## Tuesday, 4 August 2015

### Hexagram

(Redirected from Hexagram (geometry))
For other uses, see Hexagram (disambiguation).
Regular hexagram
A regular hexagram
TypeRegular polygonal figure
Edges and vertices6
Schläfli symbola{6}, {6/2}, 2{3} or {{3}}
Coxeter diagram or
Symmetry groupDihedral (D6)
Internal angle (degrees)60°
Dual polygonself
Propertiesstar, compound, cyclic, equilateral, isogonal, isotoxal
A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or {{3}}. It is the compound of two equilateral triangles. The intersection is a regular hexagon.
It is used in historical, religious and cultural contexts, for example in Hanafism,[1] Raelianism,[2] Jewish identity, Hinduism and occultism.

## Group theory

In mathematics, the root system for the simple Lie group G2 is in the form of a hexagram.

## Construction by compass and a straight edge

A six-pointed star, like a hexagon, can be created using a compass and a straight edge:
• Make a circle of any size with the compass.
• Without changing the radius of the compass, set its pivot on the circle's circumference, and find one of the two points where a new circle would intersect the first circle.
• With the pivot on the last point found, similarly find a third point on the circumference, and repeat until six such points have been marked.
• With a straight edge, join alternate points on the circumference to form two overlapping equilateral triangles.

## Origins and shape

It is possible that as a simple geometric shape, like for example the triangle, circle, or square, the hexagram has been created by various peoples with no connection to one another.
The hexagram is a mandala symbol called satkona yantra or sadkona yantra found on ancient South Indian Hindu temples. It symbolizes the nara-narayana, or perfect meditative state of balance achieved between Man and God, and if maintained, results in "moksha," or "nirvana" (release from the bounds of the earthly world and its material trappings).[citation needed]
Another theory, though apparently not very substantiated, about the origin of the shape is that it is simply 2 of the 3 letters in the name David: in its Hebrew spelling, David is transliterated as "D-V-D." In Biblical Hebrew, the letter "D" (Dalet) was written in a form like an upside-down and backwards "L," but when seen in the Greek, the letter "Delta" (Δ) is a triangle. The symbol may have been a simple family crest formed by flipping and juxtaposing the two most prominent letters in the name. The letter "W" in this case could reference the compositing operation of the two Deltas.[citation needed]
Some researchers have theorized that the hexagram represents the astrological chart at the time of David's birth or anointment as king. The hexagram is also known as the "King's Star" in astrological circles.
In antique papyri, pentagrams, together with stars and other signs, are frequently found on amulets bearing the Jewish names of God, and used to guard against fever and other diseases. Curiously the hexagram is not found among these signs. In the Greek Magical Papyri[citation needed] (Wessely, l.c. pp. 31, 112) at Paris and London there are twenty-two signs side by side, and a circle with twelve signs, but neither a pentagram nor a hexagram.

## Usage by Hinduism and Eastern religions

Diagram showing the two mystic syllables Om and Hrim
Six pointed stars have also been found in cosmological diagrams in Hinduism, Buddhism, and Jainism. The reasons behind this symbol's common appearance in Indic religions and the West are unknown. One possibility is that they have a common origin. The other possibility is that artists and religious people from several cultures independently created the hexagram shape, which is a relatively simple geometric design.
Within Indic lore, the shape is generally understood to consist of two triangles—one pointed up and the other down—locked in harmonious embrace. The two components are called "Om" and the "Hrim" in Sanskrit, and symbolize man's position between earth and sky. The downward triangle symbolizes Shakti, the sacred embodiment of femininity, and the upward triangle symbolizes Shiva, or Agni Tattva, representing the focused aspects of masculinity. The mystical union of the two triangles represents Creation, occurring through the divine union of male and female. The two locked triangles are also known as 'Shanmukha'—the six-faced, representing the six faces of Shiva & Shakti's progeny Kartikeya. This symbol is also a part of several yantras and has deep significance in Hindu ritual worship and history.
In Buddhism, some old versions of the Bardo Thodol, also known as The "Tibetan Book of the Dead", contain a hexagram with a Swastika inside. It was made up by the publishers for this particular publication. In Tibetan, it is called the "origin of phenomenon" (chos-kyi 'byung-gnas). It is especially connected with Vajrayogini, and forms the center part of Her mandala. In reality, it is in three dimensions, not two, although it may be portrayed either way.
In the endocrine system, Anahata is associated with the thymus gland, located in the chest. This gland produces T-cells, that combat disease, and bring equilibrium to the body. The functioning of the thymus is greatest before puberty and is impaired by the appearance of sex hormones in the blood stream from puberty onwards.[citation needed]
Many Western occultists[who?] associate this central chakra with the central sephirah, Tiphereth, in the kabbalistic tree of life. Christian kabbalists in particular associate this sephirah with love, healing, and knowledge.
The Shatkona is a symbol used in Hindu yantra that represents the union of both the male and feminine form. More specifically it is supposed to represent Purusha (the supreme being), and Prakriti (mother nature, or causal matter). Often this is represented as Shiva - Shakti.[3]
The Shatkona is a hexagram and looks exactly like the Star of David in Semitic lore.

### Anahata: The Heart Chakra

 Anahata (also known as Anahata-puri, or Padma-sundara) is related to the thymus, located in the chest. The thymus is an element of the immune system as well as being part of the endocrine system. It is the site of maturation of the T cells responsible for fending off disease and may be adversely affected by stress. Anahata is symbolized by a lotus flower with twelve petals. (See also heartmind). Anahata is related to the colors green or pink. Key issues involving Anahata involve complex emotions, compassion, tenderness, unconditional love, equilibrium, rejection and well-being. Physically Anahata governs circulation, emotionally it governs unconditional love for the self and others, mentally it governs passion, and spiritually it governs devotion.[4]

## Usage by the Abrahamic religions

### Usage by Jews

Main article: Star of David
The Star of David in the oldest surviving complete copy of the Masoretic text, the Leningrad Codex, dated 1008.
The Magen David is a generally recognized symbol of Judaism and Jewish identity and is also known colloquially as the Jewish Star or "Star of David." Its usage as a sign of Jewish identity began in the Middle Ages, though its religious usage began earlier, with the current earliest archeological evidence being a stone bearing the shield from the arch of a 3–4th century synagogue in the Galilee.[5] A more enduring symbol of Judaism, the menorah, has been in use since BC.[citation needed]

### Usage by Christians

The hexagram may be found in some Churches and stained-glass windows. An example of this is one embedded in the ceiling of the Washington National Cathedral. In Christianity it is often called the star of creation.
In Orthodox Christian churches, for example in Balkan countries, hexagrams can be found more often than in Roman Catholic churches.

#### Latter-day Saints (Mormons)

Star of David on the Salt Lake Assembly Hall
Main article: Mormonism and Judaism
The Star of David is also used less prominently by The Church of Jesus Christ of Latter-day Saints, in the temples and in architecture. It symbolizes God reaching down to man and man reaching up to God, the union of Heaven and earth. It may also symbolize the Tribes of Israel and friendship and their affinity towards the Jewish people. Additionally, it is sometimes used to symbolize the quorum of the twelve apostles, as in Revelation 12, wherein the Church of God is symbolized by a woman wearing a crown of twelve stars. It is also sometimes used to symbolize the Big Dipper, which points to the North Star, a symbol of Jesus Christ.

#### Rastafari

A black star of David is used to identify the black population, in Africa or otherwise, with one of the Tribes of Israel.[citation needed]

### Usage by Muslims

The symbol is known in Arabic as نجمة داوود, Najmat Dāwūd (Star of David) or خاتم سليمان Khātem Sulaymān (Seal of Solomon), but "Seal of Solomon" may also refer to a pentagram or a species of plant.
In various places in the Qur'an, it is written that David and King Solomon (Arabic, Suliman or Sulayman) were prophets and kings and therefore they are revered figures by Muslims. The Medieval pre-Ottoman Hanafi Anatolian beyliks of the Karamanids and Jandarids used the star on their flag.[1] The symbol also used on Hayreddin Barbarossa flag. Even today, the star can be found in mosques and on other Arabic and Islamic artifacts.
Professor Gershom Sholem theorizes[citation needed] that the "Star of David" originates in the writings of Aristotle, who used triangles in different positions to indicate the different basic elements. The superposed triangles thus represented combinations of those elements. From Aristotle's writings those symbols made their ways into early, pre-Muslim Arab literature.

## Usage in heraldry

In heraldry and vexillology, a hexagram is a fairly common charge employed, though it is rarely called by this name. In Germanic regions it is known simply as a "star." In English and French heraldry, however, the hexagram is known as a "mullet of six points," where mullet is a French term for a spur rowel which is shown with five pointed arms by default unless otherwise specified.

## Usage in theosophy

The Star of David is used in the seal and the emblem of the Theosophical Society (founded in 1875). Although it is more pronounced, it is used along with other religious symbols. These include the Swastika, the Ankh, the Aum, and the Ouroboros. The star of David is also known as the Seal of Solomon that was its original name until around 50 years ago.

## Usage in Raelism

The International Raelian Movement (IRM) uses a hexagram. The root of this symbol, according to the founder of the IRM, Rael, can be attributed to its use by genetic engineers from extrasolar planets who are allegedly the same entities referred to as Elohim. According to Rael, these space travellers came to Earth and synthesized life from non-living matter in 7 laboratory bases which contained the symbol.
Some meanings which involve particular variations of this symbol are supported by the IRM, such as "well being" (where "swastika" means "well being" in Sanskrit) and "infinity in time" (as Hindus see the swastika as a symbol for "eternal" cycles). In Raelism, the upper and lower triangles represent "as above, so below", which refers to either the likeness between the creators' past and created's future or the repeating fractal hierarchical structure in the universe. "As above so below" is also well known in Wicca as the last statement of an invocation or ritual in order to bring the change of events from the upper world to the lower world (our world).
The IRM has long-term plans to build a temple complex or embassy that would, at around the time of a Technological Singularity, and before 2035, support the arrival of prophets of major and some minor religions after a spectacular descent from an interstellar journey. Rael (or the Elohim, as Rael would put it) requires that the embassy contain the "symbol of the Elohim." The symbol initially used by the Raelian movement was the source of considerable controversy linked to a proposal to build the Raelian embassy in Israel since it resembled a hexagram with the image of a Swastika embedded in its center.

## Usage in Occultism

The hexagram, like the pentagram, was and is used in practices of the occult and ceremonial magic and is attributed to the 7 "old" planets outlined in astrology.
The six-pointed star is commonly used both as a talisman[6] and for conjuring spirits and spiritual forces in divers forms occult magic. In the book The History and Practice of Magic, Vol. 2, the six-pointed star is called the talisman of Saturn and it is also referred to as the Seal of Solomon.[7] Details are given in this book on how to make these symbols and the materials to use.
Double hexagram.
Traditionally, the Hexagram can be seen as the combination of the four elements. Fire is symbolized as an upwards pointing triangle, while Air (its elemental opposite) is also an upwards pointing triangle, but with a horizontal line through its center. Water is symbolized as a downwards pointing triangle, while Earth (its elemental opposite) is also a downwards pointing triangle, but with a horizontal line through its center. When you combine the symbols of Fire and Water, a hexagram (six-pointed star) is created. The same follows for when you combine the symbols of Air and Earth. When you combine both hexagrams, you get the double-hexagram. Thus, a combination of the elements is created.[8]
In Rosicrucian and Hermetic Magic, the seven Traditional Planets correspond with the angles and the center of the Hexagram as follows, in the same patterns as they appear on the Sephiroth and on the The Tree of Life. Saturn, although formally attributed to the Sephira of Binah, within this frame work nonetheless occupies the position of Daath.[9]

## Usage in Freemasonry

"The interlacing triangles or deltas symbolize the union of the two principles or forces, the active and passive, male and female, pervading the universe ... The two triangles, one white and the other black, interlacing, typify the mingling of apparent opposites in nature, darkness and light, error and truth, ignorance and wisdom, evil and good, throughout human life." – Albert G. Mackey: Encyclopedia of Freemasonry
The hexagram, one of the world's most ancient symbols, is featured within and on the outside of many Masonic temples as a decoration. It may have been found within the structures of King Solomon's temple, from which Freemasons are inspired in their philosophies and studies. Like many other symbols in Freemasonry, the deciphering of the hexagram is non-dogmatic and left to the interpretation of the individual.

## Other uses

Aerial photograph of Heathrow Airport, London, 1955
• A six-point interlocking triangles has been used for thousands of years as an indication a sword was made, and "proved," in the Damascus area of the Middle East. Still today, it is a required "proved" mark on all official UK and United States military swords though the blades themselves no longer come from the Middle East.
• The Ulster Banner flag of Northern Ireland, used from 1953-1972. The six pointed star, representing the six counties that make up Northern Ireland.
• In Unicode, the "Star of David" symbol is U+2721.
• There is a plant named Solomon's seal (Polygonatum multiflorum) in the lily family.
• In alchemy, the two triangles represent the reconciliation of the opposites of fire and water. Non-Jewish Kabbalah (also called Christian or Hermetic Kabbalah) interprets[citation needed] the hexagram to mean the divine union of male and female energy, where the male is represented by the upper triangle and the female by the lower one. Moreover, it derives four triangular symbols from it (two triangles crossed like a capital letter A and two uncrossed) to represent the four elements: water, fire, air, and earth. This use of the symbol was used as an important plot point in Dan Brown's popular novel The Da Vinci Code and the Da Vinci Code film cites this as the origin of the star.
• It was also used as a sign for quintessence, the fifth element.[citation needed]
• In southern Germany the hexagram can be found as part of tavern anchors. It is symbol for the tapping of beer and sign of the brewer's guild. In German this is called "Bierstern" (beer star) or "Brauerstern" (brewer's star).
• A six-point star is used as an identifying mark of the Folk Nation.
• The main runways and taxiways of Heathrow Airport were arranged roughly in the shape of a hexagram.[10]
• A hexagram in a circle is incorporated prominently in the supports of Worthing railway station's platform 2 canopy (UK).[11]
Flag of Dardania
• A hexagram appears on the Dardania Flag, proposed for Kosovo by the Democratic League of Kosovo.
• An extremely large, free-standing wood hexagram stands in the central park of the Municipality of El Tejar, Guatemala. Additionally, every year at Christmastime the residents of El Tejar erect a giant fake Christmas tree in front of their municipal building, with a hexagram sitting at its peak.
• The Indian sage and seer Sri Aurobindo used it -- eg on the cover of his books -- as a symbol of the aspiration of humanity calling to the Divine to descend into life (the triangle with the point at the top), and the descent of the Divine into the earth's atmosphere and all individuals in response to that calling (the triangle with the point at the bottom). (This was explained by the Mother, his spiritual partner in Her 14-volume Agenda and elsewhere by Sri Aurobindo in his writings.)

## Other hexagrams

Other hexagrams can be constructed as a continuous path.
D2 symmetry
unicursal
D3 symmetry
isogonal
D3 symmetry
isotoxal

## Footnotes

1. The Muslim Empires of the Ottomans, Safavids, and Mughals, By Stephen F. Dale, 2009
2. Jump up ^ Delhi: Adventures In A Megacity (PB) - Page 91, Sam Miller, Sam - 2010
3. Jump up ^ sivasakti.com: Iintroduction to Yantra
4. Jump up ^ The Chakra Bible, Patricia Mercier, Octopus Publishing Group Ltd., 2007, p. 199
5. Jump up ^ "King Solomon's Seal", MFA, King Solomon-s Seal.
6. Jump up ^ P299 (and throughout) of The Complete Goldendawn by Israel Regardie. ISBN 978-0875426631
7. Jump up ^ "The History and Practice of Magic" (Secaucus, NJ: University Books, published by arrangement with Lyle Stewart, 1979), Vol. II, p. 304
8. Jump up ^ P315-316 of The Wicca Bible by Ann-Marie Gallagher. ISBN; 978-1-84181-250-2. Same information also found in many other books.
9. Jump up ^ P31 of The Ritual Magic Manual: A Complete Course in Practical Magic by David Griffin. ISBN 978-0965840897
10. Jump up ^ bbc.co.uk
11. Jump up ^ wikipedia.org image Worthing railway station platform 2 canopy

## References

• Grünbaum, B. and G. C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1.
• Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
• Graham, Dr. O.J. The Six-Pointed Star: Its Origin and Usage 4th ed. Toronto: The Free Press 777, 2001. ISBN 0-9689383-0-2
• Wessely, l.c. pp. 31, 112