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This article will begin a five-part series on the arithmetic, geometry and symbolism of the Tetrad.

Elements of Four Extreme Inner Piece

Following that we will focus on one extremely important aspect of the Tetrad – that is, the five Platonic solids. We will spend twelve articles on the Platonic solids before we move onto the Pentad.

 

 

The Tetrad: 4: Fourness – The Realm of Manifestation

The Tetrad is a symbol of the Earth and the natural world.

cartography 1801579

 

It is expressed in geometry and nature as volume, form, or 3D space.

xyz axis

 

As Micheal Schneider relates, “Form in nature is a graph of forces.”

 

3D space or volume can be looked at:

  • as points, lines, areas, volume

volume lowres

 

 

  • or as corners, edges, faces and from the center outwards (as seen with the cube below)

cube 002 cube 003 cube 004 cube inverted red

 

The Tetrad is born through the vesica piscis as the square, tetrahedron and cube.

square lr tetrahedron1 cube lines

 

 

The Tetrad manifests in nature at the borderline between nonliving and living forms such as:

  • structures of atoms (Lithium, Beryllium, Boron, Carbon, Nitrogen, Oxygen, Fluorine)

Cubical atom 1.svg

 

  • molecules (diamond on left and center; NaCl or table salt on right)

Diamond cubic animation diamond cubic.svg crystal structure 148812 1280

 

  • crystals (bismuth, pyrite, flourite)

crystal bismuth crystal cube crystal cube flourite

 

  • living cells (plant stem, leaf and root cells)

plant cells4 root onion

 

Atomic Number 4 = Beryllium

Beryllium in Black

Beryllium is highly toxic, carcinogenic, and has a sweet taste. It is the 44th most abundant element on Earth. It is also the lightest alkaline earth metal known. Beryllium is immune to attacks of nitric acid and it is not magnetic at all.

 

Atomic Number 40 = Zirconium

Zirconium

Zirconium is a strong, malleable, lustrous grayish-white metal. It is exceptionally resistant to corrosion except by hydrofluoric acid. It is used to make superconductive magnets, among other things.

 

Atomic Number 44 = Ruthenium

Ruthenium crystal

 

Ruthenium is toxic and carcinogenic. It is a very rare, hard, lustrous, brittle silvery-white metal that does not tarnish at room temperature. It is unaffected by air, water and acids.

 

Iamblichus writes, “The tetrad comprehends the principle of the soul as well as that of corporeality; for they say that a living creature is ensouled in the same way that the whole universe is arranged according to harmony.”

spacehasshape lowres

 

 

Arithmetic Properties of the Tetrad

Four (4) is the first born material ‘thing’.

We write the numeral 4 as if in a 2 x 2 grid (in red).

number 4 1 number 4 2

 

Divisors – 1, 2, 4

4 is the first product of procreation: two twos.

4 spheres

 

It is the first number formed by the addition of equals (4 = 22 = 2 +2 = 2 x 2)

4 is the 2nd square number.

 

There are 4 vertices and faces of a tetrahedron.

Tetrahedron rotating

 

There are 4 sides of a square. Ingrained in the square is the 90º angle.

Square geometry.svg copy

 

A quadrilateral is a four-sided plane figure.

This includes: kites, rhombi, rectangles, and squares (pictured below).

Kite definition.svg Rhombus 1.svg A sample of the transparent rectangle.svg square lr

 

The Tetrad represented ‘justice’ to the Pythagoreans because four is the first product of equal values, a number divisible every way into equal parts.

balance 2858897 1280

 

4 = 2 + 2

4 = (1+1) + (1 + 1)

A number is a multiple of 4 if its last two digits are a multiple of 4.

For example: 1092; 92 = 4 x 23

Four is the only cardinal number in the English language that has the same number of letters as its number value.

F – O – U – R

 

 

Powers of Four

42=16

16 icon.svg

Divisors of 16: 1, 2, 4, 8, 16

Atomic Number 16 = Sulfur

Sulfur sample

 

 

43=64

64.svg

Divisors of 64: 1, 2, 4, 8, 16, 32, 64

64 represents an octave of octaves.

It is the smallest number with exactly 7 divisors.

It is the first perfect square and perfect cube.

Atomic Number 64 = Gadolinium

64 gadolinium Gd enhanced Gadolinium 4

 

64 Tetrahedral Grid – Isometric Vector Matrix

IVM circles

 

64 6-part codons specify amino acids in DNA.

codon chart

 

64 Transformations in the I-Ching

I Ching.svg

 

64 Arts of the Kama Sutra

kamasutra 1200x520

 

64 Squares on a Chess board.

chess board

 

44=256

256 number.svg

256 is a perfect square.

One octet (one byte) = 8 bits or 256 possible values.

256 different color values in each color channel of a digital color image.

256color chart.svg

 

256 characters in the new Braille 8-dot system.

256 Hz is the frequency of Middle C.

 

 

45=1024

1024 is the square of 32.

It is the smallest number with exactly 11 divisors.

1024 pixels is a common standard of display resolution.

 

 

 

Theology of Arithmetic – Iamblichus

As we have been doing, we will now take a look at the meaning of the Tetrad as prepared by Neo-Platonist philosopher, biographer of Pythagoras, Greek mystic and mathematician Iamblichus (245-325 AD).

Everything in the universe turns out to be completed in the natural progression up to the tetrad.

The tetrad provides the limit of corporeality and three-dimensionality – a complete result in three dimensions exists in four terms.

Represented by the Quadrivium of mathematical sciences: “Four are the foundations of wisdom – arithmetic, music, geometry, astronomy – ordered 1, 2, 3, 4.”

quadrivium lowres

 

The most elementary plane is contained by a triad, either of angles or of lines, and when depth is added, from this a base to a single point, then in turn the most elementary of solids, the pyramid, is formed by at least four angles or surfaces and is fitted together by virtue of three equal dimensions.

The tetrad is the first to encompass minimal and most seminal embodiment.

Tetrahedron rotating

 

There are four elements – fire, air, water, and earth; and their powers – heat, cold, wetness, and dryness.

4 elements lr

 

The heavenly realm has been assigned four centers – the one over the zenith, the one at the ascendant, the one at right angles under the Earth, and the one at the descendant.

There are four limits – north, south, east, and west.

compass 73112 1920

 

There are four parts of a sphere – center, diameter (axis), circumference, and area.

There are many ninety-degree divisions of the zodiac.

zodiac lowres

 

There are four traditional seasons of the year – spring, summer, autumn and winter; and four traditional winds.

the four seasons

 

There are four measures – eternity, time, critical time, and passing time.

There are four distinct senses in living creatures (taste, smell, sight, hearing); for touch is a common background to the other four, which is why it alone does not have a location or a regular organ.

There are four kinds of plants – trees, shrubs, vegetables and herbs.

4plants

 

There are four kinds of virtues – wisdom in the soul and keen sensibility of the body; moderation in the soul and health in the body; courage, strength and power; and justice, beauty and friendship (justice, temperance, fortitude and prudence).

virtues castel nuovo triumphal arch entrance naples italy

 

There are also four seasons for man – childhood, youth, adulthood and old age.

4 ages of man

 

Man is divided into four – head, trunk, arms, legs.

There are four sources of a rational creature – brain, heart, navel and genitals.

4 human copy

 

The tetrad is the foundation of natural plurality and accretion.

Four is a measure and complete boundary of every number. (1 + 2 + 3 + 4 = 10)

tetrakyts lr

 

It is secure, because of the stability of the number 4, a stability which binds everything in a pyramidal manner to secure bases.

tetrahedron1

 

Squares are not easy to shake from their disdain – they are steadfast.

grid 684983 1920

 

Four is the prerequisite of the general orderliness of the universe.

It is called a “custodian of Nature”.

The tetrad gives rise to duration, since eternity, time, critical time and passing time are four, as moreover are year, month, night and day; and morning, midday, evening and night.

dawn noon dusk night

 

It is also called the “enduring one.”

It is called “justice”, since the square which is based on it is equal to the perimeter; for the perimeter of squares before it is greater than the area of those squares and the perimeter of squares after it is less than the area, but in its case the perimeter is equal:

(42 = 16 = 2 x 2 x 2 x 2)

balance 2858897 1280

 

The tetrad is the first to display the nature of solidity: the sequence is point, line, plane, solid.

volume lowres

 

Some say all things are organized by four aspects – substance, shape, form and principle.

The tetrad comprehends the principle of soul, as well as that of corporeality; for they say that a living creature is ensouled in the same way that the whole universe is arranged, according to harmony.

If the universe is composed out of soul and body in the number 4, then it is also true that all concords are perfected by it.

 

 

Four in Music

music lowres

 

There are Four Beautiful Harmonies. They all arise from ratios involving the first four numbers.

tetraktys harmony lr

 

Unison (1:1)

Octave (2:1)

Fifth (2:3)

Fourth (4:3), the complement of the fifth inside the octave.

 

The Tetrad also appears as the third overtone 4:1, which is two octaves.

 

 

 

Four in Religion

Buddhism

  • Four Noble Truths – Suffering, Cause of Suffering, Cessation of Suffering, the Path leading to the Cessation of Suffering

fournoble truths

 

  • Four Sights of the Buddha – old man, sick man, dead man, ascetic
  • Four Great Elements
  • Four Heavenly Kings
  • Four Foundations of Mindfulness – contemplation of the body, feelings, the mind, and mental objects
  • Four Right Exertions
  • Four Bases of Power
  • Four Divine Abidings – loving-kindness, compassion, sympathetic joy, equanimity

4buddhist abidings

 

 

Judeo-Christianity

  • Tetragrammaton – the four letter name of God YHWH

Tetragrammaton Tetractys

 

  • Four Matriarchs of Judaism – Sarah, Rebekah, Leah, Rachel
  • Passover (Judaism) – 4 cups of wine, 4 questions to be asked, 4 sons to be dealt with, 4 expressions of redemption
  • Four Gospels – Matthew, Mark, Luke, John
  • Four Horsemen of the Apocalypse

4 horsemen 2

 

 

Hinduism

  • 4 Vedas – Rigveda, Samaveda, Yajurveda, Atharvaveda
  • Four aims of human life – Dharma (virtuous life), Artha (purposes and means of life), Kama (emotional fulfillment), Moksha (liberation, self-actualization)

4hindu aims

 

  • 4 Castes of Society

 

Islam

  • Eid al-Adha (The Sacrifice Feast) lasts for four days.
  • Four Arch Angels in Islam
  • Four Sacred Months

Untitled 1

 

  • Four Sunni schools of fiqh
  • Four major sunni Imams

4 imams 864x400 c

 

  • Four books in Islam – Torah, Zaboor, Injeel, Quran

 

And as Black Elk of the Oglala Sioux said regarding the four quarters of the universe: “The black one is for the west where the thunder beings live to send us rain; the white one for the north, whence comes the great white cleansing wind; the red one for the east, whence springs the light and where the morning star lives to give men wisdom; the yellow for the south, whence come the summer and the power to grow.

black elk 4 directions

But these four spirits are only one Spirit after all…”

black elk

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