#47 The holistic parameter 48 in sacred geometries

The number 48 is the number value of Kokab, the Mundane Chakra of Hod. Its astrological association is the planet Mercury. It is a conspicuous parameter of holistic systems possessing sacred geometry. This is illustrated below:

Outer Tree of Life
The outer form of the Tree of Life consists of 16 triangles with 10 corners and 22 sides, that is, 48 geometrical elements.

Inner Tree of Life
The inner form of the Tree of Life consists of seven regular polygons with 48 corners and 48 sides. They also have 48 sectors.

Plato's Lambda Tetractys
The 10 integers making up the tetractys extrapolation of Plato's Lambda comprise those at its corners that sum to 36, the central integer 6 and the six integers at the corners of a hexagon with sum 48.

Tetrahedron
There are 48 hexagonal yods in the four faces of the tetrahedron when they are regarded as tetractyses. The faces of the tetrahedron, octahedron, cube & icosahedron have 480 hexagonal yods, i.e., the number of yods in 48 tetractyses.

1-tree
When its 19 triangles are turned into tetractyses, the 1-tree has 80 yods, where 80 is the number value of Yesod. There are 48 red yods up to Chesed, the first Sephirah of Construction, which marks the commencement of the metaphysically "objective" aspect of the Tree of Life.

The Type A & Type B dodecagons
Divided into its sectors, a Type A dodecagon has 13 corners & 24 sides of 12 triangles, i.e., 49 geometrical elements, where 49 is the number value of EL CHAI, the Godname of Yesod. 48 geometrical elements surround its centre.

A Type B dodecagon consists of 36 triangles with 25 corners & 60 sides. Two joined Type B dodecagons have 72 triangles with 48 corners.

3-dimensional Sri Yantra
Transformation of the 43 triangles in the 3-dimensional Sri Yantra into tetractyses generates 168 red hexagonal yods lining the sides of the 21 triangles in its upper half, 168 blue hexagonal yods lining the sides of the 21 triangles in its lower half and 48 black yods either at the centres of the 42 triangles or hexagonal yods surrounding the centre of the central tetractys.

2nd-order tetractys
There are 36 green yods lining the perimeter of a 2nd-order tetractys and 48 pink yods inside it surrounding its centre. Alternatively, there are 36 white yods either making up the 1st-order tetractyses at its corners or at the corners of the seven tetractyses corresponding to the seven Sephiroth of Construction, leaving 48 coloured yods in the latter tetractyses that surround the centre of the 2nd-order tetractys symbolizing the most physical aspect of Malkuth.

The 3-torus
The 3-torus can be constructed by sticking the square faces of six square antiprisms onto the square faces of four triangular prisms. Their 56 triangular faces generate the {3,7} tessellation of the Klein Configuration on the 3‑torus. The 32 black corners of the 24 triangular sectors of the eight faces of the four triangular prisms at the apices of the tetrahedron correspond to the 32 black yods in the 1-tree above Chesed. The 48 red vertices of the triangular sectors of the 48 faces of the six square antiprisms correspond in the 1-tree to the 48 red yods up to Chesed. For more details, see Article 43. See also the discussion in #50 of the 3-torus constructed from triangular prisms and square antiprisms.

Disdyakis triacontahedron
The 62 vertices of the disdyakis triacontahedron consist of 30 A vertices, 12 B vertices (vertices of an icosahedron) & 20 C vertices (vertices of a dodecahedron). The arrow represents an axis passing through two diametrically opposite A vertices. The central plane perpendicular to this axis contains 12 vertices (four of each type). Above and below it are 50 vertices, where 50 is the number value of ELOHIM, the Godname of Binah. 48 vertices above and below the central plane surround the axis. They comprise two sets of 24 vertices (12 A, four B & eight C).

The holistic parameter 48 manifests arithmetically in superstring theory as the first 48 odd integers after 1 which sum to 2400:

49² − 1 = 3 + 5 + 7 + ... + 97 = 2400.

This is the number of space-time components of the 240 10-dimensional gauge fields mediating unified superstring forces that are associated with the 240 roots of E₈, the rank-8, exceptional Lie group that belongs to one of the two symmetry groups describing these forces. The number 48 is the number of yods surrounding the centre of an octagon whose sectors are tetractyses (see here). When these sectors are 2nd-order tetractyses, the octagon contains 496 hexagonal yods that symbolize the 496 roots of E₈×E₈, the symmetry group of the interactions between one of the two types of heterotic superstrings (see here).

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