Symmetry Pentagon

symmetries of regular pentagon. vertices colored symmetry positions. blue mirror lines drawn through vertices , edges. gyration orders given in center.

the regular pentagon has dih5 symmetry, order 10. since 5 prime number there 1 subgroup dihedral symmetry: dih1, , 2 cyclic group symmetries: z5, , z1.

these 4 symmetries can seen in 4 distinct symmetries on pentagon. john conway labels these letter , group order. full symmetry of regular form r10 , no symmetry labeled a1. dihedral symmetries divided depending on whether pass through vertices (d diagonal) or edges (p perpendiculars), , when reflection lines path through both edges , vertices. cyclic symmetries in middle column labeled g central gyration orders.

each subgroup symmetry allows 1 or more degrees of freedom irregular forms. g5 subgroup has no degrees of freedom can seen directed edges.