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The tetractys (greek τετρακτύς), or tetrad, is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mystical symbol, it was very important to the secret worship of the Pythagoreans.
Pythagorean symbol 
- The first four numbers symbolize the harmony of the spheres and the Cosmos as: (1) Unity, (2) Dyad - Power - Limit/Unlimit (peras/apeiron), (3) Harmony (4) Kosmos.
- The four rows add up to ten, which was unity of a higher order (The Dekad).
- The Tetractys symbolizes the four elements — fire, air, water, and earth.
- The Tetractys represented the organization of space:
A prayer of the Pythagoreans shows the importance of the Tetractys (sometimes called the "Mystic Tetrad"), as the prayer was addressed to it.
- "Bless us, divine number, thou who generated gods and men! O holy, holy Tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all".
As a portion of the secret religion, initiates were required to swear a secret oath by the Tetractys. They then served as novices for a period of silence lasting three years.
The Pythagorean oath also mentioned the Tetractys:
- "By that pure, holy, four lettered name on high,
- nature's eternal fountain and supply,
- the parent of all souls that living be,
- by him, with faith find oath, I swear to thee."
It is said that the Pythagorean musical system was based on the Tetractys as the rows can be read as the ratios of 4:3 (perfect fourth), 3:2 (perfect fifth), 2:1 (octave), forming the basic intervals of the Pythagorean scales. That is, Pythagorean scales are generated from combining pure fourths (in a 4:3 relation), pure fifths (in a 3:2 relation), and the simple ratios of the unison 1:1 and the octave 2:1. Note that the diapason, 2:1 (octave), and the diapason plus diapente, 3:1 (compound fifth or perfect twelfth), are consonant intervals according to the tetractys of the decad, but that the diapason plus diatessaron, 8:3 (compound fourth or perfect eleventh), is not.
- "The Tetractys [also known as the decad] is an equilateral triangle formed from the sequence of the first ten numbers aligned in four rows. It is both a mathematical idea and a metaphysical symbol that embraces within itself—in seedlike form—the principles of the natural world, the harmony of the cosmos, the ascent to the divine, and the mysteries of the divine realm. So revered was this ancient symbol that it inspired ancient philosophers to swear by the name of the one who brought this gift to humanity."
Kabbalist symbol 
There are some who believe that the tetractys and its mysteries influenced the early kabbalists. A Hebrew Tetractys in a similar way has the letters of the Tetragrammaton (the four lettered name of God in Hebrew scripture) inscribed on the ten positions of the tetractys, from right to left. It has been argued that the Kabbalistic Tree of Life, with its ten spheres of emanation, is in some way connected to the tetractys, but its form is not that of a triangle. The well known occult writer Dion Fortune mentions:
- "The point is assigned to Kether;
- the line to Chokmah;
- the two-dimensional plane to Binah;
- consequently the three-dimensional solid naturally falls to Chesed."
( We must note that the first three-dimensional solid is the tetrahedron. )
The relationship between geometrical shapes and the first four Sephirot is analogous to the geometrical correlations in Tetraktys, shown above under Pythagorean Symbol, and unveils the relevance of the Tree of Life with the Tetraktys. "Which of the two was the first one to emerge?", is a question that has been asked by many but still remains to be answered.
Tarot card reading arrangement 
In a Tarot reading, the various positions of the tetractys provide a representation for forecasting future events by signifying according to various occult disciplines, such as Alchemy.  Below is only a single variation for interpretation.
The first row of a single position represents the Premise of the reading, forming a foundation for understanding all the other cards.
The second row of two positions represents the cosmos and the individual and their relationship.
- The Light Card to the right represents the influence of the cosmos leading the individual to an action.
- The Dark Card to the left represents the reaction of the cosmos to the actions of the individual.
The third row of three positions represents three kinds of decisions an individual must make.
- The Creator Card is rightmost, representing new decisions and directions that may be made.
- The Sustainer Card is in the middle, representing decisions to keep balance, and things that should not change.
- The Destroyer Card is leftmost, representing old decisions and directions that should not be continued.
The fourth row of four positions represents the four Greek elements.
- The Fire card is rightmost, representing dynamic creative force, ambitions, and personal will.
- The Air card is to the right middle, representing the mind, thoughts, and strategies toward goals.
- The Water card is to the left middle, representing the emotions, feelings, and whims.
- The Earth card is leftmost, representing physical realities of day to day living.
The tetractys occurs (generally coincidentally) in the following:
- the baryon decuplet
- an archbishop's coat of arms
- the arrangement of pins in ten-pin bowling
- a chinese checkers board
See also 
- The Theosophical Glossary, p. 302
- The Pythagorean Sourcebook and Library, Kenneth Guthrie
- Danzig, Thomas (, 2005) Number. The Language of Science. p.42
- Introduction to Arithmetic - Nicomachus of Gerasa
- The Musical Order of the World: Kepler, Hesse, Hindemith-Siglind Bruhn
- A Dictionary of Greek and Roman Antiquities(1890) - William Smith, LLD, William Wayte, G. E. Marindin, Ed.
- Plutarch, De animae procreatione in Timaeo - Goodwin, Ed.(lang.:English)
- Sacred Architecture of London - Nigel Pennick
- The Mystical Qabalah, Dion Fortune, Chapter XVIII, 25
Further reading 
- von Franz, Marie-Louise. Number and Time: Reflections Leading Towards a Unification of Psychology and Physics. Rider & Company, London, 1974. ISBN 0-09-121020-8
- Fideler, D. ed. The Pythagorean Sourcebook and Library. Phanes Press, 1987.
- The Theoretic Arithmetic of the Pythagoreans - Thomas Taylor