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Concept
Hexagramme (géométrie)
Résumé
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 a compound figure of two equilateral triangles. 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 various religious and cultural contexts and as decorative motifs. The symbol was used as a decorative motif in medieval Christian churches and Jewish synagogues. The hexagram is thought to have originated in Buddhism and was also used by Hindus. It was used by Muslims as a mystic symbol in the medieval period, known as the Seal of Solomon, depicted as either a hexagram or pentagram.
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 onto the plane .
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