Hexagon has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations. The 6 roots of the simple Lie group , represented by a Dynkin diagram , are in a regular hexagonal pattern. The two simple roots have a 120° angle between them. The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between
Pappus's hexagon theorem Pappus's hexagon theorem In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points "A", "B", "C", and another set of collinear points "a", "b", "c", then the intersection points "X", "Y", "Z" of line pairs "Ab" and "aB", "Ac" and "aC", "Bc" and "bC" are collinear, lying on the "Pappus line". These three points are the points of intersection of the "opposite" sides of the hexagon "AbCaBc". It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem"