classical empty set. In other words, it is not to be or
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not to be but rather to be and not to be at the same time.
This fundamental mathematical conclusion was rein-
forced vigorously by yet more convincing experimental
results shown in a beam splitter set up that we have a
quantum object which shows simultaneously in a fuzzy
way both attributes of a particle and a wave [33-36]. This
squares with our theory and stresses the physical reality
of the wave function because an empty set is not only the
beginning of mathematics but also the starting point of
physics [31].
In our theory we are using a space-time geometry de-
scribed not only by a topological dimension but also by a
Hausdorff dimension. The basic core dimension of E-
infinity is [22-31]. This has two
consequences. First this Hausdorff dimension is a meas-
urement for disorder, i.e. entropy. This brings our work
near to that of T. Jacobson [37,38] and we may remind
our readers of what was termed by some authors as the
Feynman-El Naschie conjecture that gravity is the fluc-
tuation of fractal time in space-time creating a force
similar to van der Waal’s force which we recognize as
gravity [42]. Second,
4 4.236067977
can be used for exact cal-
culation although it has an infinite decimal expansion
[22-31]. This infinity is not rejected or artificially banned
from our theory but rather it is included as an important
integral part of it [41,42]. In this context we must reiter-
ate that fractal logic must be used as fractal counting in
evaluating Higgs’ experiments [39,40]. The graviton and
the Higgs could not be counted as integer numbers for
the reasons mentioned on many previous occasions [22-
31]. We reasoned elsewhere that the 12 messenger parti-
cles of the standard model are in fact 14 particles when
counted classically using integers and crisp symmetry
groups. However using fractal weight-logic and fuzzy
symmetry groups we have
0137.082039325 11.708033989
 particles in-
cluding the graviton and the Higgs [22-31]. Seen from a
classical view point this seems incomprehensible but
fractally this makes a great deal of sense and explains
why the Higgs mass and spin were not found in a water-
tight conclusive way in the recent CERN experiments. It
is part of the nature of the Higgs and the Higgs field may
also be an approximate way of looking at our empty set-
zero set quantum space-time.
Finally we address the global nature of the Cantorian
geometry used to arrive at our present conclusions and
21 22ED mc is not only 95.5% of the total
energy of the universe but also that it is a negative energy
creating a negative repulsive gravity leading to the ob-
served acceleration rather an deceleration of the expan-
sion of the cosmos. Globally our fractal-Cantorian space-
time is a material space like the space of the theory of
elasticity, only more sophisticated [60,61]. The nearest
known theory to that would be Cartan theory which in-
cludes torsion tensors or even better, the theory of polar
media of the French electrical engineering brothers
Cosserat [62].
In such material space, anticlastic curvature is a natu-
ral outcome of any induced curvature [61]. Thus as in the
simple elementary demonstration with a long cylindri-
cally rolled paper sheet of Figure 14, Ref. [24], we have
anticlastic curvature at the extremity of the cylinder. In
the holographic projection of the Penrose-Klein modular
curve hyperbolic universe, the extremity is a ramified
circular edge encircling the curve. That is where negative
energy produces antigravity and antigravity in the form
of anticlastic curvature produces negative energy in a
circulatory logic blurring the distinction between cause
and effect. The situation may be made analogous to the
horizon of a black hole where Hawking’s vacuum fluc-
tuation produces the negative energy of the vacuum [42-
45]. As things stand now we may naively define classi-
cality or Newtonian mechanics as an averaging compro-
mise between the wave aspect and the particle aspect so
that the energy is half of that given by Einstein’s equa-
 
. On the
other hand Einstein’s energy could be thought of equally
naively as a perfectly Newtonian kinetic energy with a
single character fault of having a constant speed v = c =
constant from the beginning to the end of the time inter-
val of the motion of a particle with mass m so that v = c =
constant can be taken out of the integration and one finds
that . It is the subtlety of this situa-
tion which may be the cause of many misunderstandings
of Einstein’s relativity as well as Newtonian mechanics
in equal measures.
9. Conclusions
In this paper we use chaos theory and fractal geometry to
utilize a formal correspondence between the Veneziano
dual resonance model and quotient non-commutative
spaces [10]. Veneziano’s model makes sense only in 26
dimensions [15]. Building a bridge between this model
and special relativity we see that includes
only 4 dimensions. This is 22 less than what the strong
interactions need in the Veneziano model in order to get
rid of the negative norms. This is also approximately the
scaling needed to elevate from a formula of
special relativity to
989Em1 22.18033c, which is
a formula taking on board things like generic Hardy’s
quantum entanglement [12], entire spectrum of the ele-
mentary messenger particles of minimally super sym-
metrically extended standard model as well as 22 extra
dimensions which we call “dark dimensions” [22-29].
This is because if dark energy really exists and we do
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believe that it exists being the absolute value of the nega-
tive energy of the quantum wave, then dark dimensions
and fractal voids are the place where it can screen its
existence. Mathematically these screened areas and voids
in space-time are the empty sets or the fat fractal con-
traparts of a KAM space-time [23,26]. We cannot answer
all these questions fully but at least the prediction of our
quantum relativity formula [14] 222Emc is in al-
most perfect agreement with cosmological measurements
[1]. The present result is in any event real and not specu-
lative in any sense because it starts from a conventional
picture and gives results in excellent agreement with ac-
tual highly accurate measurements [1,27,30].
Taken as a whole, the present results as well as the re-
sults and conclusions of previous publications [22-30]
suggest the following picture based on the stringent ma-
thematical logic of transfinite set theory [31,32]. Space-
time is the origin of both the quantum wave and the
quantum particle. The first is essentially a five dimen-
sional empty set [31]. The second is also five dimen-
sional but a zero set [27-31]. The empty set is the co-
bordism of the zero set and may be likened to a Dirac’s
hole with negative energy. Since interfering with an
empty set, for instance at measurement, renders it non-
empty, wave collapse is a natural result of any quantum
measurement. This is the only mathematically well de-
fined meaning of the quantum wave collapse on meas-
urement [26,31]. That is why we cannot detect the energy
of the quantum wave and quantum mechanics inaccu-
rately concluded that the quantum wave is devoid of any
energy [11,22-31]. In reality a quantum wave is devoid
of any ordinary positive energy but it has a considerable
amount of negative energy. This fact was discovered
experimentally not in any high energy laboratory but in a
completely different field using the entire cosmos as an
experimental set up [1-4]. There are other relatively re-
cent experimental and theoretical works which support
our conclusion in broad general terms [33-40]. For in-
stance, recent experiments with beam splitters shows
neither quantum particle behaviour nor quantum wave
behaviour but a little of both together [33-36]. This
agrees completely with our mathematical definition
which says that the wave is just the out surface (cobord-
ism) of the particle which means there is no way to really
separate the one from the other and that the wave is just
as real as the particle [31]. Furthermore, our picture of
the anticlastic space-time curvature produced by the dark
energy of the wave at the edge of the universe causing
negative gravity agrees in principle with some features of
the work of T. Jacobson [37,38]. In addition L. Krauss
[39,40] and others connecting the Higgs to dark energy is
not that far short from our concept of a KAM space-time
which resembles a Higgs field [22-30]. Finally our theory
can easily embrace ideas about the importance of Basian
probability [41]. Never the less, the theory of probability
is not the best way to start when you need to define a
particle and a wave. It is set theory and fractal Cantor set
theory [54] with its zero and empty set which must be
our first step as we did in the present work and earlier
publications [38,39].
[1] L. Amendola and S. Tsujikawa, “Dark Energy, Theory
and Observations,” Cambridge University Press, Cam-
bridge, 2010.
[2] B. Carr, “Universe or Multiverse?” Cambridge University
Press, Cambridge, 2010.
[3] Y. Baryshev and P. Teerikorpi, “Discovery of Cosmic
Fractals,” World Scientific, Singapore, 2011.
[4] L. Nottale, “Scale Relativity,” Imperial College Press,
London, 2011.
[5] G. Barenblatt, “Scaling,” Cambridge University Press,
Cambridge, 2003.
[6] J. Mageuijo and L. Smolin, “Lorentz Invariance with an
Invariant Energy Scale,” arXiv: hep-th/0112090V2, 18
December 2001.
[7] G. Veneziano, “Ward Identities in Dual String Theories,”
Physics Letters B, Vol. 167, No. 4, 1986, pp. 388-392.
[8] Y. Nambu, “Quark Model and the Factorization of
Veneziano Amplitude, In: R. Choud, Ed., Symmetries and
Quark Models, Gordon and Breach, New York, 1970, pp.
[9] V. Vladimirov. I. Valovich and E. Zelenov, “P-Adic Ana-
lysis and Mathematical Physics,” World Scientific, Sin-
gapore, 1994.
[10] A. Connes, “Non-Commutative Geometry,” Academic
Press, San Diego, 1994.
[11] R. Penrose, “The Road to Reality,” Jonathan Cape, Lon-
don, 2004.
[12] L. Hardy, “Non-Locality of Two Particles without Ine-
qualities for Almost All Entangled States,” Physical Re-
view Letters, Vol. 71, No. 11, 1993, pp. 1665-1668.
[13] C. Nash and S. Sen, “Topology and Geometry for Physi-
cists,” Academic Press, San Diego, 1983.
[14] I. Buchbinder, S. Odintsov and I. Shapiro, “Effective
Action in Quantum Gravity,” Institute of Physics Pub-
lishing, Bristol, 1992.
[15] M. Green, J. Schwarz and E. Witten, “Superstring The-
ory,” Cambridge University Press, Cambridge, 1987.
[16] M. Kaku, “Introduction to Superstrings and M-Theory,”
Springer, New York, 1999.
[17] J.-H. He, “Hilbert Cube Model for Fractal Space-Time,”
Chaos, Solitons & Fractals, Vol. 42, No. 5, 2009, pp.
[18] J.-H. He, “Twenty Six Dimensional Polytope and High
Open Access AM
Energy Spacetime Physics,” Chaos, Solitons & Fractals,
Vol. 33, No. 1, 2007, pp. 5-13.
[19] D. Joyce, “Compact Manifold with Special Holonom,”
Oxford Press, Oxford, 2003.
[20] T. Hübsch, “Calabi-Yau Manifolds,” World Scientific,
Singapore, 1994.
[21] E. Charpentier, A. Lesne and N. Nikolski, “Kolmo-
gorov’s Heritage in Mathematics,” Springer, Berlin, 2007.
[22] M. S. El Naschie, “Quantum Entanglement: Where Dark
Energy and Negative Gravity plus Accelerated Expansion
of the Universe Comes From,” Journal of Quantum In-
formation Science, Vol. 3, No. 2, 2013, pp. 57-77.
[23] M. S. El Naschie, “The Missing Dark Energy of the
Cosmos From Light Cone Topological Velocity and
Scaling the Planck Scale,” Open Journal of Microphysics,
Vol. 3, No. 3, 2013, pp. 64-70.
[24] M. S. El Naschie and A. Helal, “Dark Energy Explained
via the Hawking-Hartle Quantum Wave and the Topology
of Cosmic Crystallography,” International Journal of As-
tronomy and Astrophysics, Vol. 3, No. 3, 2013, pp. 318-
[25] M. S. El Naschie, “The Quantum Gravity Immirzi Pa-
rameter—A General Physical and Topological Interpreta-
tion,” Gravitation and Cosmology, Vol. 19, No. 3, 2013,
pp. 151-155.
[26] M. S. El Naschie, “What Is the Missing Dark Energy in a
Nutshell and the Hawking-Hartle Quantum Wave Col-
lapse,” International Journal of Astronomy and Astro-
physics, Vol. 3, No. 3, 2013, pp. 205-211.
[27] L. Marek-Crnjac, “Modification of Einstein’s E = mc2 to
E = mc2/22,” American Journal of Modern Physics, Vol.
2, No. 5, 2013, pp. 255-263.
[28] M. S. El Naschie, “A Resolution of the Cosmic Dark
Energy via a Quantum Entanglement Relativity Theory,”
Journal of Quantum Information Science, Vol. 3, No. 1,
2013, pp. 23-26.
[29] M. S. El Naschie, “Dark Energy from Kaluza-Klein
Spacetime and Noether’s Theorem via Lagrangian Multi-
plier Method,” Journal of Modern Physics, Vol. 4, No. 6,
2013, pp. 757-760.
[30] M. S. El Naschie, “Determining the Missing Dark Energy
of the Cosmos from a Light Cone Exact Relativistic
Analysis,” Journal of Modern Physics, Vol. 2, No. 2,
2013, pp. 18-23.
[31] M. S. El Naschie, “Towards a General Transfinite Set
Theory for Quantum Mechanics,” Fractal Space-Time
and Non-Commutative Geometry in High Energy Physics,
Vol. 2, No. 2, 2012, pp. 135-142.
[32] R. Elwes, “Ultimate Logic,” New Scientist, Vol. 211, No.
2823, 2011, pp. 30-33.
[33] V. Jacques, et al., “Delayed-Choice Test of Quantum
Complementarity with Interfering Single Photons,” Phy-
sical Review Letters, Vol. 100, No. 22, 2008, Article ID:
[34] L. Li, N. L. Liu and Z. X. Yu, “Duality Relations in a
Two Path Interferometer with an Asymmetric Beam
Splitter,” Physical Review A, Vol. 85, No. 5, 2012, Arti-
cle ID: 054101.
[35] M. F. Schriber, “Another Step Back for Wave-Particle
Duality,” Physics, Vol. 4, No. 102, 2011, Article ID:
[36] J.-S. Tang et al., “Revisiting Bohr’s Principle of Com-
plementarity Using a Quantum Device,” arXiv: 1204.5304-
V1[quant-ph], 24 April 2012.
[37] T. Jacobson, et al., “Increase of Black Hole Entropy in
Higher Curvature Gravity,” arXiv: gr-qc/9503020V1, 11
March 1995.
[38] V. Vedral, “In from the Cold,” New Scientist, Vol. 216,
No. 2886, 2012, pp. 33-37.
[39] L. M. Krauss, “A Higgs-Saw Mechanism as a Source of
Dark Energy,” arXiv:1306.3239V1[hep-ph], 13 June
[40] L. Grossman, “Dark Energy May Spring from the Higgs
Boson,” New Scientist, Vol. 219, No. 2931, 2013, p. 11.
[41] D. Mermin, “Quantum Mechanics: Fixing the Shifty Sp-
lit,” Physics Today, Vol. 65, No. 7, 2012, pp. 8-10.
[42] M. S. El Naschie, “A Note on Quantum Gravity and Can-
torian Spacetime,” Chaos, Solitons & Fractal, Vol. 8, No.
1, 1997, pp. 131-133.
[43] M. S. El Naschie, “Complex Vacuum Fluctuation as a
Chaotic ‘Limit’ Set of Any Kleinian Group Transforma-
tion and the Mass Spectrum of High Energy Particle Phy-
sics via Spontaneous Self-Organization,” Chaos, Solitons
& Fractals, Vol. 17, No. 4, 2003, pp. 631-638.
[44] M. S. El Naschie, “VAK, Vacuum Fluctuation and the
Mass Spectrum of High Energy Particle Physics,” Chaos,
Solitons & Fractals, Vol. 17, No. 4, 2003, pp. 797-807.
[45] M. S. El Naschie, “The VAK of Vacuum Fluctuation,
Spontaneous Self-Organization and Complexity Theory
Interpretation of High Energy Particle Physics and the
Mass Spectrum,” Chaos, Solitons & Fractals, Vol. 18, No.
2, 2003, pp. 401-420.
[46] J.-H. He, “A Note on Elementary Cobordism and Nega-
tive Space,” International Journal of Nonlinear Sciences
and Numerical Simulation, Vol. 11, No. 12, 2010, pp.
[47] M. S. El Naschie, “Average Symmetry, Stability and Er-
godicity of Multidimensional Cantor Sets,” Il Nuovo Ci-
mento, Vol. 109, No. 2, 1994, pp. 149-157.
Open Access AM
Open Access AM
[48] M. S. El Naschie, “Mathematical Foundations of E-In-
finity via Coxeter and Reflection Groups,” Chaos, Soli-
tons & Fractals, Vol. 37, No. 5, 2008, pp. 1267-1268.
[49] M. S. El Naschie, “Removing Spurious Non-Linearity in
the Structure of Micro-Space-Time and Quantum Field
Renormalization,” Chaos, Solitons & Fractals, Vol. 37,
No. 1, 2008, pp. 60-64.
[50] M. S. El Naschie, “On ’t Hooft Dimensional Regulari-
zation in E-Infinity Space,” Chaos, Solitons & Fractals,
Vol. 12, No. 5, 2001, pp. 851-858.
[51] O. E. Rössler, et al., “Hubble Expansion in Static Space-
Time,” Chaos, Solitons & Fractals, Vol. 33, No. 3, 2007,
pp. 770-775.
[52] M. Pusey, J. Barrett and T. Randolph, “On the Reality of
Quantum State,” Nature Physics, Vol. 8, June 2012, pp.
[53] M. S. El Naschie, “Mohamed El Naschie Answers a Few
Questions about This Month’s Emerging Research Front
in the Field of Physics,” 2004.
[54] M. S. El Naschie, “This Month’s New Hot Paper in the
Field of Engineering: On a Fuzzy Kähler-Like Manifold
Which Is Consistent with the Two Slit Experiment,” In-
ternational Journal of Nonlinear Sciences and Numeri-
cal Simulation, Vol. 6, No. 2, 2005, pp. 95-98.
[55] M. Persinger and C. Lavellee, “Theoretical and Experi-
mental Evidence of Macroscopic Entanglement between
Human Brain Activity and Photon Emission,” Journal of
Consciousness Exploration & Research, Vol. 1, No. 7,
2010, pp. 785-807.
[56] M. S. El Naschie, “COBE Satellite Measurement, Can-
torian Space and Cosmic Strings,” Chaos, Solitons &
Fractals, Vol. 8, No. 5, 1977, pp. 847-850.
[57] L. Marek-Crnjac, “The Physics of Empty Sets and the
Quantum,” Nonlinear Science Letters B, Vol. 1, No. 1,
2011, pp. 13-14.
[58] J.-H. He, “The Importance of the Empty Set Underpin-
ning the Foundation of Quantum Physics,” Nonlinear Sci-
ence Letters B, Vol. 1, No. 1, 2011, pp. 6-7.
[59] M. S. El Naschie, “Penrose Universe and Cantorian Space-
time as a Model for Noncommutative Quantum Geome-
try,” Chaos, Solitons & Fractals, Vol. 9, No. 6, 1998, pp.
[60] M. S. El Naschie, “Stress, Stability and Chaos in Struc-
tural Engineering,” McGraw Hill, London, 1990.
[61] D. Horrockos and W. Johnson, “On Anticlastic Curvature
with Special Reference to Plastic Bending,” The Interna-
tional Journal of Mechanical Sciences, Vol. 9, No. 12,
1967, pp. 835-861.
[62] E. Cosserat and F. Cosserat, “Theorie des Corps Defor-
mables,” Lavoisier S.A.S., Paris, 1909.
[63] M. S. El Naschie, “A Fractal Menger Sponge Spacetime
Proposal to Reconcile Measurements and Theoretical Pre-
dictions of Cosmic Dark Energy,” International Journal
of Modern Nonlinear Theory and Application, Vol. 2, No.
2, 2013, pp. 107-121.
[64] M. S. El Naschie, “On the Philosophy of Being and Noth-
ingness in Fundamental Physics,” Nonlinear Science Let-
ters A, Vol. 2, No. 1, 2011, pp. 5-6.
[65] M. S. El Naschie, “On the Mathematical Philosophy of
Being and Nothingness in Quantum Physics,” Fractal
Space-Time & Non-Commutative Geometry in Quantum
and High Energy Physics, Vol. 2, No. 2, 2012, pp. 103-