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Halayudha, Hexagons,Hypercubes, Hermes &Plato:


Halayudha was a Jaina mathematician (ca. 300-200 BC) credited by John McLeish, author of The Story of Numbers, published by Fawcett Columbine in 1991, with discovering the Binomial Triangle that is often called the Yang Hui triangle or Pascal's triangle.  Ernest G. McClain, author of The Myth of Invariance, published by Nicolas-Hays in 1976, says that Halayudha was actually only explaining what Pingala had said about 200 BC, that the mathematical Binomial Triangle represented Mount Meru, or Meru Prastara, the Holy Mountain, and could be used to describe the number of forms of long or short syllables that can be formed from a given number of syllables. It seems likely to me that the Binomial Triangle is actually very much older than about 200 BC, and that our discovery of written records of it only (as of now) goes back that far.   The Nth row of the Binomial Triangle is the expansion of 2^N = (1+1)^N  The powers-of-2 triangle rows through N=3 are:  N 0 1 2^0 = 11 1 1 2^1 = 22 1 2 1 2^2 = 4 3 1 3 3 1 2^3 = 8 The triangle of row N has the graded structure of the exterior algebra underlying the Clifford algebra Cl(0,N) of the Lie algebra Spin(0,N).  The Jth entry of row N is the number (N,J) = N! / (N-J)! J! of J-dimensional sub-simplexes of an N-dimensional simplex, or hypertetrahedron. Their total number is 2^N.  2^N is the dimension of the vector space of the Clifford algebra Cl(0,2^N) with Lie algebra Spin(0,2^N).  Note that for N=3, 2^N = 2^3 = 8 = 6+2 = 2N+2.  Note that 1, 2, 4, and 8 are the dimensions of the real division algebras: Real Numbers; Complex Numbers; Quaternions; and Octonions.  
 Hexagonal Numbers of order N are the number of points on a hexagonal pattern nested N times.  The hexagonal patterns through N=3 are:   3 3 3 3  2 2 2 3  1 1 2 3  0 1 2 3   1 1 2 3   2 2 2 3  3 3 3 3 The hexagonal numbers through N=3 are:  N  0 1 = 0 + (0 + 1 + 0) 1 6 = 1 + (2 + 1 + 2) 2 15 = 6 + (4 + 1 + 4) 3 28 = 15 + (6 + 1 + 6)  The hexagonal numbers of order N have the structure of the D(N+1) Lie algebra Spin(0,2N+2) of the Clifford algebra Cl(0,2N+2).  Note that 6 and 28 are both perfect numbers: 6 = 1x2x3 = 1+2+328 = 1x2x4x7x14 = 1+2+4+7+14 
 Hypercubes in N dimensions have 2^N vertices.  There are 2^(N-J) (N,J) sub-hypercubes of dimension J in an N-dimensional hypercube.  The total number of sub-hypercubes in an N-dimensional hypercube is 3^N = (2+1)^N.  It is a power of 3 triangle, while the Halayudha triangle is a power of 2 triangle.  The rows of the power of 3 triangle are the string product of  a spinor string (so called because it is related to spinors) and a wedge string (so called because it is related to the exterior product).  ( 2^N, 2^(N-1), 2^(N-2), ... , 2^(N-J), ... , 4, 2, 1 ) Spinor String ( 1 , N , N(N-1)/2, ... , (N,J) , ... , N(N-1)/2, N, 1 ) Wedge String The spinor string is a string of decreasing powers of 2. The first term, 2^N, is the number of vertices of the hypercube. It is the number of choices you have for the origin, if you want to use the hypercube to build a hypercubic lattice. N edges meet at the origin vertex.There are 2^(N-1) - 1 other vertices at which N edges meet and which have no edges in common. N(N-1)/2 faces of 2-dim meet at the origin vertex. There are 2^(N-2) - 1 other vertices at which N(N-1)/2 faces of 2-dim meet and which have no 2-dim faces in common. ...N faces of (N-1)-dim meet at the origin vertex. There is 2 - 1 = 1 one other vertex at which N faces of (N-1)-dim meet and which have no (N-1)-dim faces in common.  The wedge string is the number of vertices, edges, faces, ... meeting the origin vertex. The number of edges, 2-dim faces, ... , J-dim faces, ... corresponds to the number of vertices, edges, ... , (J-1)-dim faces, ... of an (N-1)-dim simplexIt is also the binomial coefficient string of (1+1)^N = 2^N with each element of the string giving the dimension of each grade of the exterior algebra underlying the Clifford algebra Cl(0,N) The total number of hypercube vertices, 2^N, is also the total dimension of the Clifford algebra Cl(0,N) If N is even, then consider the middle term 2^(N/2) x N! / (N/2)!(N/2)!  The factor 2^(N/2) is the dimension of the full spinors of Cl(0,N) and also is the number of vertices in a hypercube of dimension N/2 The factor N! / (N/2)!(N/2)! is the middle term of grade N/2 of the exterior algebra underlying the Clifford algebra Cl(0,N)  The Nth row of the hypercube 3^N (2+1)^N triangle is the string product of a power of 2 string decreasing from 2^N and theNth row of the simplex Halayudha 2^N = (1+1)^N triangle.  For N=3, the string product is:  2^3 2^2 2^1 2^0 times 1 3 3 1  which is equal to the string 8x1 4x3 2x3 1x1 or 8 12 6 1  The hypercube powers-of-3 triangle rows through N=3 are:  N 0 1 3^0 = 11 2 1 3^1 = 32 4 4 1 3^2 = 9 3 8 12 6 1 3^3 = 27 The number of vertices, or the left-hand side of the triangle,  N 0 1 1 2 2 4 3 8  is the total dimension of the Clifford algebra Cl(0,N) and also the dimension of each half-spinor of the Lie algebra Spin(0,2N+2) of the Clifford algebra Cl(0,2N+2).  
 Hermes Trismegistus was the Greek name for Thoth, the Egyptian god of learning. Hermopolis, or Khmunu, the City of Eight, was the ancient center of the cult of Thoth.  About 1350 BC Akhenaton founded Amarna on the opposite bank of the Nile.  In 1945 AD, the Nag Hammadi papyri were rediscovered. They may bave been written about 100-300 AD, and contained Hermetic works such as The Ogdoad Reveals the Ennead (that is, Eight leads to Nine).  That 8 leads to 9 is also evident in the Chinese Lo Shu magic square:   4 9 2   3 5 7   8 1 6  There are 8 boundary cells of a 9-cell 3x3 magic square.  There are 8 ways (1 diagonal, 3 rows, 3 columns, 1 opposite diagonal) that sets of 3 of the 9 numbers can be added to get 15.  If you take into account the direction in which you add each of the 8 ways, and add all directed ways together you get a total of 16x15 = 240 which is the number of vertices of a Witting polytope.  All 9 numbers sum to 1+2+3+4+5+6+7+8+9 = 45 = 28 + (8 + 1 + 8)  

Prof. Dr. Helmut Moritz (Physical Geodesy, Graz TechnicalUniversity Austria) (in e-mail correspondence) said:

"... the book "Ein Parmenides-Kommentar", Koehler, Stuttgart 1959, would be of interest ... In his dialogue Parmenides, Plato constructs mathematics and the world from the One and the Other, or from 1 and 0. (Speiser was an excellent mathematician.) ...".
Plato's Parmenides might be interpreted as constructing math and the world (or at least all the natural numbers) from 1 and 0, somewhat like the set theoretical 0 = phi = empty set = 0{0} = 1{{0},0} = 2{{{0},0},{0},0} = 3... etc ... .According to an English translation of Plato's Parmenidesby B. Jowett (Anchor 1973):"... the one cannot touch itself ... It cannot. ...... Two things, then, at the least are necessary to make contactpossible? They are.And if to the two a third be added in due order, the number ofterms will be three, and the contacts two? Yes.And every additional term makes one additional contact,whence it follows that the contacts are one less in numberthan the terms ... True. ...... if the things which are other than the one were neither onenor more than one, they would be nothing. True. ..."To me, those quotes seem like the natural number line0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - ...Further, these quotes"... even if a person takes that which appears to be thesmallest fraction, this, which seemed one, in a momentevanesces into many, as in a dream, and from being thesmallest becomes very great, in comparison with the fractionsinto which it is split up? Very true. ...... And there will appear to be a least among them; and eventhis will seem large and manifold in comparison with the manysmall fractions which are contained in it? Certainly. ...... each separate particle yet appears to have a limit in relationto itself and other. How so? ...... prior to the beginning another beginning appears ... And soall being ... must be broken up into fractions ... And such beingwhen seen indistinctly and at a distance, appears to be one;but when seen near ... every single thing appears to be infinite ..."seem to me to describe the process of making rational numbersfrom the natural numbers by fractions,and the process of making real numbers from rational numbersby taking limits.What I do not see in Parmenides is a higher-dimensional structurethan the 1-dimensional string of natural numbers or dust ofrational numbers or line of real numbers, because the statementof connectivity (contacts one less in number than the terms) isclearly 1-dimensional and NOT like the connectivity of, forexample, a simplex in which every term is connected to everyother term, which would lead to higher-dimensional structuresand to Clifford algebras, etc., but maybe Plato's Timaeus couldbe seen as describing such things:Plato's Timaeus (about 400 BC) describes a cosmogony using both the powers of 2 (left line) and the powers of 3 (right line).  Timaeus identifies the powers of 3 with the limited, corresponding to spinor fermions obeying the exclusion principle of Fermi-Dirac statistics.  When the hexagonal numbers are added as a central column(Plato did not use them here, but I have added them in.) you get the following through N=3: N  0                        1 1                  2     6    3   = 2 +1   2             4         15         9   = 4 +4+1 3        8              28              27   = 8 +12+6+1   The hexagonal 28 is the dimension of the D4 Lie algebra Spin(0,8).  The 2^3 = 8 is the dimension of the vector representation of the D4 Lie algebra Spin(0,8). Let (8 + 1 + 8) be the complexification of the vector representation of D4, plus a complex U(1). The hexagonal 28 plus (8 + 1 + 8) is 45  (also a hexagonal number),  the sum of the numbers of the Lo Shu magic square, which is the dimension of the D5 Lie algebra Spin(0,10).  The 8 of 27=8+12+6+1 is the dimension of one half-spinor representation of the D4 Lie algebra Spin(0,8).  Let (8+8 + 1 + 8+8) be the complexification of both half-spinor representations of D4, plus a complex U(1).  Then 45 plus (8+8 + 1 + 8+8) is 78 (not a hexagonal number), which is the dimension of the E6 Lie algebra. E6 is related to the Tai Hsuan Ching.  The D4, D5, and E6 Lie algebras are used to construct the D4-D5-E6 physics model.   Plato's Timaeus used the N=3 numbers 1, 2, 3, 4, 8, 9, 27 to construct a musical scale covering almost 5 octaves: 1  4/3  3/2  2  8/3  3  4  9/2  16/3  6  8  9  27/2  18  27  Plato recognized that the N=3 numbers were incomplete, so he extended the system to N=8 for 2^N and N=5 for 3^N, to get (again, I have added the central column): N                                          0                      1         1                    2 6 3   =  2 + 12                  4  15   9   =  4 + 4+1             3                8    28     27   =  8 + 12+6+1           4             16      45       81   = 16 + 32+24+8+1          5           32                   243   = 32 + 80+80+40+10+1        6         64                            7      128                              8    256                                 Plato used the additional numbers 256 and 243 to form the ratio 256/243, which, along with 9/8, lets him construct the the first octave as:   1    9/8    81/64    4/3    3/2    27/16    243/128     2  by using the multiplicative intervals:   9/8    9/8   256/243   9/8    9/8      9/8    256/243   Plato recognized that this was still approximate, but said that the Demiurge had to stop at this point in constructing the World Soul, because "by this time the mixture from which he was cutting off these portions was all used up."  (see James, pp. 46-48)  

Note that he stopped at N=8 for 2^N and N=5 for 3^N, like theFibonacci sequence

1 1 2 3 5 8 13 21 ...........

If you take that seriously and extend to 4^N and 5^N, you wouldstop at 3 for 4 (4^3 = 64) and stop at 2 for 5 (5^2 = 25). Of course,you could also stop at 3 because 4 is a composite number, and notprime.

 Plato's cut-off at this point is similar to the use of a finite-length HyperDiamond lattice in the D4-D5-E6 physics model.   Plato's extended system corresponds to the D4-D5-E6 physics model as follows:  The 256 of the left-hand side is the dimension of the Clifford algebra Cl(0,8).  The 28 of the center column is the dimension of the D4 Lie algebra Spin(0,8).  Compare the 28 Hsiu. The 45 of the center column is the sum of the 28 of the center column and the 16 of the 81 of the right-hand side, plus 1 for a U(1) for complex structure.The 45 is also the dimension ofthe D5 Lie algebra Spin(0,10). The 16 form an 8-dim complex space with 8-dim real Shilov boundary.Compare the 8 Immortals.  The 32 of the 243 of the right-hand side, plus 1 for a U(1) for complex structure, when added to the 45 of the center column, gives 78 - the dimension of E6 of the D4-D5-E6 physics model.  The 32 form a 16-dim complex space with 16-dim real Shilov boundary. Compare the 16 original members of the 18 Lohan.   
78 is also the number of cards in a Tarot deck.Tarot may have originated in the Egypt of Thoth/Hermes, and been carried to Europe by returning Crusaders.  The South Part of the Temple of Luxor may represent Tarot.  The 78-card Tarot is related to the 78-dim Lie algebra E6  similarly to the way that  the 64-card I Ching is related to either of the two 8x8=64-dim square halves of the even subalgebra of the 256-dim Clifford algebra of Spin(0,8); and to the 8x8 real matrix algebra of which the Lie algebra Spin(0,8) (with commutator product) is the 28-dim antisymmetric part.                   -   -  16  11   7   4   2   1     x   -   -  17  12   8   5   3         The 21 Major Arcana x   x   -   -  18  13   9   6         naturally form a Spin(7) x   x   x   -   -  19  14  10         21-dim subgroup of  x   x   x   x   -   -  20  15         28-dim Spin(8) = D4 x   x   x   x   x   -   -  21      x   x   x   x   x   x   -   -               x   x   x   x   x   x   x   -     8   8   8   87   7   7   7                         The first 8 numbers 6   6   6   6                         of the 4 suits 5   5   5   5                         of the Minor Arcana 4   4   4   4                         naturally form  3   3   3   3                         the 32-dim space of 2   2   2   2             1   1   1   1                           E6 / (D5 x U(1))     K   K   K   K   Q   Q   Q   Q        The KQkj of the 4 suits k   k   k   k   j   j   j   j        naturally form                                      the 16-dim space of                                         D5 / (D4 x U(1))  That leaves the 10 and 9 of the 4 suits (s,w,c,p), 8 cards; plus the FOOL=0, 1 card; for a total of 9 more cards.   Use 10p for the U(1) in D4 x U(1) and  0  for the U(1) in D5 x U(1). That leaves 7 of the 9 and 10 cards, to form the 7-sphere that is Spin(8) / Spin(7), to get: -  9s  16  11   7   4   2   1     x   -  10s 17  12   8   5   3         The 21 Major Arcana x   x   -  9w  18  13   9   6         plus the 7 cards x   x   x   -  9c  19  14  10         naturally form  x   x   x   x   -  1Ow 20  15         28-dim Spin(8)   =   D4 x   x   x   x   x   -  9p  21      x   x   x   x   x   x   -  1Oc               x   x   x   x   x   x   x   -     How about decomposition of 28-dim Spin(8) = D4 under dimensional reduction, into 16-dim U(4) plus 12-dim SU(3)xSU(2)xU(1) ?    9s      11       4       1            1Os      12       5             The 16 elements            9w      13       6         of the rows 1,3,5,7                9c      14             naturally form                     1Ow     15         16-dim U(4)                        9p                                     1Oc                                                       16       7       2                    17       8       3         The 12 elements                 18       9             of the rows 2,4,6                   19      10         naturally form                         20             12-dim SU(3)xSU(2)xU(1)                            21                                                                                   So, by using the 9 and 10 minor arcana cards in an unconventional (from the European point of view) way,  you can get the structure of the D4-D5-E6 model  from the Tarot.   A Tarot spread that follows the D4-D5-E6 model is:           Ks   Kw   Kc   Kp   Qs          16   17   18   19         Qw       7    8    9         20        Qc    2    3         10         21        Qp      1          6          15       10c      10p          4    5         14         9p        jp          11   12   13        10w        jc         9s   10s   9w   9c       jw                 ks   kw   kc   kp   js   1s 2s 3s 4s 5s 6s 7s 8s   1c 2c 3c 4c 5c 6c 7c 8c                         0 1w 2w 3w 4w 5w 6w 7w 8w   1p 2p 3p 4p 5p 6p 7p 8p    The Tarot shows the Lie algebra structure of the D4-D5-E6 model, while the I Ching  and the Tai Hsuan Ching show its Clifford algebra structure.   
 If the right-hand side of Plato's extended system were extended further to the 8-dim hypercube, the strings would be  256 128 64 32 8+8 8 4 2 1 Spinor String  1 8 28 56 35+35 56 28 8 1 Wedge String The wedge string vector grade-1 term is 8-dimThe wedge string bivector grade-2 term is 28-dim (hexagonal number) The spinor string full Clifford algebra is 256-dim The spinor string full spinors are 8+8-dim (half-spinors are 8-dim) The numbers of sub-cubes of the 8-dim hypercube are the hypercube string, given by the string product of the spinor string and the wedge string 256 1,024 1,792 1,792 1,120 448 112 16 1 Hypercube String Their total is (2+1)^8 = 3^8 = 6,561 The 1,120 sub-hypercubes of the middle term are 4-dim, each with 8+8=16 vertices. If the 8 vertices of the 4-dim hyper-octahedron are added to such a 4-dim hypercube, you get the (16+8)=24-cell,
the root vector polytope of the D4 Lie algebra of Spin(0,8)  After dimensional reduction, the Spin(0,8) gauge group breaks down into a Spin(0,6) = SU(4) for conformal gravity, with the central cuboctahedron of the 24-cell 
as its root vector polytope, plus one of the 2 octahedra of the 24-cell
which can be projected into 2-space to represent the 6 root vector vertices of color force SU(3) plus the other of the 2 octohedra of the 24-cell 
which can represent by its 2 central vertices, the 2 Cartan subalgebra elements of color force SU(3), and by its 4 square outer vertices the 3 elements of weak force SU(2) and the electromagnetic U(1) photon. If 2 opposite corners of the square represent the W+ and W-, then the other 2 are on the orthogonal diagonal line, and represent the Cartan subalgebra W0 and the abelian photon.   The full 8-dim hypercube has 256 = 240 + 16 vertices, more than the 240 = 224 + 16 of the Witting polytope corresponding to one E8 lattice, but less than the 480 corresponding to all 7 E8 lattices.    
 If PLATO had known about higher-dimensional polytopes, he might have liked the only polytope that is centrally symmetric and self-dual, the 24-cell.
 The 4-dim 24-cell is the root vector polytope of Spin(8). Spin(8) has 24+4 = 28 dimensions.  Spin(8) has 8 REAL Clifford algebra Gammas {G1,G2,G3,G4,G5,G6,G7,G8}.  Koji Miyazaki at Kyoto has described the use of the cuboctahedron as a symbol in Asia.  Buckminster Fuller liked the cubocthedron, which he called the vector equilibrium.  
 The 3-dim cuboctahedron is the root vector polytope of Spin(6)=SU(4). SU(4) has 12+3 = 15 dimensions.  SU(4) has 4 COMPLEX Gammas {G1 + iG2,   G3 + iG4,   G5 + iG6,   G7 + iG8}. LI Hongzhi uses a symbol that contains an octagon of 8 directions of the compass.  When up and down are added, there are 10 directions.   
 The 2-dim octagon is the root vector polytope of Spin(5)=Sp(2). Sp(2) has 8+2 = 10 dimensions.  Sp(2 has 2 QUATERNION Gammas {G1 + iG2 + jG3 + kG4,   G5 + iG6 + jG7 + kG8}.  A single OCTONION Gamma {G1 + iG2 + jG3 + kG4 + EG5 + IG6 + JG7 + KG8} corresponds to a 7-sphere S7.  S7 is NOT a Lie group, but EXPANDS to become Spin(8).   A CYCLE OF DIVISION ALGEBRAS, Real - Complex - Quaternion - Octonion, BEGINS and ENDS with SPIN(8): G1,   G2,   G3,   G4,   G5,   G6,   G7,   G8       Spin(8)    24-cell + 4 G1 + iG2,   G3 + iG4,   G5 + iG6,   G7 + iG8   SU(4)=Spin(6) cuboctahedron+3 G1 + iG2 + jG3 + kG4,   G5 + iG6 + jG7 + kG8   Sp(2)=Spin(5)   octagon + 2 G1 + iG2 + jG3 + kG4 + EG5 + IG6 + JG7 + KG8    S7 EXPANDS back to Spin(8)            
 The REAL sequence  4-dim 24-cell 3-dim cuboctahedron 2-dim octagon has a COMPLEX counterpart:   Start with the 2(complex)-dim complex octagon - 8 vertices, 8 edges:
 27 complex octagons can form the 3(complex)-dim Hessian polyhedron - 27 vertices, 72 edges, 27 faces: 
 240 Hessian polyhedra can form the 4(complex)-dim Witting polytope - 240 vertices, 2160 edges, 2160 faces, 240 cells: 
 
 If you consider one of the 2160   1(complex)-dim edges of the Witting polytope, and then define the 2(complex)-dim hyperplane containing both that edge and the center of the polytope, you can look at the figure made by the vertices lying in that hyperplane.  Those vertices form a 2(complex)-dim polygon, called the van Oss polygon of the Witting polytope.  It has  24 vertices (0(real)-dim)        24 edges    (2(real)-dim)  It is just the complex-space version of the 4(real)-dim 24-cell.  
 

 REFERENCES: Aldred, Cyril, Akhenaten, Thames and Hudson (1988)  Coxeter, H. S. M., Regular Complex Polytopes, 2nd ed, Cambridge (1991) Green, Thomas M., and Hamberg, Charles L., Pascal's Triangle, Dale Seymour Publications (1986)  James, Jamie, The Music of The Spheres, Copernicus (Springer-Verlag) (1993) McLeish, John - The Story of Numbers, Fawcett Columbine (1991).  

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