Stephen M PhillipsMar 25, 20172 min readTree of life superstring theory part 96Updated: Nov 28, 2020 The 3-torus is a hyperbolic surface of genus 3, that is, it has negative curvature and three holes. As discussed in Article 43, it may be constructed by sticking the square faces of six square antiprisms to the square faces of four triangular prisms. The resulting 56 hyperbolic triangles are the {3,7} mapping of the 168 automorphisms of the famous mathematical equation known as the Klein quartic: where X, Y & Z are complex variables. The 56 triangles consist of the four pairs of triangles at the "vertices" of the tetrahedrally-deformed 3-torus and six sets of eight triangles that form its "edges." Suppose that each triangle is transformed into a tetractys. As each triangle is on a hyperbolic surface, the tetractyses will, of course, be deformed. But they, still, have four yods along each of the 84 edges of the prisms and antiprisms. The yod composition of the 56 tetractyses is shown below: We see that they contain 248 yods made up of eight red hexagonal yods at the centres of the opposite faces of the four triangular prisms, 72 black hexagonal yods on the 36 sides of their eight tetractyses, 24 white corners of tetractyses and 144 hexagonal yods (48 red, 96 black) in the 48 tetractyses of the six square antiprisms: Turned inside-out, the 3-torus comprises 56 hyperbolic triangles that are the {3,7} mapping of the 168 anti-automorphisms of the Klein quartic. As tetractyses, they have the same numbers of yods distributed in the same subsets as those in the 3-torus. Both 3-tori contain (248+248=496) yods, where 496 is the number value of Malkuth. Their yod compositions are analogous, therefore, to the root structure of E8×E8, one of the two possible gauge symmetry groups of the heterotic superstring (see here under heading "Superstring gauge symmetry group").The number 248 is a parameter of holistic systems, being the number value of Raziel, the Archangel of Chokmah. Its presence in the {3,7} tiling of the 3-torus and its version turned inside-out, as well as the fact (discussed above) that it displays the other parameters and patterns of holistic systems, strongly suggests that both tori are connected to the physics of the E8×E8 heterotic superstring in some way to be determined. This is further strongly suggested by the fact that there are 192 yods lining the sides of the 56 tetractyses and that this number divides up into the 24 vertices of the prisms & antiprisms and the 168 hexagonal yods on the 84 sides of the tetractyses. This 24:168 division is characteristic of holistic systems, as explained in The holistic pattern, as it reflects the difference between the function of the Supernal Triad (symbolized by yods at corners of tetractyses) and the seven Sephiroth of Construction (symbolized by hexagonal yods). Many examples are discussed on this website

The 3-torus is a hyperbolic surface of genus 3, that is, it has negative curvature and three holes. As discussed in Article 43, it may be constructed by sticking the square faces of six square antiprisms to the square faces of four triangular prisms. The resulting 56 hyperbolic triangles are the {3,7} mapping of the 168 automorphisms of the famous mathematical equation known as the Klein quartic: where X, Y & Z are complex variables. The 56 triangles consist of the four pairs of triangles at the "vertices" of the tetrahedrally-deformed 3-torus and six sets of eight triangles that form its "edges." Suppose that each triangle is transformed into a tetractys. As each triangle is on a hyperbolic surface, the tetractyses will, of course, be deformed. But they, still, have four yods along each of the 84 edges of the prisms and antiprisms. The yod composition of the 56 tetractyses is shown below: We see that they contain 248 yods made up of eight red hexagonal yods at the centres of the opposite faces of the four triangular prisms, 72 black hexagonal yods on the 36 sides of their eight tetractyses, 24 white corners of tetractyses and 144 hexagonal yods (48 red, 96 black) in the 48 tetractyses of the six square antiprisms: Turned inside-out, the 3-torus comprises 56 hyperbolic triangles that are the {3,7} mapping of the 168 anti-automorphisms of the Klein quartic. As tetractyses, they have the same numbers of yods distributed in the same subsets as those in the 3-torus. Both 3-tori contain (248+248=496) yods, where 496 is the number value of Malkuth. Their yod compositions are analogous, therefore, to the root structure of E8×E8, one of the two possible gauge symmetry groups of the heterotic superstring (see here under heading "Superstring gauge symmetry group").The number 248 is a parameter of holistic systems, being the number value of Raziel, the Archangel of Chokmah. Its presence in the {3,7} tiling of the 3-torus and its version turned inside-out, as well as the fact (discussed above) that it displays the other parameters and patterns of holistic systems, strongly suggests that both tori are connected to the physics of the E8×E8 heterotic superstring in some way to be determined. This is further strongly suggested by the fact that there are 192 yods lining the sides of the 56 tetractyses and that this number divides up into the 24 vertices of the prisms & antiprisms and the 168 hexagonal yods on the 84 sides of the tetractyses. This 24:168 division is characteristic of holistic systems, as explained in The holistic pattern, as it reflects the difference between the function of the Supernal Triad (symbolized by yods at corners of tetractyses) and the seven Sephiroth of Construction (symbolized by hexagonal yods). Many examples are discussed on this website