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**The Cosmic Tetractys**

**The 2nd-order tetractys**

In **Sacred geometry/Tree of Life**,
we discussed various methods of transforming examples of sacred geometry in order to decode the scientific and
spiritual information that they embody. The next level of decoding sacred geometry after the Pythagorean
tetractys is through its next-higher version — the 2nd-order tetractys, in which each of the 10 yods of the
tetractys is replaced by another tetractys. This generates 85 yods, which is the sum of the first four integer
powers of 4:

85 = 4^{0} + 4^{1} + 4^{2} + 4^{3}.

The yod at the centre of the 2nd-order tetractys denotes Malkuth of the central tetractys, which
itself corresponds to this Sephirah. It is surrounded by 84 yods. The 2nd-order tetractys therefore expresses the
fact that 84 Sephirothic degrees of freedom in a holistic system exist *above* Malkuth — its physical
form. Of these, (7×7 − 1 = **48***) degrees are pure differentiations of Sephiroth of Construction
symbolized by coloured, hexagonal yods in the seven 1st-order tetractyses that are not at the corners of the
2nd-order tetractys. The remaining **36** degrees are denoted by both the
**15** white yods at the corners of the 10 tetractyses (these yods formally symbolize the
Supernal Triad) and the **21** coloured, hexagonal yods that belong to the tetractyses at the
three corners of the 2nd-order tetractys and which, therefore, also refer to the Supernal Triad of Kether, Chokmah
& Binah. YAH (יה), the older version of the Godname YAHWEH (יהוה) assigned to
Chokmah, has the number value **15** and prescribes the **15** corners of
the 10 1st-order tetractyses. ELOHA (אלה), the Godname of Geburah with number
value **36**, prescribes both the **36** yods lining the sides of the 2nd-order
tetractys and the **36** yods just discussed. The number 84 is the sum of the squares of
the first *four* odd integers:

84 = 1^{2} + 3^{2} + 5^{2} + 7^{2}.

As

n^{2} = 1 + 3 + 5 + ....+ 2n–1,

is the sum of the first n odd integers, 84 is the sum of (1+3+5+7=16=4^{2}) odd
integers:

The Tetrad determines the number of yods surrounding the centre of a 2nd-order tetractys. These
yods include **15** corners of 1st-order tetractyses, where

**15** = 2^{0} + 2^{1} + 2^{2} + 2^{3} = 1 + 2 + 4 +
8

(the number value of YAH) is the sum of the first *four* integer powers of 2. There
are (85–**15**=70) hexagonal yods, where 70 = 10×7 = (1+2+3+4)×fourth odd/prime number. These two
properties illustrate again how the Tetrad determines properties of the next higher-order tetractys above the
1st-order tetractys. In mathematics, triangular numbers (1, 3, 6, etc) can be represented by triangular arrays of
dots and tetrahedral numbers (1, 4, 10, etc) can be represented as a tetrahedral pile of these arrays. The piles
representing tetrahedral numbers can themselves be piled up into 4-dimensional "tetrahedral numbers": 1, 5,
**15**, 35, 70, etc. The *fourth*, non-trivial example of these numbers is 70. This is the
number of hexagonal yods in the 2nd-order tetractys. Once again, the Tetrad determines both a class of
number and a specific member of this class that is a parameter*,* or measure*,* of the
Pythagorean representation of Wholeness. It is an example of how the Tetrad Principle governs the mathematical
nature of holistic patterns and systems (for more details, see Article 1).

As another illustration of this principle, the four integers 1, 2, 4 & 8 are the first four terms in the geometric series:

1, 2, 4, 8, 16, 32, ....

in which each term is twice the previous one. They appear in what is known as **Plato's Lambda**. In
his treatise on cosmology called "Timaeus," Plato has the Demiurge marking a strip of the substance
of the World Soul into sections measured in length by the numbers 1, 2, 4 & 8 on one side of
it and the numbers 3^{1} (=3), 3^{2} (=9) & 3^{3} (=27) on its other
side (see here). The number of corners of the 10 1st-order tetractyses in the 2nd-order
tetractys is

**15** = 2^{0} + 2^{1} + 2^{2} + 2^{3} = 1 + 2 + 4 +
8,

whilst the number of yods in the 2nd-order tetractys is

85 = 1^{2} + 2^{2} + 4^{2} + 8^{2}.

The number of yods surrounding its centre is

84 = 2^{2} + 4^{2} + 8^{2},

and the number of hexagonal yods is

85 − **15** = 70 = (2^{2} − 2) + (4^{2} − 4) + (8^{2} −
8).

This illustrates the power of the integers 2, 4 & 8 to generate properties of the 2nd-order
tetractys. In mathematics, there are only *four* orders of normed division algebras**: the
**1**-dimensional scalar numbers, the **2**-dimensional complex numbers, the
**4**-dimensional quaternions and the **8**-dimensional octonions. Such is its archetypal
power as the arithmetic counterpart of sacred geometries, the Lambda and its complete, tetrahedral generalisation
(see here) generate not only the tone ratios of the notes of the Pythagorean
musical scale but also the dimensions of the four types of algebras permitting division! The four integers 1, 2,
4 & 8 spaced along the first raised edge of this tetrahedron generate as their sum the
**15** corners of the 10 1st-order tetractyses in the 2nd-order tetractys, whilst the four integers
1 (=4^{0}), 4 (=4^{1}), 16 (=4^{2}) & **64** (=4^{3}) spaced
along its third raised edge generate as their sum its 85 yods:

1 + 4 + 16 + **64** = 85.

We see that the first raised edge of the tetrahedral array of 20 integers that we call the
Tetrahedral Lambda in the section **Plato's Lambda** generates the "skeleton" of
the 2nd-order tetractys in terms of a basic, triangular array of **15** points, namely, the corners
of 1st-order tetractyses, whilst its third raised edge generates the complete "body" of the 2nd-order tetractys
comprising 85 yods. This demonstrates the character of the number **15** of YAH, the Godname of
Chokmah, as the fifth triangular number.

The correspondences between the 2nd-order tetractys, the 1-tree and the Sri Yantra are discussed here.

* Numbers in **boldface** are the number values of either the Hebrew names of the
Sephiroth or their manifestation in the four Worlds of Atziluth, Beriah, Yetzirah & Assiyah (see here).

** See here for a definition of normed division algebras.

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