A hexagram with equidistant points is unique in geometry because it cannot be drawn unicursally – that is, without lifting and repositioning the pen. Instead, overlapping two individual triangles form the hexagram.
A unicursal hexagram is possible – the creation of a six pointed shape without lifting the pen.
The Star of DavidThe most common depiction of the hexagram is the Star of David, also known as the Magen David. This is the symbol on the flag of Israel, which Jews have been commonly using as a symbol of their faith for the last couple of centuries. This is also the symbol that multiple European communities have historically forced Jews to wear as identification, most notably by Nazi Germany in the 20th century.
The evolution of the Star of David is unclear. In the Middle Ages, the hexagram was often referred to as the Seal of Solomon, referencing a Biblical king of Israel and son of King David. The hexagram also came to have Kabbalistic and occult meaning. In the 19th century, the Zionist movement adopted the symbol. Because of these multiple associations, some Jews, particularly some Orthodox Jews, do not use the Star of David as a symbol of faith.
Read more: Star of David
The Seal of SolomonThe Seal of Solomon originates in medieval tales of a magical signet ring possessed by King Solomon and having the power to bind and control supernatural creatures. Often the seal is described as a hexagram, but some sources describe it as a pentagram.
Duality of the Two TrianglesIn Eastern, Kabbalistic, and occult circles, the hexagram's meaning is commonly closely tied to the fact that it is composed of two triangles pointing in opposite directions. This relates to the union of opposites, such as male and female. It also commonly references the union of the spiritual and the physical, with spiritual reality reaching down and physical reality stretching upward.
This intertwining of worlds can also be seen as a representation of the Hermetic principle "As above, so below," referencing how changes in one world reflect changes in the other.
Finally, triangles are commonly used in alchemy to designate the four different elements. The more rarified elements – fire and air – have point down triangles, while the more physical elements – earth and water – have point up triangles.
Read more: Elemental Symbols
Modern and Early Modern Occult ThoughtBecause the triangle is such a central symbol in Christian iconography, representing the trinity and thus spiritual reality, the use of the hexagram in Christian occult thought is fairly common. In the 17th century, Robert Fludd produced an illustration of the world in which God was an upright triangle, and the physical world was his reflection and thus downward pointing. The triangles only slightly overlap, thus not creating a hexagram of equidistant points, but the structure is still present.
Likewise, in the 19th century Eliphas Levi produced his Great Symbol of Solomon, "The Double Triangle of Solomon, represented by the two Ancients of the Kabbalah; the Macroprosopus and the Microprosopus; the God of Light and the God of Reflections; of mercy and vengeance; the white Jehovah and the black Jehovah."
Read more: Levi's Great Symbol of Solomon
"Hexagram" in Non-Geometric ContextsThe Chinese I-Ching (Yi Jing) is based off of 64 different arrangements of broken and unbroken lines, with each arrangement having six lines. Each arrangement is referred to as a Hexagram.
Unicursal HexagramThe unicursal hexagram is a six-pointed star that can be drawn in one continuous movement. Its points are equidistant, but the lines are not of equal length (unlike a more standard hexagram). It can, however, fit inside a circle with all six points touching the circle.
Cycle through the images at the top of this article to see a diagram of the construction of a unicursal hexagram.
The meaning of the unicursal hexagram is largely identical to that of a standard hexagram: the union of opposites. The unicursal hexagram, however, more strongly emphases the intertwining and ultimate unity of the two halves, rather than two separate halves coming together. Furthermore, occult practices often involve the tracing of symbols during ritual, and a unicursal design better lends itself to this practice.
The unicursal hexagram is commonly depicted with a five-pedaled flower in the center. This is a variation created by Aleister Crowley, and is most strongly associated with the religion of Thelema. Another variation is the placement of a small pentagram in the hexagram's center.