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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,

where n is any positive integer, n^{2} is the sum of the first n odd integers, so that
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 (see here) 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:

The Tetrahedral Lambda is the generalisation of Plato's Lambda, which served as the basis of his cosmology. |

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, which we call the
Tetrahedral Lambda in the section **Plato's Lambda**, generates the
number **15** measuring the "skeleton" of the 2nd-order tetractys in terms of a basic, triangular
array of **15** points, namely, the corners of 10 1st-order tetractyses. Its third raised edge
generates the complete "body" of the 2nd-order tetractys comprising 85 yods. This illustrates the character of
the number **15** of YAH, the Godname of Chokmah, as the fifth triangular number.

The sum of the seven integers on the first and third raised edges of the Tetrahedral Lambda = 1
+ 2 + 4 + 8 + 4 + 16 + **64** = 99. As the sum of all its 20 integers is 350, the sum of the remaining
13 integers is 251. This is the number of yods in the 1-tree when its 19 triangles are Type A (see here). It is embodied in the UPA as the number of space-time coordinates of
points on the 10 whorls as 10 closed curves in **26**-dimensional space-time: 10×25 + 1 = 251.
Notice that the sum of the squares of the four integers on the first raised edge:

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

is the same as the sum of the four integers on the third raised edge:

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

This means that the number **168** which, being a parameter of holistic systems,
always displays the division **168** = 84 + 84 (see here), can be expressed as:

2^{2} + 4^{2} + 8^{2} + 4^{1} + 4^{2} +
4^{3}.

The holistic parameter 336 = 2×**168**, which is discussed in numerous places on
this website, can be expressed as 4×84 = 4^{2} + 4^{3} + 4^{4}. As 336 = 350 − 14, where 14
is the sum of the integers 2, 4 & 8 on the first raised edge, we see that the sum of the squares of these three
integers is 84, which is the sum of the nine integers on the boundary of the first face of the Tetrahedral Lambda,
whilst the sum of all its integers except 2, 4 & 8 is 336, which is 4×84, i.e., the sum of the integers 4
assigned to all the 84 yods surrounding the centre of a 2nd-order tetractys.

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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