Hexagram
A hexagram is a six-pointed plane figure formed by the intersection of two congruent equilateral triangles sharing the same center, with one triangle rotated 180 degrees relative to the other.[1][2] In geometry, it is classified as the star polygon with Schläfli symbol {6/2}, equivalent to a compound of two equilateral triangles.[3] The symbol, also termed the Star of David (Magen David) in Jewish tradition and the Seal of Solomon in Islamic and esoteric contexts, appears in artifacts and architecture across ancient and medieval cultures, including Hindu, Jain, and Byzantine influences, often as a decorative or apotropaic motif denoting harmony, protection, or cosmic balance.[4] Its adoption as a primary emblem of Judaism emerged prominently in the 17th century among European Jewish communities, later becoming central to Zionist and Israeli iconography, though earlier uses in synagogues and manuscripts were sporadic and non-exclusive.[5] Mathematically, the hexagram exhibits symmetries of the dihedral group D6 and relates to structures like the root system of Lie algebra G2, underscoring its recurrence in crystallography and higher-dimensional projections.[3] Despite its ubiquity, interpretations vary, with some traditions attributing mystical properties such as warding off evil, while geometric analyses emphasize its construction via extending hexagon sides or intersecting triangles.[6]Geometry
Definition and Properties
A hexagram is a six-pointed geometric figure formed by the compound of two equilateral triangles, one pointing upward and the other downward, rotated 180 degrees relative to each other. It is classified as a regular star polygon with Schläfli symbol {6/2} or equivalently 2{3}. The intersection of the triangles creates a regular hexagon at the center.[3][7] The regular hexagram has six vertices equally spaced on a circumscribed circle and consists of twelve line segments outlining the two triangles. Its symmetry group is the dihedral group D_6, of order 12, comprising six rotational symmetries (by multiples of 60 degrees) and six reflection symmetries across axes passing through opposite vertices and midpoints of opposite sides. For a side length a of the constituent equilateral triangles, the circumradius (distance from center to vertex) is a / \sqrt{3}, and the side length of the central hexagon is a/3.[3][7] The area of the hexagram, taken as the union of the two triangles, is \frac{\sqrt{3}}{3} a^2, accounting for the overlap in the central hexagon whose area is \frac{\sqrt{3}}{6} a^2. This figure has a density of 2, meaning the winding number around the center is 2, distinguishing it from simple polygons.[3][7]Constructions
A regular hexagram, formed as the compound of two equilateral triangles rotated by 60 degrees relative to each other, is constructible with compass and straightedge. The process begins by inscribing the vertices of a regular hexagon in a circle, as the hexagram's points coincide with those of the hexagon.[8] To construct it:- Draw a circle centered at point O using the compass.[8]
- Mark an arbitrary point A on the circumference.[9]
- Adjust the compass width to the circle's radius (distance OA), place the point at A, and mark the intersection point B with the circle.[8]
- Repeat the process successively from B to C, C to D, D to E, and E to F, yielding six equally spaced points on the circle separated by 60-degree arcs.[9][8]
- Using the straightedge, connect A to C to E to A, forming one equilateral triangle.[10]
- Connect B to D to F to B, forming the second equilateral triangle.[10]