## Special Symmetric hexagons (h)

These hexagons result by applying a 2x2 non-singular matrix to the vertices of the unit-regular-hexagon (h6) (whose vertices are the 6-ths roots of the unit). The matrix defines a linear transformation (affinity) (f) and the vertices of the hexagon (h) are the images of the vertices of (h6) under f. If A and B are the points (1,0) and (cos(pi/3), sin(pi/3)) (6-th primitive root), then their images A' = f(A) and B' = f(B) span the hexagon h in the same way A and B span the hexagon h6. i.e. the vertices of h are A', B', B'-A', -A', -B', A'-B'. The hexagon has in- and circum- scribed ellipses, which are the images of the in- and circum- circles of h6, correspondingly. The two ellipses are similar and their similarity ratio is r = sqrt(3)/2.
Not every symmetric hexagon is of this kind. A necessary and sufficient condition to be of this kind is that, placed with its center at (0,0) it has three consecutive vertices of the kind A', B', B'-A' (or A', B', A'+B').