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288 Sparks of Genesis and Ternary Sedenions

According to Bernard Pick's 1913 work The Cabala on a sacred-texts.com web page:

"... the Zohar is ... [a] production ... of ... thirteenth century ... Spain ... by Moses de Leon (1250-1305) ...".

According to web pages by Rabbi Moshe Miller about Arizal on the www.safed-kabbalah.com web site:

"... Rabbi Yitzchak Luria (the Arizal) ... 1534-1572 c.e. ... set out to explain ... the kabbalistic literature ... particularly Zohar ...

There are five areas of focus in the Arizal's teachings ... :

• the concept of tzimtzum (G-d's self-contraction, so to speak) through its various stages ...

  Prior to creation, there was only G-d and His infinite revelation of Himself, the Or Ein Sof, filling all existence ...[ corresponding to

the letter Aleph (first of the Hebrew alphabet) placed before the First Verse of the First Book Genesis of the Torah

and

the dimensionless Empty Set ...]

the tzimtzum ... established a radical distinction between Creator and created (from the viewpoint of the created, although not from the viewpoint of the Creator ... ), ... so that creation comes about by way of a "quantum leap" ...[ corresponding to the first word of the First Verse that begins with Genesis letter number 1 Bet and ends with Genesis letter number 6 Tav

and

the 0-dimensional Natural Numbers created from the Empty Set by the Peano unitizer operation 0 -> {0} = 1 as described by David Finkelstein in his book "Quantum Relativity" (Sprimger 1996)

the 1-dimensional Real Numbers created from the Natural Numbers by ratios and completion

the 2-dimensional Algebraically Complete Complex Numbers created from the Real Numbers by Cayley-Dickson Doubling

the 4-dimensional Associative NonCommutative Quaternions created from the Complex Numbers by Cayley-Dickson Doubling

the 8-dimensional Alternative NonAssociative Octonions created from the Quaternions by Cayley-Dickson Doubling

the 16-dimensional Sedenions created from the Quaternions by Cayley-Dickson Doubling

 

• the process of shevirat hakeilim (the shattering of the vessels in the world of Tohu) ...

  The first "world" (plane of existence) that came into being after the tzimtzum is called Adam Kadmon. ... the light in Adam Kadmon ... manifested as ... sefirot [that] compose the world of Tohu (chaos or disorder) ... Due to the intensity and exclusivity of the lights ... the vessels of the lower sefirot of Tohu shattered ...[ corresponding to the First Verse

that ends with Genesis letter number 28 Zadi-final

and

the shattering of Division Algebra structure at the formation of the Sedenions, which have non-trivial Zero Divisors is described by Guillermo Moreno in arXiv math/0512517 as "... For ...[Sedenions]... the set of zero divisors ... of fixed norm can be identified with V7,2 the real Stiefel Manifold of two frames in R7 and the singular set of (x,y) with xy = 0 and ||x|| = ||y|| = 1 is homeomorphic to G2 the exceptional simple Lie group of rank 2. ...". If the x and y coordinates are allowed to expand/contract from unit norm, then the set of Zero Divisors can be seen as having two more dimensions, equivalent to two scalar dimensions, so that the Full Zero Divisors of Sedenions have 16 dimensions equivalent to R + G2 + R .

(see also the descriptions by Robert P. C. de Marrais in his papers including arXiv 0804.3416)

The new natural product rule at the Sedenion level is described by Jaak Lohmus, Eugene Paal, and Leo Sorgsepp said in their book "Nonassociative Algebras in Physics" (Hadronic Press 1994) as "... a ternary algebra ... the nonassociativity may appear only when there are three elements of the algebra combined, therefore the binary multiplication rule cannot account for nonassociativity ... and we must introduce ... a ternary algebra ...[in which]... two sedenions A , C , ... are represented by L- , R-type 16x16-matrices and the third one, B , ... by 16-columns ...".

The 16x16-matrices of Real Numbers are a regular birepresentation of the left and right actions of the Sedenions and form the 256-dimensional Real Clifford Algebra Cl(8).

The left action is represented by a Left-Action 16-column SpL16 corresponding to a copy of the Cl(8) Spinors.

The right action is represented by a Right-Action 16-column SpR16 corresponding to a copy of the Cl(8) Spinors.

Taken together, SpL16 + SpR16 represent a 32-Real-dimensional space E6 / D5xU(1) which is 16-Complex-dimensional and corresponds to a Complex Bounded Domain whose Shilov Boundary is 16-Real-dimensional and which physically corresponds to 8 Fundamental First Generation Fermion Particles plus 8 Fundamental First Generation Fermion AntiParticles.

The total representation of the Ternary Sedenions that emerge after the shattering of Divsion Algebra structure is

SpL16 + Cl(8) + SpR16

have 16L + 256 + 16R = 288 dimensions. ...]

 

These [ 288 dimensional representing Ternary Sedenions ] are referred to by the Arizal as the 288 nitzotzin ("sparks") - the initial number of fragments from the vessels that broke. The ... process is alluded to in .... [ Genesis 1 : 2 from www.mechon-mamre.org web site

with the central three letters (Genesis numbers 68, 69, and 70) of the word "hovered" indicated by a red box. Note that 70 is the dimensionality of the central part of the graded structure of Cl(8) 16x16 = 256 = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1

and that the 16 dimensions of the (norm one) Sedenion Zero Divisors that caused the shattering of the Division Algebra structure are represented by R + G2 + R with graded structure

1 + 4 + 6 + 4 + 1

which live in the bold-faced grades of the Cl(8) grading 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1

]... The Arizal explains that the word ... hovered (merachefet) ... is actually a compound of two words: met and rapach - signifying that 288 (the numerical value of rapach) fragments had died (met) ...[in]... the shattering of the vessels of Tohu into 288 initial sparks ... The shattering of the sefirot of Tohu ... serves a very specific and important purpose, which is to bring about a state of separation or partition of the light into distinct qualities and attributes, and thereby introduce diversity and multiplicity into creation ...".

 

• the Tikkun (rectification) of that shevira through birur hanitzotzot (elevating the sparks) ...

  [The]... process of extracting the sparks is called birur, which is part of a larger cosmic plan called Tikkun - rectification or restoration of the broken vessels ... When the sparks ... are rebuilt into the vessels of Tikkun ... the repaired vessels will be able to contain the light ...[ For the 288 nitzotzin spark components of the Ternary Sedenions to be rebuilt, they must be transformed into a form that allows them to be interactively compounded with each other. The Cl(8) part of 288 = 16 + Cl(8) + 16 is easily compounded because of the 8-Periodicity of Real Clifford Algebras:

  Cl(N8) = Cl(8) x ...(N times tensor product)... x Cl(8)

  However, the 16L + 16R parts and the resulting mixed terms result in complications when you try to compound:

  ( 16L + Cl(8) + 16R )x( 16L + Cl(8) + 16R ) = 16Lx16L + Cl(16) + 16Rx16R + ????

  where the 16Lx16L and 16Rx16R terms do correspond to the Spinors of the Cl(16) Real Clifford algebra of 256x256 matrices but the ???? mixed terms are problematic. To avoid such problems, go to the Arizal's next point: ]...

• the concept of partzufim ... compound structures ... in arrays that interact with each other ...

  the partzufim are compound structures of the sefirot ... In the universe of partzufim, it may be said that the chief dynamic of creation is not evolution (hishtalshelut), but rather interaction (hitlabshut). ...[ To transform the 288 spark Ternary Sedenions into partzufim that naturally interact with each other:

  Start with the Cl(8) part of the 288 spark Ternary Sedenions and note that the 16-dimensional R+G2+R Zero Divisor space of the Sedenions is represented by G2 plus two scalars, with total graded structure

  1 + 4 + 6 + 4 + 1

  which lives in the bold-faced grades of the Cl(8) grading 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1

  Then, delete the Zero Divisor space from the Sedenions, leaving the graded structure

  8 + 24 + 56 + 64 + 56 + 24 + 8

  which is the graded structure of the 240 Root Vectors of the Lie Algebra E8.

  Since the 16L part of the 288 spark Ternary Sedenions correspond to a Complexification of generators of 8 Fundamental Fermion Particles, and since the first 8+56 = 64 = 8x8 of the E8 Root Vectors correspond to an Octonification of generators of 8 Fundamental Fermion Particles (physically, the 8 covariant components with respect to 8-dimensional Kaluza-Klein spacetime for each of the 8 Particles), the 16L can be projected into 8+8 of the first 8+56.

  Since the 16R part of the 288 spark Ternary Sedenions correspond to a Complexification of generators of 8 Fundamental Fermion AntiParticles, and since the second 56+8 = 64 = 8x8 of the E8 Root Vectors correspond to an Octonification of generators of 8 Fundamental Fermion AntiParticles (physically, the 8 covariant components with respect to 8-dimensional Kaluza-Klein spacetime for each of the 8 AntiParticles), the 16R can be projected into 8+8 of the second 56+8.

  After the Zero Divisor deletions and the Fermion Projections, the 288 spark Ternary Sedenions are transformed into the 240 Root Vectors of the E8 Lie Algebra. ,

  which (by effectively reintroducing 8 = 4+4 of the G2 Zero Divisors and adding them to the two 28 parts of Cl(8)) generate the full 248-dimensional E8 Lie Algebra with the 7-graded structure of Thomas Larsson:

  8 + 28 + 56 + 64 + 56 + 28 + 8

  Physically, the central 64 = 8x8 corresponds to 8 Kaluza-Klein spacetime dimensions each with 8 dual momenta

  and the first 28 = D4 Lie Algebra contains 16-dimensional U(2,2) = Spin(2,4)xU(1) subalgebra which gives Conformal Gravity by the MacDowell-Mansouri mechanism

  and the second 28 = D4 Lie Algebra contains the 12-dimensional SU(3)xSU(2)xU(1) Standard Model subalgebra.

  Therefore, the Partzufim is the E8 Lie Algebra, which looks like this:

  

  By the above construction, the 248-dimensional E8 Lie Algebra lives inside the 256-dimensional Cl(8) Real Clifford Algebra, so by the 8-Periodicity of Real Clifford Algebras

  Cl(N8) = Cl(8) x ...(N times tensor product)... x Cl(8)

  the E8 Partzufim can be compounded to form extended chains woven into nets that fill spacetime

  

  beginning with the compound of the first two E8 Partzufim, to form the first 248+248 =

  496-dimensional Compound E8 Partzufim

   

  Note that 0, 1, 6, and 28 are Perfect Numbers, and that the only others below 33,550,336 (related to the Mersenne Prime 8,191 = 2^13 -1) are the Perfect Numbers

  496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 (related to the Mersenne Prime 31 = 2^5 -1) and

  8,128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064 (related to the Mersenne Prime 127 = 2^7 -1).

  Thus, the first Compound E8 Partzufim at the very Beginning of the Inflationary Expansion of Our Universe (a non-unitary process due to the non-unitarity of nonassociative Octonions and Sedenions) corresponds to the Perfect Number 496,

  which is consistent with Genesis letter number 496 being the last Shin in Chapter 1 Verse 11

 

Shin looks like a Tree of Life that Grows, Multiplies by its Seeds, and Evolves.

Since Perfect Number 496 corresponds to the Beginning of Inflation, it seems that Perfect Number 8,128 should correspond to the End of Inflation, and therefore indicate the duration of Inflation, the fundamental unit of mass, and the number of particles created during Inflation.

Genesis letter number 8,128 is the fourth Ayin in Chapter 7 Verse 4, that is, the word 40 of 40 days in the statement

"... I will cause it to rain upon the earth 40 days ... "

which rain (see verse 23): "... blotted out every living substance ... from the earth; and Noah only was left, and they that were with him in the ark ...".

The 2 branches of Ayin look like a fork in the Road of History leading to two alternative futures:

the Death Future of most of Life on Earth and the Life Future of the Ark beings.

 

Note that 8,128 = 64 x 127 = 64 x (128 - 1).

As Paola Zizzi said in gr-qc/0007006: "... during inflation, the universe can be described as a superposed state of quantum ... [ qubits ]. The self-reduction of the superposed quantum state ... reached at the end of inflation ... corresponds to a superposed state of ... [ 10^19 = 2^64 qubits ] ... ...[at]... the decoherence time ... [ Tdecoh = sqrt(10^19) Tplanck = 10(-34) sec ]. ...", so 2^64 tells us the duration of Inflation.

2^64 qubits corresponds to the Clifford algebra Cl(64) = Cl(8x8). By the periodicity-8 theorem of real Clifford algebras that Cl(K8) = Cl(8) x ... tensor product K times ... x Cl(8),

we have: Cl(64) = Cl(8x8) = Cl(8) x Cl(8) x Cl(8) x Cl(8) x Cl(8) x Cl(8) x Cl(8) x Cl(8)

Therefore, Cl(64) is the first ( lowest dimension ) Clifford algebra at which we can reflexively identify each component Cl(8) with a vector in the Cl(8) vector space. This reflexive identification/reduction causes decoherence. It is the reason that our universe decoheres at N = 2^64 = 10^19 which Decoherent Collapse into the Many Worlds of the Many-Worlds Quantum Theory

led to our World being only one of the Many.

At the time Tdecoh = 10^(-34 sec) at the End of Inflation, the number of qubits is Ndecoh = 10^19 = 2^64 .

Each qubit at the end of inflation corresponds to a Planck Mass Black Hole, which undergoes decoherence and, in a process corresponding to Reheating in the Standard Inflationary Model, each qubit transforms into 2^64 = 10^19 elementary first-generation fermion particle-antiparticle pairs.  

The resulting 2^64 x 2^64 = 2^128 = 10^19 x 10^19 = 10^38 fermion pairs populating the Universe Immediately After Inflation constitutes a Zizzi Quantum Register of order n_reh = 10^38 = 2^128.

Since, as Paola Zizzi says in gr-qc/0007006, ( with some editing by me denoted by [ ] ): "... the quantum register grows with time. ... At time Tn = (n+1) Tplanck the quantum gravity register will consist of (n+1)^2 qubits. [ Let N = (n+1)^2 ] ...", we have the number of qubits at Reheating:

Nreh = ( n_reh )^2 = ( 2^128 )^2 = 2^256 = 10^77

Since each qubit at Reheating should correspond, not to Planck Mass Black Holes, but to fermion particle-antiparticle pairs that average about 0.66 GeV, we have the result that the number of particles in our Universe at Reheating is about 10^77 nucleons, so (2^128)^2 tells us the number particles in our universe.

After Reheating, our Universe enters the Radiation-Dominated Era, and, since there is no continuous creation, particle production stops, so the 10^77 nucleon Baryonic Mass of our Universe has been mostly constant since Reheating, and will continue to be mostly constant until Proton Decay.

The present scale of our Universe is about R(tnow) = 10^28 cm, so that its volume is now about 10^84 cm^3, and its baryon density is now about 10^77 protons / 10^84 cm^3 = 10^(-7) protons/cm^3 = 10^(-7-19-5) gm / cm^3 = 10^(-31) gm / cm^3 = roughly the baryonic mass density of our Universe.

Since the critical density of our Universe is about 10^(-29) gm / cm^3, it is likely that the excess of the critical mass of our Universe over its baryonic mass is due to a cosmological constant as described by Conformal Gravity in the E8 Physics model which gives

a ratio of Dark Energy : Dark Matter : Ordinary Matter of 0.753 : 0.202 : 0.045 . ]...

 

• the nature of the soul, the purpose of its descent into this world, and its relationship with the higher realms and ultimately with G-d. ...

  the soul is both part of the Creator and at the same time it is created - its luminous essence is "a tiny spark of G-dliness," and the sheath in which it is clothed is a created being, albeit a spiritual being and not physical. As the soul emanates from the Ein Sof - the Infinite One - eventually to be clothed in the physical body ... Thus man is a microcosm of creation and his actions have cosmic significance ... He is able to affect the balance of the universe, both spiritual and physical, by his kavanot (mystical intentions) and yichudim (unifications of the sefirot). ...".

  Since Ndecoh = 2^64 = 10^19 qubits is just an order of magnitude larger than the number of tubulins Ntub = 10^18 of the human brain, and Conscious Thought is due to superposition states of those 10^18 tubulins, and since a brain with Ndecoh = 10^19 tubulins would undergo self-decoherence and would therefore not be able to maintain the superposition necessary for thought, it seems that the human brain is about as big as an individual brain can be. The Zizzi Self-Decoherence can be compared to GRW decoherence. Thus the Mind of Man seems to be an image of the Mind of Our Universe.

   

   

   

   

 Some Details about Ternary Sedenions:

Richard Kerner, in math-ph/0004031, said:

"... the two copies of the Hilbert space that have been used to produce general linear operators ... L(V,V) ... (containing the algebra of observables) by means of the tensor product

is utterly different from

the role of the third copy ... V ... serving as the space of states ...".

Jaak Lohmus, Eugene Paal, and Leo Sorgsepp said in their book "Nonassociative Algebras in Physics" (Hadronic Press 1994):

"... Ternary sedenions ...

the nonassociativity may appear only when there are three elements of the algebra combined, therefore the binary multiplication rule cannot account for nonassociativity ... and we must introduce some kind of ternary operation ... the new meaninfully generalized algebra will be a ternary algebra ...

Let us define the corresponding ternary *-associator product for arbitrary three sedenions

A = x + X e , B = y + Y e , C = z + Z e ; x , y , z, X, Y, Z in O:

 *( A , B , C ) = *( A B ) C - A ( B C )* =

*[( x + X e )( y + Y e)](z + Z e) - ( x + X e )[( y + Y e)](z + Z e)]* =

= ( x y ) z - Ybar( X z ) - Zbar ( Y x ) - Zbar ( X ybar ) -

- x ( y z ) + ( x Zbar ) Y + ( ybar Zbar ) X + ( ( z Ybar ) X +

[( Z x ) y - Z ( Ybar X ) + ( Y z ) zbar + ( X ybar ) zbar -

- ( Z y ) x - ( Y zbar ) x - ( X zbar ) ybar + X ( Ybar Z )] e .

(3.27)

The difference between the *-product *( A , B , C ) and the associator ( A , B , C ) computed in the binary sedenion algebra ... lies in the position of brackets in the underlined terms (see also Fig. 1).

We call the 16-dimensional (sedenion) algebra with the ternary *-product (3.27) ternary sedenion algebra and its elements ternary sedenions.

*-associator has the common linearized alternativity property

*( A , B , C ) = (-1)^s *( PA , PB , PC )

... where P represents some permutation and s is the parity of the permutation. So it may be said that the alternativity property is restored, and so is the antiassociativity property for the basic units (now in terms of *-product).

... There is a connection with the binary operation through the half-ternary products *( A B ) C and *A ( B C )

which may be traced from (3.27)

*( A B ) e_0 = A ( B e_0)* = A B

... where A B is the product of A and B in the binary sedenion algebra.

... these properties of our algebra are not purely ternary, because this modification has been derived from binary algebra. The viewpoint of purely ternary operation demands the consideration of associativity relations involving five elements, etc. We shall not discuss this problem here. ...

... the R_i matrices of the octonion ... regular birepresentation ... regbirep ... are the anticommuting antisymmetric matrices forming the 64-dimensional Clifford algebra C6 with 6 generic elements

( R_1 , R_2 , R_3 , R_4 , R_5 , R_6 for example, as R_1 R_2 R_3 R_4 R_5 R_6 = R_7 ).

... The R_i-matrices corresponding to different ... octonion ... multiplication tables are different. The transition between two tables may be carried out by L_i-matrices. ... and the transition formula looks like

- L(0)_i R(0)_j L(0)_i = R(i)_j

... where the ... indices in parentheses denote the numbers of table modification.

...

For the binary sedenion algebra regular birepresentation (regbirep) L , R-operators can be constructed easily. The operators representing the binary sedenion units e_i , i = 0 , 1 , ... , 15

e_i -> L_i , R_i : L_i x = e_i x , R_i x = x e_i , x in BS

... If L_i , R_i are thought of as 16x16 matrices, then the elements x, , e_i x , x e_i must be 16-columns (vectors).

A general element A = x + Xe of the binary sedenion algebra is then represented by 16x16 matrices

 

       L_x    -Rbar_X               R_x    -L_Xbar
L_A =                   ,    R_A = 
       Lbar_X  R_x                  L_X     R_xbar
   

where L , R , Lbar , Rbar are the corresponding regrep 8x8 matrices and their conjugates for octonions.

We can introduces a mixed representation where one sedenion in the product is represented by a 16x16-matrix and the other one, by a 16-column.

For ... the ternary algebra this gives a nontrivial possibility to construct a ... mixed representation for ternary sedenions. In this representation two sedenions A , C , in the ternary "half-products" *(AB)C , A(BC)* in (3.27) are represented by L- , R-type 16x16-matrices

and the third one, B , and the half-products themselves, by 16-columns (underlined):

*( A B ) C = *R_C L_A B

A ( B C )* = *L_A R_C B

... where *R_C L_A , *L_A R_C are special nonassociative products of L-, R-matrices calculated by means to ternary product (3.27):

 

               R_z L_x - L_Zbar L_X           - Rbar_Xz - L_Zbar R_x
*R_C L_A  =
               L_Zx + R_zbar Lbar_X           - L_Z R_X + R_zbar R_x
   
   
   
               L_x R_z - Rbar_X L_Z           - L_xZbar - Rbar_x R_zbar
*L_A R_C  = 
               Lbar_Xzbar + R_x L_Z           - L_X L_Z + R_x R_zbar
   

Here we make the following replacements (corresponding to underlined terms in (3.27))

R_z Rbar_X   ->   Rbar_Xz 
   
L_Z L_x      ->   L_Zx
   
L_x L_Zbar   ->   L_xZbar
   
Lbar_X R_z   ->   Lbar_Xzbar

where

Rbar_Xz     =   R_z Rbar_X + R_X Lbar_z - L_z Rbar_X 
   
L_Zx        =   L_Z L_x + [ L_Z , R_x ]
   
L_xZbar     =   L_x L_Zbar + [ L_x , R_Zbar ]
   
Lbar_Xzbar  =   L_X Lbar_zbar + L_X Xbar Rbar_zbar - R_zbar Lbar_X

...".

 Zero Divisors, Graded Structures, and Associative Triples

There are multiple ways in which the (14+2) = 16-dimensional Zero Divisor space of the Sedenions represented by R+G2+R with graded structure 1 + 4 + 6 + 4 + 1 can be deleted from the Cl(8) grading 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 leaving the graded structure 8 + 24 + 56 + 64 + 56 + 24 + 8 of the 240 Root Vectors of the Lie Algebra E8.

There is only one way that the scalar 1+1 of R+G2+R can be deleted from the scalar 1+1 of Cl(8).

Just as the 28-dimensional D4 Lie algebra has 7 independent sets of 4 Cartan subalgebra generators, there are 7 different ways that the first 4 of the 1+4+6+4+1 Zero Divisors can be deleted from the first 28 of the 1+8+28+56+70+56+28+8+1 of Cl(8).

Choosing which of the 7 ways for the first 4 and 28 fixes, by dualities in the graded structures, the choices for the second 4 to be deleted from the second 28 and for the middle 6 to be deleted from the middle 70, so there are 7 ways that R+G2+R can be deleted from Cl(8).

Those 7 ways correspond to the 7 different independent E8 lattices in 8-dimensional Euclidean space, each of which has its own 240-vertex Witting Polytope configuration as the first shell around the origin, and therefore its own set of 240 Root Vectors.

The also correspond to the 7 Associative Triples of the Octonions.

Quaternions have only one Associative Triple { i , j , k }
For the octonions, 
6 new associative triple cycles appear  
{ i , J , K }
{ I , j , K } 
{ I , J , k }
{ i , E , I }
{ j , E , J }
{ k , E , K }
  
They correspond to the Lie algebra Spin(4).  
The other 35 - 7 = 28 triples are not cycles.  
 
Denote the 7+8 = 15 sedenion Imaginary basis elements 
by { i, j, k, E, I, J, K, S, T, U, V, W, X, Y, Z } .
 
The sedenions correspond to a tetrahedron, 
a 3-dimensional simplex, 
4 vertices v of the tetrahedron corresponding to  EIJK ; 
6 edges e of the tetrahedron corresponding to    ijkTUV ; 
4 faces f of the tetrahedron corresponding to     WXYZ ; 
and the 1 entire tetrahedron T corresponding to    S  . 
There are 4+6+4+1 = 15 things.  
There are 35 (projective) lines each with 3 things and  
they correspond to the 35 associative triples of the sedenions.  
Geometrically, they are of the form: 
3+4 = 7 corresponding to the 7 associative triples of octonions:  
3  like eTe (where e is opposite e on the whole tetrahedron T); 
4  like vTf (where v is opposite f on T); 
and 
16+12 = 28 corresponding to 28 new ones formed at the sedenions: 
4  like eee (where eee are all on the same face);
6  like vev (these are the edges); 
6  like fef (where the edge of e is not on f or f, 
             that is, f and f are opposite to e); 
12 like vfe (where v is opposite e on face f).  
 
   
The sedenion multiplication table is 16x16
so it has 256 = 2^8 entries and can be written 
as a 16x16 matrix: 
 
   r i j k E I J K S T U V W X Y Z
 r x x x x x x x x x x x x x x x x
-i x x 
-j x   x q
-k x     x
-E x       x o o o
-I x         x o o
-J x           x o
-K x             x 
-S x               x s s s s s s s
-T x                 x s s s s s s
-U x                   x s s s s s
-V x                     x s s s s
-W x                       x s s s
-X x                         x s s
-Y x                           x s
-Z x                             x
 
The 16+15+15 = 46 x entries denote the "real" products that 
cannot belong to an associative triple cycle of the type ijk.  
 
For the ri part of the table, the complex numbers, 
there are no associative triple cycles. 
 
For the rijk part of the table, the quaternions, 
there is only one associative triple cycle, 
the ijk triple itself, denoted by the q entry.  
 
The 6 o entries represent the 6 new 
associative triple cycles that come with the octonions.  
 
The 28 s entries represent the 28 new 
associative triple cycles that come with the sedenions.  
The 28 new associative triple cycles of the sedenions 
are related to the 28-dimensional Lie algebra Spin(0,8), 
and to the 28 different differentiable structures on 
the 7-sphere S7 that are used to construct 
exotic structures on differentiable manifolds. 
   
WHAT ABOUT GOING UP TO HIGHER DIMENSIONS?
 
For 32-ons, we get 120 new associative triple cycles, 
and they represent the Lie algebra Spin(0,16) of 
the Clifford algebra Cl(0,16).  
 
HOWEVER, NOTHING REALLY NEW HAPPENS BECAUSE OF 
THE PERIODICITY PROPERTY OF REAL CLIFFORD ALGEBRAS. 
 
The periodicity theorem says that 
Cl(0,N+8) = Cl(0,8) x Cl(0,N)     (here x = tensor product) 
 
That means that the Clifford algebra Cl(0,16) of the 32-ons
is just the tensor product of two copies 
of the Clifford algebra Cl(0,8).  
 
So, everything that happens in the 32-on Clifford algebra 
is just a product of what happens with Cl(0,8).  
 
That point is emphasized by the fact 
(see Lohmus, Paal, and Sorgsepp, 
Nonassociative Algebras in Physics (Hadronic Press 1994))
that the derivation algebra of ALL Cayley-Dickson algebras 
at the level of octonions or larger, 
that is, of dimension 2^N  where N = 3 or greater, 
is the exceptional Lie algebra G2, 
the Lie algebra of the automorphism group of the octonions.
 
The exceptional Lie algebra G2 is 14-dimensional, 
larger than the 8-dimensional octonions, 
but smaller than the 16-dimensional sedenions.  

The Associative Triples are discussed by Guillermo Moreno (who uses the term "special triple" for them) in math/0512516

"... For n > 4 , Eakin-Sathaye showed that Aut(A_n) = Aut(A_(n-1)) x S_3 Where S_3 is the symmetric group of order 6 ...

We will describe the set M( A_m , A_n ) = { F : A_m -> A_n | F algebra monomorphism } ...

[ special triple = associative triple = associative triangle ]...

For a special triple {a,b,c} in A_n and n > 3 ... f(a,b,c) = Span{ e_0 , a , b , ab , c(ab) , cb , ac, c } ... is an eight-dimensional vecor subspace isomorphic, as algebra, to A_3 = f the octonions and M( A_3 , A_n ) = { (a,b,c) in (A_n)^3 | {a,b,c} special triple } ...

Suppose that n > 4 and that {a,b,c} is a special triple in A_n ... any orthonormal triple {x,y,z} of pure elements in f(a;b;c) with z perpendicular to (xy) is also a special triple in A_n ...

The main result of this paper is ...[that]... the set of type II monomprphsims from A_3 to A_(n+1) can be described by the set of zero divisors in A_(n+1) for n > 4 ... this set is ... complicated to describe ...".

Guillermo Moreno had written some background in an earlier paper "The higher dimensional Cayley-Dickson algebras" that he sent to me around the summer of 2000 for which I have no publication reference

"... the group of automorphisms ...[of]... the Cayley-Dickson algebras, denoted by A_n = R^(2^n) ... for n > 4 ... is isomorphic to G2 x F_n where F_n is a finite group, in fact F_n is the product of ( n - 3 ) copies of S_3 the symmetric group of order 6 (see ... Eakin-Sathaye, On automorphisms and derivations of Cayley-Dickson algebras, Jornal of Pure and Applied Algebra, 129, 263-278 (1990) ...) ...

question ... the real Stiefel manifold ... V_( 2^n - 1 , 2 ) consists ... of zero divisors an A_(n+1) for n > 3 ? ...

Any zero divisor in A_n is double pure ...

The zero divisors in A_n , n > 4 form real algebraic variety in ... R^(2^n - 2) ...

The set of non-zero divisors in A_n ( n > 4 ) is a open dense subset ...

singular elements ... are the zero divisors and regular elements are the non-zero divisors and ... the rank ... is r = 2 ...

The real algebraic variety defined by the zero divisors in A_n for n > 4 has at most ( 2^(n-2) - 2 ) irreducibles components ...

For ... n > 4 ... alpha = (a,b) ... =/= 0 ... is a zero divisor if and only if F(alpha) is a zero divisor ... for all F in O(2) ...

If alpha ... is a Stiefel element ... then F(alpha) is also Stiefel element for all F in O(2) ...".

 

In a later paper at math/0512517 Guillermo Moreno wrote about the "complicated to describe" zero divisors

"... For n = 4 the set of zero divisors in A_4 of fixed norm can be identified with V_(7,2) the real Stiefel Manifold of two frames in R^7 and the singular set of (x,y) = 0 and ||x|| = ||y|| = 1 is homeomorphicto G2 the exceptional simple Lie group of rank 2. ... The description of the zero divisors in A_4 is given by the known fibration G2 -pi-> V_(7,2) with fiber S3 [the 3-sphere] since all the nontrivial annihilators are 4 dimensional.

For n > 5 there is NO analogous description. We will show that the zero divisors are in A_(n+1) and V_( 2^n - 1 , 2 ) are related, but they are not equal and the corresponding singular set has (unknown) complicated description. ...

Any zero divison in A_n is double pure ... we define a suitable O(2)-action on the double pure elements of A_n ...

in contrast with the case n = 4 where the zero divisors must have coordinates in A_3 of equal norm ... this is no the case for A_5 ... Therefore the zero divisors in A_5 are "very far" to be described s in A_4 where they can be identified with V_(7,2) the Stiefel Manifold ...[ SO(7) / SO(5) ]... But also ... the set of zero divisors in A_(n+1) has some subset which can be describe in terms of the Stiefel Manifold V_( 2^n - 1 , 2 ) for n > 3 ...

... the set of Stiefel elements ... with entries of norm one ... can be seen as the real Stiefel manifold V_( 2^n - 1 , 2 )

... Is any Stiefel element ... a zero divisor ? ... Open Question ...".

M. Nakahara describes Stiefel Manifolds in "Geometry, Topology and Physics" (Adam Hilger (1990)

"... The Stiefel manifold V(m,r) is ... SO(m) / SO(m-r) ... The Stiefel manifold is, in a sense, a generalization of a sphere ... V(m,1) = S^(m-1) ... and dim V_(m,r) = [r(r-1)]/2 + r(m-r) ...". Therefore, dim V_(m,2) = 1 + 2( m - 2 ) = 2m -3 .

Note that Lie Spheres are described by the Conformal Group symmetric space SO(m) / ( SO(m-2) x SO(2) ) of dimension 2m - 4 .

Consider the Stiefel Manifolds V_(N,2) = SO(N) / SO(N-2) and Conformal Lie Spheres SO(N) / ( SO(N-2) x SO(2) ) and the fibrations:

V_(7,2) = SO(7) / SO(5)   ->          G2                  -> S3 = SU(2)
    dim = 11                        dim = 14                  dim = 3
   
   
SO(7) / (SO(5)xSO(2))     ->   V_(7,2) = SO(7) / SO(5)    -> SO(2) = U(1) = S1 
    dim = 10                        dim = 11                  dim = 1

 The Zero Divisors of A_n consist of a number of Irreducible Components, the maximum possible number of which is 2^(n-2) - 2 .

Each Zero Divisor Irreducible Component for A_n is related to the Stiefel Manifold V_( 2^(n-1) - 1 , 2 ) .

Consider the following sequence of Cayley-Dickson algebras A_n :

n   CDA          Im        AssocTriple      dim Aut(A_n)  dimLieSph maxIrComp MaxTotal

1  Complex       1             0               0 = Z/2                    0        
 
2  Quaternion    3             1               3 = SU(2)                  0

3  Octonion      7         1+6 =     7        14 = G2                     0

4  Sedenion     15        7+28 =    35       84 = 14x1x6       10         2       20

5  32-ons       31      35+120 =   155      168 = 14x2x6       26         6      156

6  64-ons       63     155+496 =   651      252 = 14x3x6       58        14      812

7  128-ons     127    651+2016 =  2667      336 = 14x4x6      122        30     3660

8  256-ons     255   2667+8128 = 10795      420 = 14x5x6      250        62    15500
 
9  512-ons     511                          504               506

Only the Quaternions, Octonions, and Sedenions have Excess Associative Triples, because only for them is the number of Associative Triples greater than the maximum number of Irreducible Zero Divison Components times the dimension of the core Lie Sphere of each Zero Divisor Component:

Quaternions have no Zero Divisors and only one Associative Triple.

Octonions have no Zero Divisors and 7 Associative Triples,

which correspond to the 7 Imaginary Octonion basis elements { i, j, k, E, I, J, K } and to a 7-vertex configuration called by Arthur Young the Heptahedron (also independently developed by Onar Aam), which is composed of the 6 vertices of an Octahedron plus a central point

as, for example, with one associative triple {i,j,k} being the vertices of a face of the Octahedron and the central element being {E}

in which alternate faces of the Octahedron correspond to associative triples. Since the Octonions have 7 associative triples, corresponding to the 7 Imaginary Octonions, the Heptaverton construction can be nested recursively

Note that the Octahedron is the Conformal Kepler representative of the two innermost planets Mercury and Venus.

The Octonion Recursive Structure corresponds to the 7 Imaginary Octonions, the 7 vertices of the Heptaverton, and the 7 Octonion Associative Triples. It is related to the Non-Unitary Physics of Octonions that is manifested in the Early Inflationary Phase of Our Universe.

Sedenions have 35 - 20 = 15 Excess Associative Triples,

which correspond to the 15 Sedenion Imaginary basis elements { i, j, k, E, I, J, K, S, T, U, V, W, X, Y, Z } and to the Rhombic Dodecahedron

which has 14 vertices plus a center (shown above as a stereo-pair image) and so the natural polytope to use, with 7 of the vertices corresponding to 6 octahedral vertices plus the center correspond to

the 7 pure Imaginary Octonionic basis elements (red dots - octahedron vertices plus center) { i, j, k, E, I, J, K } that represent 7 Excess Associative Triples that are beyond the 28 of the two Irreducible Component copies of 14-dim G2 (including the 10-dim LieSphere(7) at the core of G2 and the enhancing S3xS1 that extends LieSphere(7) to G2) - in other words, they correspond to the 7 A_3 Octonion Associative Triples while the 28 = 2x14 = 2 x dim(G2) correspond to the 28 Associative Triples that newly emerge with at the A_4 level of Sedenions -

while the remaining 8 cube-type vertices correspond to

the remaining 8 Sedenion basis elements (blue dots - cube vertices) { S, T, U, V, W, X, Y, Z } that represent two Irreducible Component copies of the non-core enhancing 4-dimensional S3 x S1 = SU(2) x U(1) = U(2) that extend/enhance the 10-dim LieSphere(7) to 14-dim G2 .

The 35 Sedenion Associative Triples correspond to the 7 Octonion Associative Triples (and therefore to the 7 Imaginary Octonions)

plus the 28 generators of Spin(8) which correspond to

the ordinary 7-sphere S7 (and therefore to a second copy of the 7 Imaginary Octonions) plus

the 21 generators of Spin(7) which correspond to

the 14 generators of G2 (and therefore 14 of the Sedenion Zero Divisors) plus

the Spin(7)/G2 7-sphere with torsion S7# (and therefore the 7-dimensional representation of G2 and, equivalently, a second copy of half of the 14 G2 Sedenion Zero Divisors).

Let the Mirror Sedenion S that maps { i, j, k, E, I, J, K } <-> { T, U, V, W, X, Y, Z } represent the first 7 of the 14 G2 Zero Divisors and be at a Rhombic Dodecahedron Cube-Vertex opposite W

and the Mirror Sedenion E that maps { i, j, k } <-> { I, J, K } represent the 7 Spin(7)/G2 Zero Divisors and be at the center of the Rhombic Dodecahedron,

and the Mirror Sedenion W that maps { T, U, V } <-> { X, Y, Z } represent second 7 of the 14 G2 Zero Divisors and be at a Rhombic Dodecahedron Cube-Vertex opposite S,

so that { S, E, W } represent the 21 Zero Divisor Sedenions of type Spin(7).

That leaves 3x4 = 12 Imaginary Sedenions { i, j, k } , { I, J, K } , { T, U, V } , and { X, Y, Z } which correspond to the remaining 12 vertices of the 14 vertices of the Rhombic Dodecahedron plus one central vertex,

so that, with { S, E, W } representing Zero Divisors, we can make a Sedenion Rhombic Dodecahedral Recursive Structure:

 Note that the Rhombic Dodecahedron is the Conformal Kepler representative of the two outermost planets Uranus and Neptune.

The Sedenion Recursive Structure corresponds to the 15 Imaginary Sedenion and the 15 Vertices of the Rhombic Dodecahedron plus Center, but it does NOT involve all of the 35 Sedenion Associative Triples.

The 35 - 15 = 20 Sedenion Associative Triples that are not involved in the Sedenion Rhombic Dodecahedron plus Center Recursive Structure correspond to the 20 dimensions of the Sedenion Zero Divisor subspace represented by two copies of the 10-dimensional LieSphere(7).

Each 10-dimensional LieSphere(7) has the structure of the Conformal Space of dimension 7 = 5+2 over a 5-dimensional manifold that has the structure RP1 x S4 of the Shilov Boundary of the Bounded Complex Domain related to the compact Hermitian Symmetric Space Spin(7 / Spin(5) x Spin(2) ) that has 21 - 10 - 1 = 10 Real dimensions and 10/2 = 5 Complex dimensions.

• The 10 dimensions of LieSphere(7) correspond to the vector space of Spin(10).
• Spin(10) is the Conformal Group over 8-dimensional Spacetime, whose symmetry group Spin(8) is a subgroup of Spin(100.
• The two copies of the LieSphere(7) in the Sedenions correspond to the two 28-dimensional components of the bosonic part of E8, each of which corresponds to a Spin(8) Lie algebra D4.
• The 5 dimensions of the related Shilov Boundary correspond to the rank of the Spin(10) Lie Algebra D5 as well as to the SU(5) subgroup of Spin(10).

Conjectural Construction of 496-dimensional E8 x H248

where E8 is the exceptional Lie Algebra and H248 acts as a Heisenberg Algebra

Sigurdur Helgason in his book "Geometric Analysis on Symmetric Spaces" (AMS 1994) said: "... The Heisenberg Group Hn .. .is the set Cn x R with the multiplication (z,t)(z',t') = ( z+z' , t+t'+Im(z*z') where z = ... column vector ... z* = ... row vector ...[and]... |z| = (z*z)^(1/2) ... The group Hn can be realized as the group of matrices

1 + (1/2)|z|^2 + it          z*            - (1/2)|z|^2 - it     

          z                  In                    z

    (1/2)|z|^2 + it          z*          1 - (1/2)|z|^2 - it

 with Lie algebra hn in sl(n+2,C) given by

          it                 z*                  -it

           z                 0                   -z 

          it                 z*                  -it

z in Cn, t in R. ... Hn is isomorphic to the group N in the Iwasawa decomposition of SU(1,n+1) ... if U is in U(n) the mapping (z,t) -> (Uz,t) is an automorphism of Hn ... ".

Barut and Raczka in their book "Theory of Group Representations and Applications" (World 1986) said: "... The Iwasawa decomposition ... for sl(n,R) is just the decomposition of an arbitrary real traceless matrix into a sum of a skew-symmetric, a traceless diagonal, and a ... nilpotent ... zero-triangular upper triangular real matrix. ... N ...".

J. M. Landsberg and L. Manivel in their paper "The Sextonions and E7 1/2" at math.RT/0402157 said: "... The adjoint variety Xad_H ...[in the projectivization]... Ph ...[of]... a complex simple Lie algebra ... h ... is the closed H-orbit in Ph , where H denotes the adjoint group of h . The adjoint variety parameterizes the highest root spaces: given a line L in h which corresponds to a point of Xad_H , we can chose a Cartan subalgebra of h and a set of positive roots, such that L = h_a , the root space of the highest root a. ...

The adjoint bariety Xad_G2 , the closed G2-orbit in Pg2 , parameterizes:

• (1) null-planes in O , i.e., planes on which the octonionic multiplication is identically zero;
• (2) rank two derivations of O , up to scalars;
• (3) six-dimensional subalgebras of O .

... we fix a six dimensional subalgebra S of O , denote it by S , and call it the sextonion algebra. ... over the real numbers ... We get a six-dimensional alternative algebra, with zero divisors. ...

the Lie algebra of type ... E7 ... has a representation V of dimension 2n ...[ 2n = 56 ]... which admits an invariant symplectic form w . Then ... E7 ... acts on the Heisenberg algebra of (V,w) and ... E7 x H56 ... denotes the semi-direct product. These algebras are not reductive and the Heisenberg algebra is the radical. ...".

Conjectural Questions:

Could you do something similar with E8:

• start with 248-dimensional E8
• then take the 248-dimensional smallest nontrivial representation of E8
• and put a symplectic structure on it
• and make a 248-dimensional Heisenberg algebra H248
• and
• then construct 496-dimensional semi-direct product E8 x H248 ?

Would its corresponding algebra be the 8+8 = 16-dimensional sedenions ?

Since the 6-dimensional sextonions inherit zero divisor structurefrom the Heisenberg algebra H_56 part of E7 x H56 could the sedenion zero-divisor structure be said to come from the Heisenberg algebra H_248 part of E8 x H248 ?

If so, could you represent 496-dim Ternary Sedenions be represented by E8 x H248, i.e, does

248-dim H248 Weyl-Heisenberg Algebra + 248-dim E8 Lie algebra

produce 496-dim Ternary Sedenions ?

 E8

E8 grading = 8 + 28 + 56 + 64 + 56 + 28 + 8

Dennis W. Marks in his paper A Binary Index Notation for Clifford Algebras (revised 27 February 2003) said: "... Duality operations generate isomorphism between grades k and n-k. There are several different duals, including

• the Clifford dual, which we will write as e_m*, defined ... as e_m* = e_m J_n,
• and the Hodge dual, which we will write as m_e_*, defined as m_e_* = (m_e) J_n = (-1)^( k_m ( k_m - 1 ) / 2 ) e_m J_n = (-1)^( k_m ( k_m - 1 ) / 2 ) e_m* ...
• bit inversion that maps e_m -> e_mbar , where mbar is the bit inverse of m ... In particular, bit inversion transforms vectors (grade 1) ... into covectors (grade n-1) ...

Complementarity between space-time and momentum-energy is achieved by bit inversion, which interconverts between position representation and momentum representation. Treating momentum as a Clifford covector has the virtue of automatically enforcing the Heisenberg commutation relation as a consequence of the commutation and anti-commutation propeerties of the Clifford elements. ...".

Odd E8 grading 8 + 56 + 56 + 8 = 128 = 8 + 56 Fermion Particles + 56 + 8 Fermion AntiParticles

Duality for the Odd part of E8 is a Particle-Antiparticle duality that represents a Heisenberg Algebra for Second Quantization Fermion Creation-Annihilation. Anthony Sudbery in his book Quantum Mechanics and the Particles of Nature (Cambridge 1986, 1989) discusses "... Dirac spinors whose components are annihilation operators and creation operators respectively ...[for]... the electron field ... are annihilation operators for electrons and ... creation operators for positrons. ... Dirac fermion... cration and annihilation operators obey cnatcommuttion relations... particles and antiparticles have opposite helicity ... in order to change to a frame of reference in which a particle has opposite helicity, it would be necessary to overtake the particle and look at it from the other side ... this is not possible if the particle is travelling at the speed of light [as can be the case for massless particles] ... the many-particle theory is called second quantization ...".

Even E8 grading 28 + 64 + 28 = 28 Position Gauge Bosons + 64 + 28 Momentum Gauge Bosons

Duality for the 28 + 28 part of the Even part of E8 is a Position-Momentum duality that represents a Heisenberg Algebra for First Quantization of Gauge Boson Position-Momentum representations. Each 28 contains 16 + 12 to describe Gravity by 16-dim U(2,2) and its Spin(2,4) subgroup plus the Standard Model by its 12-dimensional Gauge Group.

The 64 part of the Even part of E8 is Self-Dual, and corresponds to U(8) whose elements, as stated above by Helgason, are automorphisms of the Heisenberg algebra H8. Each of the 8 of the U(8) Even Part elements (a Cartan subalgebra) will combine with each of the 8 Fermion Particle-AntiParticle Odd Part sets to produce 8 H8 Heisenberg Algebras for Second Quuantization Fermion Creation-Annihilation within E8.

E8 has automorphism-like action on H8 and therefore on H248.

H248

H248 grading = 8 + 28 + 56 + 64 + 56 + 28 + 8

odd = 8 + 56 + 56 + 8 = 128 = 1x8 + 7x8  +  7x8 + 1x8 


even = 28 + 64 + 28 = 120 = 7+21 + 63+1 + 7+21 = 7+1+7 + 21+63+21 = 
     = 15 + 105 = 15 + 7x15 = 15 + 21 + 63 + 21 = G2 + U(1) + Sp(3) + SU(8) + Sp(3) = 
     = G2 + U(1) + C7

C7 = Sp(7) = 7x15 = 105 = 21 + 63 + 21 = Sp(3) + SU(8) + Sp(3) = C3 + SU(8) + C3


C3 = Sp(3) = 3x7 = 21 contains Sp(2) = Spin(5) = 10-dim deSitter/Poincare so
the first C3 gives MacDowell-Mansouri gravity

     Sp(3) / U(3) is 21-9 = 12-dim rank 3 space with isotropy rep = U(3)xU(3) 
     U(3)xU(3) contains the Standard Model, so 
the second C3 contains the Standard Model


8 +     28 +  56 +    64    +  56 + 28     + 8   = H248
 
8 +     28 +  56 + 16+32+16 +  56 + 28     + 8

        28       + 16+ 2+16       + 28           correspond to 28+16+1 + 1+16+28 = D5 + D5 
                                                 two position-momentum dual copies of D5 
                                                 Bounded Complex Domain for D5 / D4xU(1) 
                                                 has Shilov Boundary = 8-dim spacetime
                                                 Shilov Boundary for D3 / D2xU(1) represents 
                                                 4-dim physical spacetime of 8-dim Kaluza-Klein
                                                 and D3 = Spin(2,4) = SU(2,2) = A3 
                                                 Conformal 6-dim space over 4-dim physical spacetime 

8 + 1+6+21 + 7x8 + 16+32+16 + 7x8 + 21+6+1 + 8

                       1                         U(1) of U(8) = SU(8)xU(1)

        21       + 16+31+16       + 21           C7 = Sp(7) = C3 + SU(8) + C3 
                                                 Nilpotent Iwasawa part of SU(8) is isomorphic 
                                                 to the H6 Heisenberg algebra of position-momentum 
                                                 spacetime first quantization related to 
                                                 6-dim Conformal space over 4-dim physical spacetime

8                    + 1                   + 8   Heisenberg algebra H8 with Complex U(1) grade 0 
                                                 of 8 covariant components of Neutrino-AntiNeutrino
                                                 Creation-Annihilation second quantization

    1      +   8            +   8       +1       Heisenberg algebra H8 with Lorentz R(1,1) grade 0 
                                                 of 8 covariant components of Electron-Positron
                                                 Creation-Annihilation second quantization

      1    +   8            +   8     +1         Heisenberg algebra H8 with Lorentz R(1,1) grade 0 
                                                 of 8 covariant components of R Up Quark-AntiQuark
                                                 Creation-Annihilation second quantization
 
      1    +   8            +   8     +1         Heisenberg algebra H8 with Lorentz R(1,1) grade 0 
                                                 of 8 covariant components of G Up Quark-AntiQuark
                                                 Creation-Annihilation second quantization
     
      1    +   8            +   8     +1         Heisenberg algebra H8 with Lorentz R(1,1) grade 0 
                                                 of 8 covariant components of B Up Quark-AntiQuark
                                                 Creation-Annihilation second quantization

      1    +   8            +   8     +1         Heisenberg algebra H8 with Lorentz R(1,1) grade 0 
                                                 of 8 covariant components of R Down Quark-AntiQuark
                                                 Creation-Annihilation second quantization

      1    +   8            +   8     +1         Heisenberg algebra H8 with Lorentz R(1,1) grade 0 
                                                 of 8 covariant components of G Down Quark-AntiQuark
                                                 Creation-Annihilation second quantization

      1    +   8            +   8     +1         Heisenberg algebra H8 with Lorentz R(1,1) grade 0 
                                                 of 8 covariant components of B Down Quark-AntiQuark
                                                 Creation-Annihilation second quantization


The Null-Spaces of Symplectic 248 come from 
the 8 Heisenberg algebras H8 each of which have Octonionic Structure.  
G2, as the automorphism group of the Octonions of the Octonionic Structure, 
corresponds to the G2 symmetry of the Sedenion Zero-Divisors. 

The Iwasawa Nilpotent SU(8) H6 Heisenberg algebra of Position-Momentum spacetime First Quantization is related to 6-dim Conformal space over 4-dim physical spacetime,

which is in turn related to the Lie Ball geometry of Bohmian Quantum Theory.

In my E8 Physics model high-energy 8-dimensional spacetime is represented by the 8-Complex-dimensional Complex Bounded Domain Lie Ball for the Symmetric Space Spin(2,8) / Spin(1,7)xU(1) Lie Ball

shown as a white Lie Ball interior for RP1 = 1-dim Imaginary Complex Numbers and as interior with Octonion Imaginary Heptagram for S7 = 7-dim Imaginary Octonions of the RP1 x S7 Lie Sphere Shilov Boundary of the Lie Ball.

At low energies, a Quaternionic substructure freezes out producing a Kaluza-Klein spacetime M4 x CP2 in which the 4-dimensional CP2 Internal Symmetry Space is represented by the 4-Real-dimensional symmetric space CP2 = SU(3) / SU(2)xU(1) carrying the Standard Model Gauge Groups Color SU(3), Weak SU(2), and ElectroMagnetic U(1) according to a geometric mechanism described by Batakis and the 4-dimensional M4 spacetime is represented by the 4-Complex-dimensional Complex Bounded Domain Lie Ball for the Symmetric Space Spin(2,4) / Spin(1,3)x U(1)

shown as a white Lie Ball interior for RP1 = 1-dim Imaginary Complex Numbers and as interior with Quaternion Imaginary Triangle for S3 = 3-dim Imaginary Quaternions of the RP1 x S3 Lie Sphere Shilov Boundary of the B4 Lie Ball.

Misha Gromov in his book "Metric Structures for Riemannian and Non-Riemannian Spaces" (Birkhauser 2001), and Stephen Semmes in Appendix B, said: "... Let B4 be equipped with its Bergman metric ... so that B4 is isometric to the complex hyperbolic plane,

The action of U(1,2) also preserves [the boundary] bB4 ... and the field of 2-planes on bB4 .... the stabilizer of a point of bB4 contains a subgroup isomorphic to the Heisenberg group ...".

R. Coquereaux says, in his paper " Lie Balls and Relativistic Quantum Fields", Nuc. Phys. B. 18B (1990) 48-52: "... In the present paper, we are mainly interested in the four dimensional (complex) Lie ball [ B4 ] that we shall denote by D. This smooth manifold can be written as SOo(4,2) / SO(4)xSO(2) or as SU(2,2) / S(U(2)xU(2)). ... D is a bounded non compact symmetric domain of type I and IV. ... The metric of D is euclidean and blows up near the boundary (as in the usual geometry of Lobachevski) but ... induces a conformal Lorentz structure on the boundary. The domain D is a Lie ball ... the Shilov boundary (compactified Minkowski Space-Time) can be defined as the Lie sphere ... The domain D also admits an unbounded realization: the future tube. ... This ... unbounded realization of the Lie ball admits a simple physical interpretation ... the imaginary part y of z = x + iy can be interpreted as the inverse of a momentum ... Points of the domain D describe therefore both the position (in space and time) and the momentum (with p^2 > 0) associated with a physical event. The domain itself becomes therefore a curved relativistic phase-space. ... The group SO(4,2) ... as the group of conformal transformations for a Lorentzian Space-Time of signature (1,3) coincides with the group of analytical diffeomorphisms of the Lie ball D ... [Using] the physical interpretation of Space-Time with the Shilov boundary ... S ... of D together with the interpretation of the imaginary part of the complex variable ... as the inverse of a momentum ... a relativistic analogue of the wavelet transform should be defined via ... the Szego kernel ... s(z,Z) ...[whereby]... any holomorphic bounded function in D can be written as

where du is a measure on S ... Physics is "simple" (and euclidean) in the domain D and ... many of the difficulties of classical or quantum physics arise because we try to go to the "boundary" and to formulate the laws of Physics there ... what we have here is rather a (Radon)-Gelfand-Graev transformation - i.e. integration over horocycles - and its physical interpretation is quite different ...[from].. the Fourier transformation ...".

R. Coquereaux and A. Jadczyk say, in their paper "Conformal Theories, Curved Phase Spaces, Relativistic Wavelets and the Geometry of Complex Domains" (Reviews in Mathematical Physics Volume 2 No 1 (1990) 1-44): "... The domain D4 has been proposed .... as a conformal relativistic phase space ... D4 is ... topologically homeomorphic to the eight-dimensional open ball B8 ... its ... topological boundary ... in C4 ... is homeomorphic to the seven-sphere S7 ... The Shilov boundary Sh4 ... also called a Lie sphere ... is a subset of ... the ... topological boundary of ... D4 ... made of points where the Euclidean distance ... is equal to the Lie distance ...

... this generalized Klein bottle (the n = 2 case) can be identified, for n = 4 with compactified space-time. ... Physically ... start with space-time Sh4 ... up to a Z2-factor it has the topology of S3 x S1 ... choose a fixed "event" and consider the corresponding 3-sphere S3 = S3 x {t_0} ... this set ... can be considered as the spatial universe at time t_0 ... we can then fill it in ...[to]... get a four-dimensional ball whose boundary is "space" at time t_0 ... If we now consider a massless scalar field on this space S3 at time t_0, we can propagate it "in time" to a later time T_1 by using the wave equation (Dalembertian), but we can also

• extend it to the "inside" of S3, t = t_0 by a solution of the Laplacian in the ball,
• then extend it to the harmonicity cell D4 by analytic continuation using a holomorphic map and
• finally consider its radial limit to the Shilov boundary at a later "time" t_1.

... if we consider a fixed S3 hypersurface t = t_0 in compactified Minkowski M = S3 (x_Z2) S1 and the two-spheres inside this S3, each two-sphere determines a causality diamond (past and future lightcones) with apexes z1 , z2 in Mikowski M3,1 . Therefore, up to a Z2 factor, the space of S2's is just M3,1 = SO(4,2) / ( P x R+ ) where the little group is the semi-direct product of the Poincare group P and dilations R+ . The conformal group acts on these spheres ... the space of two-spheres in B4 coincides with the eight-dimensional Cartan domain D4 ... up to a discrete Z2 factor ... The space of two-spheres in the three-sphere S3 can be identified with its Shilov boundary, the compactified Minkowski space-time M(3,1) ( up to a discrete Z2 factor ) ... A light-cone in (compactified) Minkowski M is fully characterized by its origin so that the space of these light-cones is M itself so it can identified with M = SO(4,2) / ( P x R+ ) . ...

The Lelong transformation ... T maps points of Cn to spheres of dimension n - 2 included into Rn. If Z is in Cn, then T(Z) is the S(n-2)-sphere centered in a = Re(Z), of radius ||b|| contained in the hyperplane of equation < u-a , b > = 0 ... that ... cuts the S(n-1)-sphere of center a and radius ||b|| along an equatorial S(n-2)-sphere. ... the map T is two-to-one ... T(Z) = T(Z') <=> Z' = Z* ... if Z = a in Rn then T(a) = a ... it is an (n-2)-sphere of dimension zero ... the space of S(n-2)-spheres of Rn is of dimension 2n since a given element is determined by the center (n parameters), the radius (1 parameter) and the choice of the (n-1) hyperplane in which it is embedded (n-1 parameters). In particular, if n = 4 then the space of S2 spheres of R4 is 8-dimensional. ... the image T(Z) of a point Z in Sh4 is a 2-sphere included in the 3-sphere S3 ... when an observer at time t_0 contemplates the sky two-sphere T(Z) ... centered on him, he should remember that this ... 2-sphere ... defines a causality diamond or .... double-cone with apexes Z and Z* .... if we take Z in D4 and not on the Shilov boundary ... the corresponding Lelong 2-sphere lies "in" the euclidean ball B4 but not "in" its boundary S3 ... In the case n = 1 , it is not possible to define D1 as the harmonicity cell of the interval B1 = ]-1,+1[ because this would lead to the entire complex plane. However, the Lie norm coincides in this case with the usual norm, therefore the Lie ball D1 coincides with the unit disk ...".

Arkadiusz Jadczyk says in his paper "Born's Reciprocity in the Conformal Domain" ( in "Spinors, Twistors, Clifford Algebras and Quantum Deformations, ed. by Z. Oziewicz et al. (Kluwer 1993 pp. 129-140): "... Max Born ... in ... 1938 ... in ... Edinburgh ... introduced his 'principle of reciprocity' ... a primary symmetry between coordinates and momenta ... He explained that ... in the lattice theory of crystals ... the motion of the particle is described in the p-space with help of the reciprocal lattice ... we assume that the fundamental arena ... in which relativistic quantum processes take place is ... the Cartan domain D4 = SU(2,2) / S(U(2)xU(2)) = SO(4,2) / S(O(4)xO(2)) ... D4 has positive-definite metric ... the Shilov boundary of D4 ... is naturally isomorphic to the (compactified) Minkowski space endowed with its indefinite conformal structure ... the points of D4 ...[are]... elementary micro event-processes ... time inversion is not a symmetry of D4 - it would change the complex structure into the opposite one ... when realizing D4 as part of the Grassmannian in C4 one gets automatically two copies of the domain ... time inversion can be thought of as the trasposition of these two copies ... D4 has constant curvature ... the primary arena of elementary event-processes is homogeneous ... topologically ... D4 is ... the future tube of the Minkowski space ... we propose to consider D4 as the replacement for space-time on the micro scale ...".

Harald Upmeier says in his paper "Weyl Quantization of Complex Domains": "... a quantization of a classical phase space M ... on a state space H .... is a linear map

Q: Fun(M) -> Op(H)             f -> Q(f)

from ... functions on M ("classical observables") to symmetric operators on H ("quantum observables"). ... The map Q .. satisfies

Q(f) Q(g) = Q(fg) + (h/i) Q({f,g}) + higher orders . Here h is ... Planck's constant ... and {f,g} is the classical Poisson bracket on M. ...

suppose G is a Lie goup, acting on M and on H ... Weyl quantization. ... with position and momentum coordinates Z = (q,p) ... 2Z in R2n ... is regarded as an element in ... the nilpotent Heisenberg group G = H_(2n+1) ... the classical Weyl quantization ... is given by ...

over the Schrodinger-quantized symmetries of the phase space M ... the "Weyl operators" make sense for every symmetric space M ... and every representation P of its symmetry group G ... Every ... bounded symmetric domain ... D ... has a Harish-Chandra realization as the open unit ball of a complex vector space V = Cn, with respect to some norm ... The vector space Z carries a ternary operation u,v,wq -> {uv*w} (the so-called Jordan triple product, symmetric bilnear in (u,w) and anti-lnear in v) and ||z|| is the spectral radius of the operator u -> {zz*u}, acting on Z. ... the Jordan triple product is a generalized anti-commmutator ... Every Jordan triple system Z carries a "determinant function" N(z,w) (specializing to N(z,w) := Det(I - zw*) for matrices), and for a certain power of N, K(z,w) := N(z,w)^(-p) coincides with the Bergman kernel function of D ... For L > p -1, the Hilbert space ... of holomorphic functions has the reproducing kernel N(z,w)^(-L) ... The natural parameter in the Weyl calculus ... is L, with L -> oo being .. the "classical limit" case. In Berezin's approach ... h = p / L ( p = genus of D ) anc consequently L -> +oo is the classical limit h -> 0.

In the Weyl calculus ... things are more complicated ... as L -> +oo , one obtains a new non-classical "Fuchs calculus" on the half-space realization ...[ D = A + iX ]... A is a symmetric cone in a real vector space X whose complexification X + iX agrees with Z. For trhe unit disk D in C, A is the half-line in X = R. ...

The invariant differential operators on D (called "higher Laplacians" in the physics literature) ... form a polynomial algebra in r generators d1, d2, ..., dr [of]... order 2, 4, ..., 2r, respectively ( d1 is the ordinary non-euclidean Laplacian on D ) ...

the Lie balls of rank 2 .. have immediate interpretations in physical terms, since D is related to I. Segal's conformal universe and also to the forward light cone ...".

Bohm and Hiley say, in their book "The Undivided Universe" (Routledge 1993, section 15.9): "... the theory of the higher spherical geometry of Lie and Klein ... relate[s to]... light cones. ... Each light cone intersects ...[a]... hyperplane ... say at t = 0 ... in a sphere. Those light cones whose vertices have t < 0 will be defined to produce spheres of positive radius r = t. The light cones whose vertices have t > 0 will also produce spheres but these will be assigned negative radii. So there is a 1 : 1 correspondence between ight cones and their intersections with any hyperplane t = const. ... Lie and Klein proposed ... an interesting parameterization of these spheres ...

[where]... the coordinates of the center are (a,b,c) and the radius is r ... introduce homogeneous coordinates by writing ...[

]... The advantage of this parameterization is that the conformal invariance of the description is now evident ...[it]... implies the invariance to six dimensional pseudo-orthogonal transformations which are ... isomorphic to the conformal group. ... the condition for two spheres to come into contact ... is

where S and Y refer respectively to the spheres in question ... if two spheres are in contact, then the vertices of their corresponding light cones are connected by a null ray ...

... Let us now consider our trajectory starting from a sphere of zero radius representing a light cone with vertex at t = 0 . The first step in our trajectory ...

... is represented by the point of contact P and a sphere of radius r.

• The corresponding null ray is in the direction of the radius of the sphere at the point P.
• The next null ray will be represented by a larger sphere contacting the first sphere at the point Q.
• The next null ray will correspond to a still larger sphere contacting the second sphere at another point R.

This procedure is to be continued indefinitely so that we obtain a complete description of the zigzag trajectory. ... we have described our trajectory entirely in terms of wave forms at any given time. ... this description is conformally invariant. ... in a time displacement, dt, the radius of each sphere changes by dt. ... in a Lorentz transformation, however, the radii and centers change together so that initially concentric spheres cease to be concentric. ... our trajectory is now defined in an implicate order. As this order unfolds in time, the radii of the spheres corresponding to incoming waves decrease while those corresonding to outgoing waves increase. The entire trajectory is now shown as a kind of enfolded geometric structure ... The law of development is implicit in this structure. ... Feynman describes his trajectories as a sequence of points in time. We describe them as sequences of spheres all enfolded at a given time. The Lagrangian is thus, in our approach, a property of the implicate order that holds at any given moment ... backwards tracks in time are replaced by tracks in which the implication parameter is decreasing. In our model, the implication parameter is just the radius of the sphere. Thus ...

... we could have the radii of spheres decrease for a while and then increase again. ... this corresponds to pair production. ...".

The physics of the Interior of the physical spacetime Lie Ball B4 is the Quantum Physics of Bohmian Quantum Theory described by mathematical structures such as Heisenberg Algebras, Bergman Kernels, Szego Kernels, Geodesics and Horospheres. If properly understood, that physics might describe Quantum Consciouness Resonant Connections between, for example, two human brains located at different points on the M4 spacetime Lie Sphere Shilov Boundary of the B4 Lie Ball:

 

Relevant mathematical techniques include the Radon Transform generalizations of Fourier Transforms.

Sigurdur Helgason, in Geometric Analysis on Symmetric Spaces (AMS 1994), describes Radon Transforms, saying:"...

let G be a connected semisimple Lie group with finite center,

• X = G / K the associated symmetric space where K is a maximal compact subgroup of G.
• Let G = K A N be an Iwasawa decomposition ... [ K compact, A abelian, N nilpotent ] ...
• let M denote the centralizer of A in K,
• let W denote the Weyl group ... and |W| its order ...
• B = K / M is identified with the set of all Weyl chambers ... K acts transitively on the set of all Weyl chambers ...

Given x in X, b in B there exists a unique horocycle passing through x with normal b. ... Given a horocycle h and a point x in h there exist exactly |W| distinct horocycles passing through x with tangent space at x equal to h_x. ... Each horocycle is a closed submanifold of X ... The group G acts transitively on the set of horocycles in X. The subgroup of G which maps ... [a] horocycle ... into itself equals M N. ... The set of horocycles in X with the differentiable structure of G / M N ... is ... the dual space of X and denoted by Z ... Horocycles are classical objects for hyperbolic spaces Hn and are considered ... [by] Gelfand and Graev ... as orbits of the conjugates of the nilpotent group N. ... X and Z have the same dimension and show many analogies reminiscent of the duality between points and hyperplanes in Rn. ... the coset space G / M N is not symmetric in general ... it does possess curves resembling geodesics. ... the case of bounded symmetric domains D. A harmonic function on D which is continuous on [the closure of] D has a Poisson integral representation involving the Shilov boundary S(D) ... [let * denote dual space] ... For a polydisk G* / K* = D* the Shilov boundary is the product B* of the (circle) boundaries. ...".

In his review (Bull. A.M.S. 32 (1995) 441-446)) of Helgason's book Geometric Analysis of Symmetric Spaces (AMS 1994), Francois Rouviere asks a question: "...What are ... natural substitutes for the Euclidean lines in the hyperbolic unit disk X = H2 = SU(1,1) / SO(2) ?

• first answer: take as Z the set of all geodesics of X (circles orthogonal to the unit circle). The corresponding Radon transform may be called a generalized X-ray transform, recalling ... the mathematical theory of tomography ...
• A second answer is: take Z as the set of all horocycles of X, that is, the "wave surfaces" orthogonal to a "parallel beam of rays" (geodesics meeting at infinity on the unit circle). The horocycles are thus all circles inwardly tangent to the unit circle.

Both settings are considered in the book ... Helgason deals with the following Radon transforms:

• integration over k-dimensional totally geodesic submanifolds in a space with constant curvature: Rn, Hn, Sn ("geodesic transform");
• Radon transform for Grassmannians, wiht a specific incidence relation between p-planes and q-planes in R(p+q+1);
• Poisson integrals for bounded domains: for example, if G = SU(1,1) acting on the complex plane and if K and H are the isotropy subgroups of the points 1 and 0 respectively, then X is the unit circle, Z is the unit disk, and Rf is the classical Poisson integral of f;
• integration over horocycles in a Riemannian symmetric space of the noncompact type ("horocycle transform") ...

... when dim X < dim Z and the range of R can be characterized as the kernel of a certain differential operator on Z ... This extends the Poisson integral example ... where X is the circle, Z is the disk, and the range of R is known to be the kernel of the Laplace operator on the disk. ... The Fourier transform ... on a Riemannian cymmetric space of the noncompact type X = G / K ... is ... f^(l,b) = INTEGRAL(X) f(x) exp( - i l + r , A(x,b) dx . It is inverted by

... as a function of x, the exponential is an eigenfunction of all G-invariant differential operators on X ... Horocycles are ... level surfaces of A(x,b) for fixed b ... the Plancherel measure | c(l) |^(-2) dl db involves Harish-Chandra's celebrated, and explicitly known, c-function. ... the Poisson kernel for the unit disk is an exponential of the function A(x,b) ...".

Frank Dodd (Tony) Smith, Jr. - 2010

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288 and 22 Letters

There are 22 letters (not including Finals) in the Hebrew alphabet.

The 4-dimensional D4+ HyperDiamond Lattice of the 4-dimensional Physical Spacetime part of 8-dimensional Kaluza-Klein Spacetime has 288 vertices of in the layer of norm 22 distance from the origin (where the norm is the square norm usually used in lattice theory, that is, the inner product x.x of the vector x).

How to visualize the 288 vertices in layer 22.:  

• Start with the 24 vertices of the 24-cell.
• Then consider 96 more vertices placed on each of the 96 edges of the 24-cell.
• Then consider 24 more vertices placed in each of the the 24 cells (octahedra) of the 24-cell.
• These 24 + 96 + 24 = 144 vertices correspond to the 144 vertices in each of layers 10, 17, and 20, and they correspond to half of the 288 vertices in layer 22.

Layer 22 with 288 vertices follows layer 21 with ( 1 + 3 + 7 + 21 ) x 8 = 256 = 16x16 = 2^8 vertices.

Here are the numbers of vertices in some of the layers of the D4+ lattice. The even-numbered layers correspond ot the even D4 sublattice:

m=norm of layer             N(m)=no. vert.
   0                                 1
   1                                 8  =    1 x 8
   2                                24  =    1 x 24
   3                                32  =  ( 1 + 3 ) x 8 
   4                                24  =    1 x 24
   5                                48  =  ( 1 + 5 ) x 8
   6                                96  =  ( 1 + 3 ) x 24 
   7                                64  =  ( 1 + 7 ) x 8
   8                                24  =    1 x 24
   9                               104  =  ( 1 + 3 + 9 ) x 8
  10                               144  =  ( 1 + 5 ) x 24
  11                                96  =  ( 1 + 11 ) x 8
  12                                96  =  ( 1 + 3 ) x 24
  13                               112  =  ( 1 + 13 ) x 8
  14                               192  =  ( 1 + 7 ) x 24 
  15                               192  =  ( 1 + 3 + 5 + 15 ) x 8 
  16                                24  =    1 x 24
  17                               144  =  ( 1 + 17 ) x 8
  18                               312  =  ( 1 + 3 + 9 ) x 24
  19                               160  =  ( 1 + 19 ) x 8
  20                               144  =  ( 1 + 5 ) x 24
  21                               256  =  ( 1 + 3 + 7 + 21 ) x 8
  22                               288  =  ( 1 + 11 ) x 24
  23                               192  =  ( 1 + 23 ) x 8
  24                                96  =  ( 1 + 3 ) x 24
  25                               248  =  ( 1 + 5 + 25 ) x 8
  26                               336  =  ( 1 + 13 ) x 24
  27                               320  =  ( 1 + 3 + 9 + 27 ) x 8
  28                               192  =  ( 1 + 7 ) x 24
  29                               240  =  ( 1 + 29 ) x 8
  30                               576  =  ( 1 + 3 + 5 + 15 ) x 24
  31                               256  =  ( 1 + 31 ) x 8 
  32                                24  =    1 x 24
  33                               384  =  ( 1 + 3 + 11 + 33 ) x 8
  34                               432  =  ( 1 + 17) x 24
  35                               384  =  ( 1 + 5 + 7 + 35 ) x 8
  36                               312  =  ( 1 + 3 + 9 ) x 24
  37                               304  =  ( 1 + 37 ) x 8 
  38                               480  =  ( 1 + 19 ) x 24
  39                               448  =  ( 1 + 3 + 13 + 39 ) x 8
  40                               144  =  ( 1 + 5 ) x 24
  41                               336  =  ( 1 + 41 ) x 8
  42                               768  =  ( 1 + 3 + 7 + 21 ) x 24
  43                               352  =  ( 1 + 43 ) x 8
  44                               288  =  ( 1 + 11) x 24
  45                               624  =  ( 1 + 3 + 5 + 9 + 15 + 45) x 8

The notation in the following table is based on the minimal norm of the D4 lattice being 1, in which case the D4 lattice is the lattice of integral quaternions. This is the second definition (equation 90) of the D4 lattice in Chapter 4 of Sphere Packings, Lattices, and Groups, 3rd edition, by Conway and Sloane (Springer 1999), who note that the Dn lattice is the checkerboard lattice in n dimensions.

m=norm of layer             N(m)=no. vert.      K(m)=N(m)/24
   1                                24                  1
   2                                24                  1
   3                                96                  4
   4                                24                  1
   5                               144                  6
   6                                96                  4
   7                               192                  8
   8                                24                  1
   9                               312                 13
  10                               144                  6
  11                               288                 12
  12                                96                  4
  13                               336                 14
  14                               192                  8
  15                               576                 24
  16                                24                  1
  17                               432                 18
  18                               312                 13
  19                               480                 20
  20                               144                  6
 
 127                             3,072                128
 128                                24                  1
 
65,536=2^16                         24                  1
65,537                       1,572,912             65,538
 
2,147,483,647           51,539,607,552      2,147,483,648
2,147,483,648=2^31                  24                  1 

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Perfect Numbers

The Perfect Numbers are numbers that are themselves the sum of their proper factors:

0 = 0 
  = dimension of the D0 Lie Algebra of Spin(0) = {-1,+1} 

1 = 1 
  = dimension of the D1 Lie Algebra of Spin(2) = U(1)

6 = 1 + 2 + 3 
  = dimension of the D2 Lie Algebra of 
  Spin(4) = Spin(3)xSpin(3) = SU(2)xSU(2) = Sp(1)xSp(1) + S3 x S3 
(related to the Mersenne Prime 3 = 2^2 -1)

28 = 1 + 2 + 4 + 7 + 14 
   = dimension of the D4 Lie Algebra of Spin(8)
(related to the Mersenne Prime 7 = 2^3 -1)

496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248  
    = dimension of the D16 Lie Algebra of Spin(32)
(related to the Mersenne Prime 31 = 2^5 -1)

8,128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064  
      = dimension of the D64 Lie Algebra of Spin(128)
(related to the Mersenne Prime 127 = 2^7 -1)

33,550,336 
(related to the Mersenne Prime 8,191 = 2^13 -1)

and larger numbers

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References:

A web site including the page at www.mechon-mamre.org/p/pt/pt0101.htm  has a side-by-side Hebrew-English version of Genesis which is used in this web page.

Frank Dodd (Tony) Smith, Jr. - 2010