Hexagram

A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or {{3}}. Since there are no true regular continuous hexagrams, the term is instead used to refer to the compound figure of two equilateral triangles shown to the right. The intersection is a regular hexagon.Morocco Fez Embroidery Horse CoverHexagram at Humayun's Tomb, Delhi, IndiaHexagram on obverse of Moroccan 4 Falus coin (1873)Hexagram on the Minaret of Arasta Mosque, Prizren, KosovoHexagram on the flag of Hayreddin BarbarossaHexagram on the flag of Karamanid beylik'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 A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or {{3}}. Since there are no true regular continuous hexagrams, the term is instead used to refer to the compound figure of two equilateral triangles shown to the right. The intersection is a regular hexagon. The hexagram is part of an infinite series of shapes which are compounds of two n-dimensional simplices. In three dimensions, the analogous compound is the stellated octahedron, and in four dimensions the compound of two 5-cells is obtained. It has been historically used in religious and cultural contexts and as decorative motifs; for example by medieval Muslims, especially Hanafi and Maliki dynasties, and later in Judaism and occultism. The symbol was used merely as a decorative motif in medieval Christian churches many centuries before its first known use in a Jewish synagogue. It was first used as a religious symbol by Arabs in the medieval period, known as the Seal of Solomon, depicted as either a hexagram or pentagram, and which was later adopted by Jewish Kabbalists. 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: 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 ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) , ( 1 , 1 , 0 ) , ( 1 , 0 , 1 ) , ( 0 , 1 , 1 ) , ( 1 , 1 , 1 ) {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 x + y + z = 1 {displaystyle x+y+z=1} . The midpoints are ( 0 , 0 , 1 / 2 ) , ( 0 , 1 / 2 , 1 / 2 ) , ( 0 , 1 , 1 / 2 ) , ( 1 , 1 , 1 / 2 ) {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.