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4-dimensional Sacred Geometries

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The Polychorons & the Gosset Polytope

 The 6 polychorons. The 10×24 division in sacred geometries & its realisation in the 421 polytope representing the 240 roots of E8. The sacred-geometrical basis of the five revolutions of each whorl of the UPA/E8×E8 heterotic superstring. The sacred-geometrical character of the 24-cell. A 2nd-order tetractys representation of the compound of two 600-cells. A 3rd-order tetractys representation of the compound of two 600-cells. The Pythagorean musical counterpart of the disdyakis triacontahedron & the compound of two 600-cells determining E8. #8 The correspondence between the 24-cell and the 3-dimensional Sri Yantra. The correspondence between the 24-cell and the two Type B dodecagons. #10 The correspondence between the 24-cell and the seven separate Type B polygons. #11 The 421 polytope as the inner form of 10 Trees of Life. #12 Arithmetic connections between the 421 polytope and the Tree of Life. #13 Square representations of some parameters of the 421 polytope. #14 Pentagramic representation of the geometrical composition of the 421 polytope. #15 The first four Platonic solids embody the number of edges of the 421 polytope. #16 The 2-d Sri Yantra embodies the number of edges of the 421 polytope. #17 The pair of Type C dodecagons embodies the number of edges of the 421 polytope. #18 The disdyakis triacontahedron embodies the number of edges of the 421 polytope. #19 The Godname EL CHAI prescribes the number of edges of the 421 polytope. #20 The 24-cell embodies the holistic parameter 672. #21 The 421 polytope as the inner form of the Tree of Life. #22 How the Type B square and octagon embody the numbers of 0-, 1-, 2- & 3-polytopes in the 24-cell. #23 How the Kabbalistic Godnames prescribe the 421 polytope. #24 The 421 polytope as the inner form of 10 Trees of Life.

The triacontagon

 The eight triacontagons in the E8 Coxeter plane projection of the 421 polytope. The Lambda Tetractys pattern of the interior & vertex angles of the triacontagon. The sum of the interior angles of the 8 triacontagons is the yod population if the inner form of 10 Trees of Life with Type C polygons. The triacontagon has the holistic pattern of the Type B dodecagon & the disdyakis triacontahedron. The pattern of interior & vertex angles in 8 sectors of the 8 triacontagons matches the yod population of the (7+7) Type B polygons. The yod population of the (4+4) Type A triacontagons matches that of the (7+7) Type B polygons. A correspondence between the 421 polytope and each half of the inner form of 10 Trees of Life. #8 The yods in the (4+4) triacontagons denote the turns in one revolution of the 10 helical whorls of the UPA. The inner form of 10 Trees of Life embodies the number of geometrical elements in the 8 Type C triacontagons. #10 How 8 sacred geometries embody the 84:84:84:84 pattern of base angles in the triacontagon. #11 Interior & vertex angles of the triacontagon conform to the I Ching pattern of 64 hexagrams. #12 The yod population of the 8 Type B triacontagons is the structural parameter of the UPA. #13 The geometrical composition of the 8 triacontagons. #14 The dodecagons in the inner form of 10 Trees of Life embody the 16800 turns in the 10 helical whorls of the UPA. #15 The 8 triacontagons embody the global structural parameter of the UPA. #16 The (4+4)×30 pattern of the 8 triacontagons in some sacred geometries. The 8 Church modes as the musical counterpart of the pattern of 8 triacontagons in the E8 Coxeter plane projection of the 421 polytope.

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