Symmetry Combinations

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In 2D, mirror-image symmetry in combination with n-fold rotational symmetry, with the center of rotational symmetry on the line of symmetry, implies mirror-image symmetry with respect to lines of reflection rotated by multiples of 180°/n, i.e. n reflection lines which are radially spaced evenly (for odd n this already follows from applying the rotational symmetry to a single reflection axis, but it also holds for even n). The symmetry group is the dihedral group of order 2n. For n 2 an example is the n-sided regular polygon and various n-sided star polygons, including complex ones, which are a combination of simple ones for a divisor of n; also we have the simple "star" of n radial line segments (for even n this is a degenerate star polygon, for odd n it is not). Also multiple regular n-sided polygons with common center, differing by arbitrary rotations, as long as these rotation angles have mirror-image symmetry, e.g. two squares differing by a rotation angle of 10°, or three squares differing by two successive rotation angles of 10°.

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High Quality Content by WIKIPEDIA articles! In 2D, mirror-image symmetry in combination with n-fold rotational symmetry, with the center of rotational symmetry on the line of symmetry, implies mirror-image symmetry with respect to lines of reflection rotated by multiples of 180°/n, i.e. n reflection lines which are radially spaced evenly (for odd n this already follows from applying the rotational symmetry to a single reflection axis, but it also holds for even n). The symmetry group is the dihedral group of order 2n. For n 2 an example is the n-sided regular polygon and various n-sided star polygons, including complex ones, which are a combination of simple ones for a divisor of n; also we have the simple "star" of n radial line segments (for even n this is a degenerate star polygon, for odd n it is not). Also multiple regular n-sided polygons with common center, differing by arbitrary rotations, as long as these rotation angles have mirror-image symmetry, e.g. two squares differing by a rotation angle of 10°, or three squares differing by two successive rotation angles of 10°.