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 collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon ADEGBZ are collinear. Lemmas XV and XVII are that, if the point M is determined as the intersection of HK and BG, then the points A, M, and D are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon BEKHZG are collinear. In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements: Pappus's hexagon theorem In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set