A regular hexagon with a vertex at the top containing an equilateral triangle with vertex at the top and centroid at the center of the hexagon, using translation directions given by vectors (1, 0) and (1, √3).

Describe a pattern to be used to fill the plane by translation in the two directions given by the specified vectors. Answer these questions in each case. 

a. Does the symmetry group contain any rotations? If so, through what possible angles θ where 0 < θ ≤ 180°?

b. Does the symmetry group contain any reflections? 

c. Does the symmetry group contain any nontrivial glide reflections?

 

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