This diagram of the Sefirot (and the English words for eachvertex) come from a figure on page 193 of the book The Jew in theLotus, by Rodger Kamenetz (HarperCollins 1994). I added the 3 colorsfor the lines. There exist other, slightly different, arrangements ofthe 22 lines, such as those shown on the webpages of Michael Graffam.

The Sefirot has some structure that is similar to the structure ofSingularities.

The 22 lines can be put in correspondence with the 22 letters ofthe Hebrew alphabet.

The Hebrew alphabet has 22 letters (compare the 0 plus 21 Major Arcana of the Tarot); 3 are Elementary (compare quaternion imaginaries)(green Sefirot lines); 7 are Double (compare octonion imaginaries)(red Sefirot lines); and the remaining 12 are Simple (compare the 12 Branches)(cyan Sefirot lines); as shown in this table that lists a correspondence with ASCII characters that can be used in an internet version of the Torah:

Elementary Double Simple Final ALEF A BET B GIMEL G DALET D HEY H VAV V ZAYIN Z HET X TET + YUD Y KAF K LAMED L MEM M NUN N SAMEH $ AYIN O PEH P TZADE C QOF Q RESH R SHIN/SIN S TAV T KAF SOFIT k MEM SOFIT m NUN SOFIT n PEH SOFIT p TZADE SOFIT c

The 12 Simple letters correspond to the 12 vertices (both concave and convex) of the Mogan David:

/\____/__\____\ / \ /\/ \//\ /\/__\____/__\\ /\/

In addition to the 22 Hebrew letters, there are 5 Finals that also appear in the Torah sequence. Stan Tenen is studying relationships between the 22 Hebrew letters and the 5 Final, and has shown how the 27 letters correspond to a Magic Cube. The Sefirot has 3+7 = 10 vertices (compare the 10 Stems). It is basically 3 columns: x x x x x x x x x x 3 vertices - 4 vertices - 3 vertices The pattern of 10 vertices could correspond to the 10-dimensional Poincare group or the 10-dimensional compact Spin(5) deSitter group, by letting the left column represent Spin(3) rotations, the right column represent Spin(3) Lorentz boosts, and the central column represent 4-dimensional spacetime. The central column has a 1+3 structure like spacetime, with the top vertex representing time. Consider the 8 vertices: x x x x x x x x Now consider the octonions, with basis {1,i,j,k,E,I,J,K}. The 8-vertex set could represent the octonions: 1 i I j J E k K This part of the Sefirot could represent the octonion vector representation space of the D4 Le algebra Spin(8) of the D4-D5-E6-E7 physics model. What about the other two vertices, F and G? 1 i I j J E k K F G If they are added to the octonionic 8 vertices, you get the 10-dimensional vector representation space of the D5 Lie algebra Spin(10) of the D4-D5-E6-E7 physics model. The D5 Lie algebra Spin(10) is 45-dimensional, representing all possible connecting lines between the 10 vertices of the Sefirot. The traditional Jewish Rabbinical Kabbala uses only 22 of the 45 lines that connect Sefirot vertices. Those 22 lines, plus the 10 vertices, form a total of 32 things, called the 32 Paths of Wisdom. The 32 Paths of Wisdom correspond to the 32-dimensional full spinor representation of D5 = Spin(10). Adding the 45 dimensions of the adjoint representation of D5 and the 32 dimensions of the full spinor representation of D5 plus 1 more dimension (mathematically, a complex U(1)), gives the 45+32+1=78-dimensional Lie algebra E6 of the D4-D5-E6-E7 physics model.Why does the Kabbala use only 22 lines ?There may be many reasons. One reason may be to show underlying Octonionic structure: 2 lines connect i to j and j to k orienting the associative triangle ijk; 6 lines connect ijk to E and E to IJK defining the coassociative square EIJK whereby E acts to mirror ijk and IJK; 3 lines connect ijk to IJK identifying the triangle subspace of the EIJK coassociative square; 2 lines connect I to J and J to K orienting the triangle subspace of the EIJK coassociative square; 2 lines connect ij to JI defining i and j as the algebraically independent generators of the quaternions; 3 lines connect 1 to iEI defining the associative triangle iEI; 2 lines connect E to F and F to G defining EFG as a mirror of IJK and of ijk; and 2 lines connect k to F and F to K orienting the triangle kFK related to the associative triangle kEK. What about the 23 lines that are not shown on the Sefirot? They also indicate octonionic structure: Since i and 1 form the complex numbers, there is a line from i to 1 but the 2 lines from j or k to 1 are omitted; Since I mirrors i, the 2 lines from J or K to 1 are omitted; Since k is not algebraically independent, the 1 line from k to i is omitted; Since K mirrors k, the 1 line from K to I is omitted; Since k and IJ are in mirror image triangles, the 2 lines from k to I and from k to J are omitted; Since K mirrors k, the 2 lines from K to i and from K to j are omitted; Since G is not octonion, and is algebraically dependent on E and F through EFG, G is connected only to F and the other 8 lines from G are omitted; and Since F, also not octonion, is the middle element of EFG, and E mirrors i and F mirrors j, the 1 line from F to 1 is omitted, but is implicitly defined through E. Since F is algebraically independent of E within EFG, the mirror property of F is shown by the kFK lines from F to k and from F to K. The 4 lines of the iFI and jFJ are omitted, but are implicitly defined through E. The 5 implicit lines from F correspond to the 5 Finals. Added to the 22 explicit Hebrew letter lines, the full 27-letter alphabet of the Torah sequence is formed. Since EFG is a mirror of IJK and of ijk, and since ijk, the imaginary quaternions, form a 3-sphere S3, and since 1 represents a point, a nice representation of the Sefirot is by a point and three 3-spheres: 1 S3 S3 S3 What if you expand the quaternionic 3-spheres S3 to octonionic 7-spheres S7 ? Then you get the configuration: 1 S7 S7 S7 with 1 + 3x7 = 22 vertices, corresponding to the 22 lines of the Sefirot, and to the 22 letters of the Hebrew alphabet, and to the 21 Major Arcana plus 0 of the Tarot, and to the 21-dimensional Lie algebra Spin(7) plus U(1). If you cannected all 22 vertices to each other, you would have 22x21/2 = 231 lines, corresponding to pairs of the 22 Hebrew letters. Stan Tenen is studying relationships between the 22 Hebrew letters and the 5 Finals. If you add the 5 final letters to the configuration, one to each of the three 7-spheres to make three octonions, OX, OY, and OZ, and the other two to the 1 to make three real numbers a, b, and c, you get the configuration: a,b,c OX OY OZ that has 3 + 3x8 = 27 vertices. To see more about this algebraically, represent the 5 Final letters by bars and dots, two letters being represented by solid and broken bars: ___ _ _ and three letters being represented by 3 sets of dots: . .. ... If the 22 Hebrew letters are broken into 3 groups of 8, 6, and 8 letters, and if the two Final lines are added to make 5 groups of 1, 8, 6, 8, and 1 letters including 2 Finals, those 24 letters and Finals represent the 24 vertices of a 24-cell, or, equivalently, 3 octonions. If the three Final sets of dots are added, and are taken to represent 3 real numbers, then the resulting 27-letter Hebrew alphabet represent the 3x3 Hermitian Octonion Matrices a OX OZ OX* b OY OZ* OY* c (where * denoted octonion conjugation) of the exceptional Jordan algebra J3(O), and correspond to the 27 lines on a cubic surface, whose symmetry is that of the Weyl group of the 78-dimensional E6 Lie algebra, corresponding to the 78 Tarot cards.

The Sefirot has some structure that issimilar to the structure of Singularities:

**Beauty/Tiferet** has 8 lines, and the bimodal singularitieshave 8 infinite series plus 14 (corresponding to the Liegroup G2) exceptional families.

**Yesod/Foundation** has 3 lines (plus 1 other line toMalkhut/Kingdom), and there are 3 H singularities (H2, H3, H4) plusthe I2(k) singularities.

**Malkhut/Kingdom** has 3 lines, and the simple singularitiesare A, D, and E (if the E is thecentral line up to Yesod/Foundation, it is a connection between E andH).

**The 6 outer vertices**, 2 sets of 3 (like ijk and IJK of the1ijkEIJK octionions, are connected toeach other by a network of 7 lines (like the 7 imaginary octonions).

According to the book Singularities of Differentiable Maps, VolumeI, by V. I.Arnold, S. M. Gusein-Zade, A. N. Varchenko (Birkhauser 1985):

- "... The quasihomogeneous
**unimodal singularities**are obtained from automorphic functions connected with 14 distinguished triangles on the Lobachevskii plane and three distinguished triangles on the Euclidean plane in precisely the same way as **simple singularities**are connected with regular polyhedra ...- ... The quasihomogeneous
**bimodal singularities**are associated with the 6 quadrilaterals and the 14 triangles on the Lobachevskii plane (in the latter case one must consider automorphic functions wiht automorphy factors corresponding to 2-, 3-, or 5- sheeted coverings) ...".

"... a non-Jew who studies the details of the seven Noahide laws,which are incumbent on him, deserves the honors due a Kohen Gadol(Sanhedrin 59a). The study of the seven Noahide laws may lead him tostudy most of the precepts of the Torah (Meiri to Sanhedrin 59a; seeResponsa, Rama section 10). ...

... Noahide Laws - the seven commandments given to Noah and hissons, which are binding upon all gentiles. These laws include theobligation to have a body of civil law, and the prohibitions againstidolatry, immorality, bloodshed, blasphemy, stealing and robbing, andeating limbs from a live animal. ...". (From Talmud Bavli / TractateChagigah, The Gemara, The ArtScrollSeries / Schottenstein Edition, Mesorah Publications Ltd. (1999) 13a3)

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