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African Journal of Physics Vol.3, 2010.

HYPERSPACE AND THE TORUS REVISITED: AN AFRICAN

PERSPECTIVE ON THE THEORY OF EVERYTHING†

A. O.E. Animalu1 and Professor Catherine O. Acholonu2 1Dept. of Physics & Astronomy, University of Nigeria, Nsukka &

Chairman/CEO, International Centre for Basic Research, Abuja

2 Catherine Acholonu Research Center, Abuja

e-mail: 1

[email protected] [email protected]

Abstract

We review the “ichi” linguistic geometry of Catherine Acholonu’s three books

titled The Adam Trilogy and published under three book titles – Gram Code of

African Adam (GCAA, 2005), They Lived Before Adam (TLBA, 2009) and Lost

Testament of Ancestors of Adam (LTAA, 2010) – based on its algebraic

representations as “Afa” Matrix : 8x8 = 64 = 28+36 = 26+38 = 32+32 = 24+40

and the following correspondence principle with elements of quantum field theory

Points<=>Particles, Lines<=>Fields, Planes<=>Currents.

As a consequence, we relate the number of the primitive elements (points,lines &

planes) of 3-dimensional projective geometry of linguistic frames of reference

(tetrahedron, square & hexagonal pyramids, Cartesian cube and cube-hexagon

hyperspace) to the Gell-Mann-Ne’emann 8x8 representation of SU(3) symmetry

and current algebra of strongly interacting particles in high energy physics. By

comparing the “ichi” linguistic geometry with Kaku’s ten-dimensional

hyperspace view of the (superstring) theory of everything (TOE) (unifying the

four basic (electromagnetic, weak, strong and gravitational) forces in nature and

Stephen Hawking’s Torus view of the universe (based on the use of quantum

fluctuations to eliminate the black-hole singularity of Einstein’s general relativity

theory of gravitation), we develop an African perspective on TOE based on cube-

hexagon hyperspace/torus and the associated current algebra and discuss the

consequences.

Key Words: Linguistics, Hyperspace, Torus, Current algebra, Theory of Everything,

† African Journal of Physics Vol.3, pp. 31-50, (2010)

ISSN: PRINT: 1948-0229 CD ROM:1948-0245 ONLINE: 1948-0237

mailto:[email protected]

mailto:[email protected]

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1. INTRODUCTION At the 2009 International Seminar on Theoretical Physics and National

Development (ISOTPAND-2009), we began under the title[1], “Ichi” linguistic

geometry and evolution, to lay the mathematical/physical foundations of

Catherine Acholonu’s three books[2] titled The Adam Trilogy, Vols 1, 2 & 3,

published under the separate titles: The Gram Code of African Adam (hereunder

abbreviated GCAA, 2005); They Lived Before Adam (TLBA, 2009); and The Lost

Testament of the Ancestors of Adam (LTAA, 2010). The mathematical foundation

of interest is the branch of topology known as singular homology[3] which is

concerned, in geometric and algebraic terms, with the triangulation and algebraic

projective geometry of linguistic frames of reference (tetrahedron, square-based

and hexagonal-based pyramids, cube, pentagon, hexagon and cube-hexagon

hyperspace) from which can be proved, as stated by Machio Kaku[4] on p.105 of

his 1995-published book titled Hyperspace, A Scientific Odyssey Through

Parallel Universes, Time Warps, and the 10th

Dimension, that “the fifth

dimension is topologically identical with a circle and the universe is topologically

identical with a cylinder”.

In order to anchor this paper on available physical data (artefacts) from

traditional African culture, let us recall[5] the prevalence of conical-shaped huts

on polygonal base throughout Africa which was adopted in the architecture of the

National Institute for Policy and Strategic Studies (NISS) at Jos, replicating the

famous groundnut pyramids on polygonal base (in Northern Nigeria) and, the

more intriguing architecture of the step pyramid at Nsude in Enugu State of

Nigeria photographed by missionaries in the 1930s and reproduced in Fig. 1

below. By enclosing the step pyramid in a cylinder whose circular section is

defined by the six tips of a hexagon and the height by the five tips of a square

pyramid (to represent growth along the cylindrical axis), one can generate the

inter-joined 5&6-quadrangle pattern indicated in Fig. 1 which is a recurrent motif

of Igbo artistic expression (LTAA, p.81) and replicate the distinctive hexagon

grid on the “enigmatic” Igbo-Ukwu bronze tyre/hyperdoughnut (TLBA, p.447),

i.e. a torus, immersed in a cube-hexagon hyperspace!. This semiology, that is,

physics and logic, of African culture goes beyond the post-colonial theme of

struggle between invading European culture (represented by a rectangular box

totem of Christian missionaries) and African culture (represented by the “creation

snake”) that got trapped in the box in Chinua Achebe’s novel, Arrow of God.[6].

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2. REVIEW OF THE ADAM TRILOGY Our primary interest in ref.[1] was to characterize the linear writing code

(preserved in Ikom Monoliths) of the African[Igbo] Mother tongue of world

languages discussed in GCAA and the various artefacts excavated by the British

archaeologist, Thurston Shaw[7], in 1950s at Igbo-Ukwu in South Eastern

Nigeria, especially the bronze hyper-doughnut(torus) (TLBA, p.447) shown in

Fig. 1 in terms of the facial scarification, (called “ichi”) discussed in TLBA &

LTAA, and the stone age axe excavated by Anozie[7]. Projective geometry comes

into play through Da Vinci codes[8] defined (like Miller indices in solid state

physics) by the ratios and total number (S) of the three primitive geometric

elements, points(vertices) :lines(edges): planes(faces), of the various geometric

objects, e.g. 4:6:4 (S=14) for the tetrahedron and 5:8:5 (S=18) for square-based

pyramid, 8:12:6 (S=26) for cube, 7:12:7 (S=26) for hexagonal-based pyramid and

12:18:8 (S=38) for hexagon, etc. Accordingly, from the statement (p. 128,

TLBA) that “Eve or Shi in plane geometry meant the side of a square or triangle”

and (pp. 155 & 177,TLBA) quoting from Nag Hammadi that “the entities Father

[Nna], mother [Nne] and Child [Nwa] exist as perceptible speech” having in them

Fig. 1

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“three powers, and abiding in three NNN, three quadrangles, secretly in ineffable

silence” [of the Ikom Monoliths], we set up a correspondence in Fig. 2.

FIG.2: Summary of “ichi” Linguistic Geometry: Row (i)

Realization of 28+36=64=8x8;(ii) 26+38=64=8x8 & (iii) 32+32=64=8x8

as Hyperspace Geometries for systems of alphabets: Arabic(28),

Igbo(36,38), English(26), & Man (Adam) symbol of Creation (32).

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The correspondence between the “ichi” (linear writing) codes and the alphabet

systems identified in ref.[1], i.e. Arabic(28), English(26), Igbo(36&38), etc and

man (Adam) as the symbol of creation (32) is based on complementary

representations of the “Afa” Matrix : 8x8 = 64 = 28+36 = 26+38 = 32+32 =

24+40, with consequential doubling of the reference objects.

Physically, linguistic geometry is based on sound as the vehicle of

communication of the first human, called “Ele Chi Ife” or “Bearer of Light” (p.30

LTAA). Ele was the father of the forest people (Bushmen) which was the original

meaning of the word Igbo who called themselves Sons of the Soil, i.e. the Adama

and the Nshi. It is the claim of The Adam Trilogy that Ele evolved from the Big

Bang before the Niphilim came and changed the course of things from evolution

to creation; that Ele lived in Ugwuele Okigwe (Imo State) and environs and his

language was the Igbo Afa word Atu (the origin of the Egyptian word Atum)

which, according to Prof. J.A. Umeh [9] on p.81 of his 1977-published book titled

After God Dibia, means “The Word of the Creator”. Afa is the Igbo word for

Oracle - the Primeval Oracle Matrix given to the first people by God; its

equivalent among the Yoruba is called Ifa. It was a language used by all growing

things (man, beast and crystalline minerals) alike, a language of symbols and of

geometry. which is why it is possible to lay the foundation for the creation of man

(“Adam”) and the theory of everything (TOE) in geometry.

Consequently, the linguistic frames of reference (tetrahedron, square-

based and hexagonal-based pyramids, cube, hexagon and pentagon) indicated in

Fig. 2 are capable of uniting, on one hand, Gell-Mann-Ne’eman[10] 8x8

representation of the (SU(3)) symmetry of the strongly interaction particles in

high energy physics from their occurrence in the product 8x8 = 1 + 8 + 17 + 8 +

10 + 10*, which (as shown on p. 31 of Y. Ne’eman’s [10] 1963-published book

entitled Algebraic Theory of Particle Physics) contains two equivalent octets and

splits into a symmetric 1 + 8 + 27 = 13+2

3+3

3 = 36 (dual pyramid) and

antisymmetric 8+10+10* = 2x(12+2

2+3

2) = 28 (dual tetrahedron) parts. On the

other hand the cube (S=26) [or hexagonal-based pyramid(S=26)] and the

hexagon (S=38) satisfy a similar relation 26+38 = 64 = 8x8 which we shall

explain in Sec. 3 of this paper after presenting the cosmological framework of the

ancient African(Igbo-Ukwu) craftsmen, for the purpose of developing in Sec. 4 an

African perspective on the string and superstring theory of everything espoused in

the articles in Cambridge University Press 1988-published Superstrings, A Theory

of Everything? edited by Davies and Brown[11] as well as Kaku’s[4] 1995-

published Hyperspace and Time Magazine[12] Dec. 31, 1999 edition on Albert

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Einstein as Person of the Century). While Witten’s supergravity (string)

theory[12], also called the M-theory, is constructed in 11-dimensional hyperspace

geometry inhabited by weird objects called branes, superstring TOE distinguishes

an open string model with gauge group SO(32) and a closed (“heterotic”) string

model with gauge group E8xE8 for grand unification of strong and electroweak

forces with gravity in hyperspaces of 10- and 26-dimensions that get compactified

to the observed O(3,1) Minkowki space-time.

The crux of the problem in these theories is the dichotomy of point-like

and extended (string-like) particles and analytically, hyperspaces of more than

four dimensions[13] used in TOE correspond to the Da Vinci Codes number (S)

of the geometrically closed (linguistic) objects of the theory while the uncertainty

principle relating energy and time in conventional quantum mechanics and energy

and extremely short distances (much smaller than the observed size of elementary

particles) in quantum theory of gravity is linked to the existence of black hole

singularity in the universe (see, M. Green’s article at p.123 of ref.[11] and

Stephen Hawking’s [14] 1988-published Bantam book entitled A Brief History of

Time from the Big Bang to Black Holes). For the geometrically closed objects,

the mathematical singularity of Einstein’s general relativity theory of gravitation

arises in the representation of interaction between strings (in Feynman space-time

picture of quantum field theory) by world sheets which are topologically

equivalent to a sphere or torus with an arbitrary number of holes. Consequently,

if we adopt Kaku’s procedure (p. 160 ref’[4]) for constructing TOE from

geometry along Einstein’s unified field programme:

Geometry =>Field Theory=>Classical theory=>Quantum theory,

the following correspondence principle with concepts of quantum field theory,

Points <=> Particles, Lines <=> Fields, Planes <=> Currents,

can be used to construct an African perspective on TOE based on cube-hexagon

hyperspace/torus and current algebra as will be presented in Sec.4. The results

will be discussed in Sec. 5 where conclusions will also be drawn.

3. AFRICAN WORLD VIEW

3.1 Cosmological Framework

As stated by Animalu[15] in his 1988 Lecture to the Nigerian Academy of

Science titled, Science, Religion and African Culture: “throughout history, human

societies have developed world views and cosmological frames of reference for

organizing their activities. These world views explain the how and why of daily

existence and therefore constitute the essential cultural ingredient of every

society. 'Usually the views are products of experiences so pregnant with drama

that they give rise to symbols or totems of some sort; the symbols or totems give

rise to thought; and thoughts give rise to customs and codes of the society. The

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world view of a society is successful to the extent that it is so internalized, from

childhood onwards, that it goes unquestioned, manifesting itself in various forms

of geometric symbols as written language and other artistic expressions”.

Consequently, as the foremost African scientific historiographer, Professor K.O.

Dike[16] put it in a paper for UNESCO in 1960:

“Africa in particular has confronted scholars with communal societies where

science cannot be separated from religion, or religion from law and politics,

where history is nothing but the history of the whole culture, a seamless

garment of such fine-woven texture, that any attempts to separate single

strands merely distort the whole.”

It is from this holistic perspective that we may look at Anya’s[17] dictum:

"Most African Societies. including the Nigerian society, operate on a

cosmological framework in which time is conceived as cyclical and space is

organized in three compartments - the heavens above, the earth below it and

the underworld beneath the earth - all conceived as contiguous and

continuous, once more in a cyclical continuum, as summarized succinctly in

Fig.3a[below], derived from Kalu[18].

Thus in ref.[15] the distinctive concept in Kalu’s representation of African cosmic

space-time continuum in Fig. 3a was identified as cyclical “African time” related

by a Mobius (non-unitary) transformation to the conventional linear time of

classical (Newtonian & special relativistic) and (Heisenberg) quantum mechanics.

Consequently, one may follow Kepler’s[16] use of Platonic solids as “archetypes”

in his reconstruction of cosmic pentagon grid around a spherical model of Sun-

centred (Copernican) planetary system, and similarly use interconnected

equilateral triangles to reconstruct Kalu’s African cosmic space and cyclic time

continuum so as to reproduce the distinctive hexagonal grid on the African (Igbo-

Ukwu) bronze torus as indicated in Fig. 3b

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Fig.3a

Fig.3b

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In addition, if we apply a similar triangulation to the nano-world of the hexagonal

crystallographic structure of graphene[19] made from two different sublattices,

we may (by employing the conventional cyclic boundary conditions of solid state

physics[20]) topologically fold the crystal lattice into either a torus with square

grid or a torus with hexagonal grid, as elaborated in Fig. 5.

Fig.5: (i) hexagonal crystallographic structure of graphene[19]

made from two different sublattices, and (ii) its topological folding

into either a torus with square grid or torus with hexagonal grid.

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A similar hexagonal grid over Earth’s spherical surface has been produced in

more recent time by the National Geographic [21] as shown in Fig.6,

underscoring the depth and contemporary relevance of the ancient African (Igbo-

Ukwu) world view and the need to analyze the geometry of the cube-hexagon

hyperspace identified earlier in Figs1 and2(ii) in order to determine its physical

content from the linguistic “Afa” Matrix relations 8x8 = 64 = 26+38=32+32 and

modern physics to which we turn next.

3.2 Characterization of the Cube-Hexagon Hyperspace

We begin by recalling the well-known fact in solids state physics (see chapter 1 of

ref.[20]) that a face-centred cubic (fcc) packing of hard spheres has the same

packing fraction(f=0.74) as an ideal hexagonal close-packed (hcp) arrangement

for which the ratio of the dimensions of axial to basal lattice cells,

633.13/8/ ac is the so-called -ratio that accounts for growth in the

universe. Thus, to construct the cube-hexagon hyperspace, in Fig. 1 one may use

the fact[22] that the plane passing through the mid-points of six sides of the cube

produces a hexagon which is the base of a hexagonal-based pyramid, and if the

Fig. 6: National Geographic hexagonal Grid Covering the Earth’s

Surface(left) and enlargement of encircled area for details (right).

(Source: Google Maps[21])

41


side of the cube is 1 unit, then the side of the hexagon is 21

. As a result we

obtain the cube-hexagon hyperspace shown in Fig.7, which is suitable for

representations of hadrons (baryons and mesons) in hexagonal (I,B,Y)-space

according to SU(3) symmetry and leptons (electron, muon and neutrino) in cubic

(Q,L,B)-space, I being the isotopic spin, Y the hypercharge, B the baryon number,

L the lepton number, and eQ the electric charge ( in unit of proton charge, e=1). It

is this apparent doubling of axes which makes the cube-hexagon hyperspace 6-

dimensional that we are after..

Fig.7: Physical Basis of the Cube-Hexagon Hyperspace as

provided by the representations of hadrons (baryons and

mesons) in hexagonal (B,Y,I)-space and leptons (electrons,

muons and neutrinos) in cubic (B,L,Q) space.

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3.3 Representation of Cube-Hexagon Hyperspace by a Torus.

Cube-hexagram duality reflects the hologrammatic nature of growth in the

universe, from replication of cubes and its hyperlink with the hexagon in all

directions into an apparently infinite lattice, which results in expansion of the

universe in accordance with the ratio, 8/5=1.6, determined by the Da Vinci Code

(5:8:5) of the square pyramid and nearly equal to the ratio 6.13/8/ ac of

the hexagonal close-packed lattice. The multiplication of cubes and the

hyperlinked hexagons is made possible in the presence of matter through

mirroring effect, as if the six sides of the cube are mirrors but this creates an

illusion of infinite lattice whose folding into a torus [like the African (Igbo-Ukwu)

bronze torus distinctive hexagonal grid] eliminates boundary effects. In terms of

propagation of spherical light wave from a point with constant speed (c) in

vacuum the mirroring may be associated with a dielectric () due to the presence

of matter that results in deformation of “point” sphere in Minkowski space-time

into a torus,

0ˆ2ˆ0 222222 trcrtcrtc (3.3.1)

where nncc ,ˆ 2 22 / being the refractive index of the medium and the torus is

characterized by the parametric equations:

,sin,sin)cos(,cos)cos( ctzctryctrx (3.3.2)

t being the time, the meridian angle, and the latitude angle . By eliminating

from the parametric equations, one obtains

sin))((2 )()( 22222 rctrzyxct (3.3.3a)

from which the quantization into a lattice results in the form,

n

rctrct

if , 0

,)(n if ),)((2 sin))((2 2

1

(3.3.3b)

i.e.,

nzyxrct

zyxrct

if , 0 & 0)()(

)(n if ,0)(22222

212222

(3.3.4)

where n = 0, 1, 2,… The only values of the angle compatibles with perfect

translational symmetry of a crystal lattice in three-dimensional space are those

for which[20] integer )1cos2( , and hence ,2 n where 1,2,3,4,6n

include cubic and hexagonal lattices, but not pentagonal lattices. The visual

43


images of the types of geometric objects represented by Eqs.(3.3.4) are easily

constructed by rewriting the first of the two equations in the form

,0)()( 22222222 rctzyxrctzyx (3.3.5a)

so that, for )(n21 , a pair of concentric spheres in r

-space of radii rct

define a spherical shell of thickness, r2 , with one sphere circumscribing the

cube and the other sphere circumscribing the hexagon as shown in,Fig.8: This

may be termed the “black hole” solution associated with “Big Bang”.

Fig.8: (i) “Black hole” representation of Eq. (3.3.5a) as two concentric

spheres of radii ( rct )defined by acube-hexagon hyperspace (ii)

(u,v) geneators of the “string” solution (3.3.5b); (iii) Kaku’s

communication “warm hole”; and (v) Igbo-Ukwu bronze vessel.

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. The second of the two equations in (3.3.4) is a ruled quadric surface with

real ),( vu line-generators in 3-dimensional projective space with homogeneous

coordinates ( ),,, yxrct given by

n

yctxr

xryct

xctyr

yrxct

if

0v

v

,0u

u

2

1

2

1

(3.3.5b)

which may be termed “string” or “worm hole” solution as shown in Fig.8(ii) &

(iii) comparable with an Igbo-Ukwu bronze vessel shown in Fig.8(iv).

These are simple but surprising results. They are simple because Eqs(3.3.4),

being homogeneous in the five variables, ),,,,( zyxrct , is invariant under the

usual scale transformation ( being a scaling parameter) and conformal

transformations so that. translational invariance and conformal invariance are

compatible on a torus! The surprise is apparent from what Carl Segan of Cornell

University said in his Introduction to Stephen Hawking’s book titled A brief

History of Time[13] about a universe that has the form of a torus:

“Hawking embarks on a quest to answer Einstein’s famous question

whether God had any choice in creating the universe. Hawking is

attempting, as he explicitly states, to understand the mind of God. And

this makes all the more unexpected the conclusion of the effort : a

universe with no edge in space, no beginning or end in time[like a torus]

and nothing for the Creator to do”.

The conclusion of the effort in question, as Hawking stated on p. 116 of his

book[13] is

“the possibility [when quantum mechanics is taken into account] that

space-time was – [like a torus] – finite but had no boundary, which means

that it had no beginning or moment of creation”.

Moreover, for small ,))((22 rcts one gets

222222 ))((2sin))((2 )()( srctrctzyxrct (3,3.6a)

which defines an O(3,2) de-Sitter group with line element

. )()( 222222 szyxrct (3.3.6b)

and a corresponding generalized Einstein’s energy-momentum-mass relation

45


.)( )( 222222 mccpcpcppE zyx (3.3.6c)

Alternatively, we may choose to replace the approximation in Eq.(3.3.5) by

Maduzemzia’s[23] ansatz for recovering the classical theory of gravitation

22

21sin

rc

GMg s (3.3.7)

In other words, both quantization in geometry and generalization to higher

dimensions lie within the ambit of Einstein’s general relativity theory on a torus

which, according to Kaku[4], is the “generic” geometric object of superstring

TOE.

4. AFRICAN PERSPECTIVE ON THE THEORY OF EVERYTHING

An important feature of any theory of everything is that it must bring

together all the elementary particles (leptons, baryons and mesons) and hydrogen

as well as all gauge fields and conserved currents. This is achieved in the cube-

hexagon representation displayed earlier in Fig. 7 through the correspondence

principle between the (points, lines & planes) of projective geometry of three-

dimensional space and the (particles, fields & currents) of quantum field theory:

Points<=>Particles, Lines<=>Fields, Planes<=>Currents.

Our (African) perspective on TOE is how to achieve these objectives through a

synthesis of linguistic (cube-hexagon) hyperspace/torus geometry in which space-

time quantization into a lattice is achieved according to modern solid state

physics and hadronic mechanics of extended (string-like) elementary particles.

For avoidance of doubt, a simple characterization of hyperspace/torus by a

6-vector in the context of a tetrahedron of reference of 3-dimensional projective

geometry is provided by Einstein’s “unified” field metric tensor (hg ) in terms of

three-vectors of both space and time analogous to electric and magnetic field

vectors, suggested by Animalu[24] such that:

(4.1.1) 0)(])[()det( 22224

12

13

23

t.rrt ccss

sttz

tsty

ttsx

zyxs

g h

46


where 2

3

2

2

2

1 tttt and 222 zyxr . The vanishing of the

discriminant of the quadratic equation in 2s expressed in the form

0)()( 22 t.rrt cc . (4.1.3)

characterizes a deformation() of the “point” sphere 0)( 22 rtc in ),( rtc -6-

vector space, into a torus whose quantization into a space-time lattice defines, as

discussed in Sec. 3.3, a cube-hexagon hyperspace among other features.

The bringing together of all elementary particles into a cube-hexagon

hyperspace with coupled double axes (B,L,Q) and (B,Y,I), is provided by a

group-extension of the Lorentz group to O(4,2) dynamical symmetry group

(Barut[25]) such that, as shown by Animalu[26], there is a charge-like Fermion

number (F) in addition to Fermion electric charge FeQ , and strangeness FS

obeying (for leptons and baryon quarks)

FF SFY );Q(F)Q(F F

2

F ,

and the spin-angular momentum (j) obeys, according to Yuval Ne’eman,(p.3 of

ref.[10]), the spin-parity relation

F32j )1((-1)

Consequently, one can transform the fractional electric charge matrix ) ( qQ of the

Gell-Mann-Zweig quark triplet (u,d,s) into the corresponding charge matrix ( Q )

the lepton triplet ),,( e by using a discrete symmetry of hyperspace defined

by an antisymmetric matrix (U) (which is non-unitary):

100

010

000

030

300

000

00

00

00

030

300

000

31

31

32

QUQU q

and shifting 2/3 of the leptonic charge to the lepton number obtain fractional

baryon number of the quarks. However, to make the non-unitary transformation

non-singular, we may, following Santilli[27], replace the matrices, U and U+, by

,

130

310

001

UIW ,

130

310

001

UIZ

47


This demands, for mathematical consistency, the use of Santilli’s Lie-admissible

generalization of the conventional Lie-algebraic structure .of quantum mechanics.

One of the consequences is that quarks can be eliminated as hadronic constituents,

and hadrons can be synthesized as “hadronic” bound states of physical particles

produced in their decays.

To unify the gauge fields from our perspective, the usual 10 independent

components of 4x4 Einstein symmetric metric tensor ( g ) can be unified with

the Maxwell’s electric and magnetic field vectors and augmented with Yang-Mills

field 6x6 metric tensor for a unified gauge (boson) fields which, in linguistic

geometry, corresponds an extension of the cube (S=26) for the 26=5x5+1 English

alphabets to the dual square pyramid (S=36) for the standard 36 Igbo alphabets.

as shown in Fig 13

.

Fig. 13: (Top) Representation of linguistic 36=6x6 Standard

Igbo alphabet as a generalization of 26=5x5+1 English

alphabet system from SU(3) 8x8=36+28 analogy with

(Bottom) 6x6=36 metric tensor of Einstein-Maxwell-Yang-

Mill gauge theory.

48


5. CONCLUSION

The finding in this paper is that all elementary particles can be represented in 6-

dimensional cube-hexagon hyperspace with axes (B,L,Q) and (B,Y,I), in

agreement with the demonstration by the Australian biologist Illert[28], winner of

the Nobel Prize in physics, that an accurate representation of the growth of

seashells occurs via the use of a six-dimensional space, that is, a space in which

each axis is doubled. This leads to the conclusion that everything in nature

including languages, nano-particles and elementary particles, exhibit the

phenomenon of growth and hence life whose time-irreversible evolution is

characterized by non-unitary discrete symmetry of six-dimensional cube-hexagon

hyperspace that relates integral-charged leptons and rational-charged quarks: we

call this the principle of complementary duality to which the next paper is

devoted.

REFERENCES

[1] Animalu, A.O.E. and Acholonu, C. “Ichi” Linguistic Geometry and

Evolution – A Commentary and They Lived Before Adam, African J.

Physics, Vol. 2, pp. 373-392 (2010)

[2] Acholonu-Olumba, Catherine (with Ajay Prabhakar & Nneka C. Egbuna),

The Gram Code of African Adam – Stone Books and Cave Libraries

(referred to as GCAA), Catherine Acholonu Research Centre (CARC)

Publication ISBN 978-31997-8-1(2005); They Lived Before Adam –

Prehistoric Origins of the Igbo (referred to as TLBA), CARC Publication

ISBN-978-2579-51-10 (2009); The Lost Testaments of the Ancestors of

Adam – Unearthing the Heliopolis/IGBO-UKWU The Celestial City of the

Gods of Ancient Egypt and India (referred to as LTAA), CARC

Publication ISBN:978-31997-3-0 ( 2010).

[3] Rowland, Todd and Weisstein, Eric W. "Homology." From MathWorld--A

Wolfram Web Resource. http://mathworld.wolfram.com/Homology.html

[4] M. Kaku Hyperspace: A Scientific Odyssey Through Parallel Universes,

Time Warps, and the 10th

Dimension

[5] A.O.E. Animalu and W.A. Umezinwa, From African Symbols to Physics

(Desk-published 1988 reprinted by Ucheakonam Foundation (Nig.) Ltd

1994). See also Charles Jenks in Signs, Symbols and Architecture ed. G.

Broadbent, R. Bunt and C. Jenks (J. Wiley & Sons Ltd. NY, 1981) p.7.

[6] Chinua Achebe, Arrow of God (Doubleday/Anchor, NY, 1969 revised

edition)

49


[7] See References to Thurston Shaw & Anozie in ref.[2] LTAA.

[8] Dan Brown, The Da Vinci Code , see also by the same author, Angels and

Demons(Washington Square Press, available on paperback, 2009).

[9] J.A. Umeh, After God Dibia (1977)

[10] See, Y. Ne’eman, Algebraic theory of Particle Physics (Benjamin Inc,

1963).

[11] Davies and Brown edit. Superstrings, A Theory of Everything?

(Cambridge Univ. Press, 1988)

[12] Time Magazine Dec 31, 1999 Edition on A. Einstein as Person of the

Century.

[13] A.O.E. Animalu and A.O. Azali, Language and Dimensionality in

Physics, The Prospect of African Renascence African J. Physics, Vol. 1, p.

255-283 (2008).

[14] Stephen Hawking, A Brief History of Time from the Big Bang to Black

Holes (Bantam, 1988).

[15] A.O.E.Animalu, Science, Religion an African Culture, Discourses of

Nigeria Academy of Science vol. 9, p.45 (1988).

[16] K.O. Dike Selected Works of Kenneth Onwuka dike, ed. C. Ifemesia

(KODIC, Awka, 1988) p. 38.

[17] A. O. Anya, in Basic Science Development in Nigeria, problem and

prospects ed. Sam Okoye and K. Motso Onuoha (Evans Brothers, Nig.

1987) p. 91

[17] O. Kalu, Precarious Vision in Readings in African Humanities, African

Cultural Development (Fourth Dimension Publishing 1987).

[18] J. Kepler in Arthur Koestler’s[3] Sleepwalkers-A history of Man’s Changing

Vision of the Universe Pelican Books (1984) p.399.

[19] M.I. Katsnelson, K.S. Novoselov, Graphene: New bridge between

condensed matter physics and quantum electrodynamics, Solid State

Communications 143 (2007) 3–13

[20] Animalu, A.O.E, Intermediate Quantum Theory of Crystalline Solids

(Prentice-Hall, 1977); see also J.M. Ziman, Electrons and Phonons

(Oxford Univ. Press, 1960).

[21] National Goegraphic Google Maps.

50


[22] www.puzzles.com – Puzzle Playground, geometrical Puzzles - Hexagon

Inside the Cube (© 2004 ThinkFun.Inc).

[23] A. Maduemezia De-Sitter Group Approach to the Theory of Gravitation,

J. Nig. Ass. Math. Phys. 3, 1 (1999).

[24] A.O.E. Animalu, Review of GAGUT available at

http://www.math.buffalo.edu/mad/physics/Animalu_Alex_GAGUTreview1.pdf

[25] A.O.Barut, Phys. Rev. Letters 20, 893 (1968).

[26] A.O.E. Animalu, Phys. Rev. D4, 1922 (1971)

[27] R.M. Santilli, Hadronic J. 1, 574 (1978).

[28] Dr. Chris Illert, “sub-physical underpinnings of matter”, Alchemy Today

volume 1, matter on the E5 level: Platonic Geometries in Nuclear Physics,

a Centennial Commemoration of Occult Chemistry 1895-1995; First

Edition 1992/5, 172 A4 pages, ISBN: 0 949357 13 8 (v1); see also C. R.

Illert and R. M. Santilli, Foundation of Theoretical Conchology, Hadronic

Press (1995), http://www.santilli-foundation.org/docs/Santilli-109.pdf

http://www.puzzles.com/

http://www.math.buffalo.edu/mad/physics/Animalu_Alex_GAGUTreview1.pdf