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4-dimensional Sacred Geometries

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The Polychorons & the Gosset Polytope

  #1

The 6 polychorons.

  #2

The 10×24 division in sacred geometries & its realisation in the 421 polytope representing the 240 roots of E8.

  #3 The sacred-geometrical basis of the five revolutions of each whorl of the UPA/E8×E8 heterotic superstring.
  #4 The sacred-geometrical character of the 24-cell.
  #5 A 2nd-order tetractys representation of the compound of two 600-cells.
  #6 A 3rd-order tetractys representation of the compound of two 600-cells.

  #7

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.
  #9 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

  #1

The eight triacontagons in the E8 Coxeter plane projection of the 421 polytope.

  #2

The Lambda Tetractys pattern of the interior & vertex angles of the triacontagon.

  #3

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.

  #4

The triacontagon has the holistic pattern of the Type B dodecagon & the disdyakis triacontahedron.

  #5

The pattern of interior & vertex angles in 8 sectors of the 8 triacontagons matches the yod population of the (7+7) Type B polygons.

  #6

The yod population of the (4+4) Type A triacontagons matches that of the (7+7) Type B polygons.

  #7

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.

  #9

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.
 #17

The 8 Church modes as the musical expression of the pattern of 8 triacontagons in the E8 Coxeter plane projection of the 421 polytope.

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