Hexagram

In mathematics, the root system for the simple Lie group G₂ is in the form of a hexagram, with six long roots and six short roots.

A six-pointed star, like a regular 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.

A regular hexagram can be constructed by orthographically projecting any cube onto a plane through three vertices that are all adjacent to the same vertex. The twelve midpoints to edges of the cube form a hexagram. For example, consider the projection of the unit cube with vertices at the eight possible binary vectors in three dimensions ${\displaystyle (1,0,0),(0,1,0),(0,0,1),(1,1,0),(1,0,1),(0,1,1),(1,1,1)}$ onto the plane ${\displaystyle x+y+z=1}$. The midpoints are ${\displaystyle (0,0,1/2),(0,1/2,1/2),(0,1,1/2),(1,1,1/2)}$, and all points resulting from these by applying a permutation to their entries. These 12 points project to a hexagram: six vertices around the outer hexagon and six on the inner.

As a derivative of two overlapping triangles, the hexagram may have developed from different peoples with no direct correlation to one another.

The oldest known depiction of a six-pointed star (dating back to the 3rd millennium BC.) was excavated in the Ashtarak burial mound in “Nerkin Naver” (in Historic Armenia).

The mandala symbol called yantra, found on ancient South Indian Hindu temples, is a geometric toolset that incorporates hexagrams into its framework. 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).

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 (Wessely, l.c. pp. 31, 112) at Paris and London there are 22 signs side by side, and a circle with twelve signs, but neither a pentagram nor a hexagram.

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. In Albanian heraldry and vexillology, hexagram has been used since classical antiquity and it is commonly referred to as sixagram. The coat of arms of the House of Kastrioti depicts the hexagram on a pile argent over the double headed eagle.

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.

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[8] and for conjuring spirits and spiritual forces in diverse forms of 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.[9] 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.[10]

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 Tree of Life. Saturn, although formally attributed to the Sephira of Binah, within this frame work nonetheless occupies the position of Daath.[11]

In alchemy, the two triangles represent the reconciliation of the opposites of fire and water.[12]

The hexagram is used as a sign for quintessence, the fifth element.

"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 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.

The figure {6/3} can be shown as a compound of three digons.

Other hexagrams can be constructed as a continuous path.

Regular compounds		D2 symmetry unicursal	D3 symmetry isogonal				D3 symmetry isotoxal
{6/2}=2{3}	{6/3}=3{2}

• Pentagram
• Star of Bethlehem
• Star of David
• Seal of Solomon
• Heptagram
• The Thelemic Unicursal hexagram
• Pascal's mystic hexagram
• Hexagram (I Ching)
• Sacred Geometry

• Scholem, Gershom (September 1949). "The Curious History of the Six-Pointed Star: How the "Magen David" Became the Jewish Symbol". Commentary Magazine. Retrieved 2018-07-10.
• 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
• 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.
• Wessely, l.c. pp. 31, 112

Wikimedia Commons has media related to Hexagrams.

• Hexagram (MathWorld)
• The Archetypal Mandala of India
• Thesis from Munich University on hexagram as brewing symbol