Solution: A regular 12-sided polygon (dodecagon) has rotational symmetry of order 12. The full set of rotational symmetries consists of rotations by multiples of $ \frac{360^\circ}{12} = 30^\circ $. These rotations are: $ 0^\circ, 30^\circ, 60^\circ, \dots, 330^\circ $. This forms an arithmetic sequence with first term $ 0^\circ $, common difference $ 30^\circ $, and 12 terms. The sum of an arithmetic sequence is given by:

["The Rotational Symmetry of a Regular Dodecagon: Understanding Its Symmetry Group", "A regular 12-sided polygon, known as a dodecagon, is not only visually striking but also rich in geometric symmetry. One of its most fascinating properties is its rotational symmetry of order 12 — a concept deeply rooted in both mathematics and geometry.", "### What Is Rotational Symmetry?", "Rotational symmetry refers to a shape’s ability to look identical after being rotated around its center by a certain angle. For a regular polygon with ( n ) sides, the rotational symmetry order is ( n ). In the case of a regular dodecagon (( n = 12 )), this means the shape maps onto itself 12 times during a full 360° rotation.", "### The Rotational Symmetries of a Dodecagon", "The rotational symmetries of a regular dodecagon are rotations by multiples of ( \frac{360^\circ}{12} = 30^\circ ). These rotations include:", "[\n0^\circ, 30^\circ, 60^\circ, 90^\circ, 120^\circ, 150^\circ, 180^\circ, 210^\circ, 240^\circ, 270^\circ, 300^\circ, 330^\circ\n]", "This sequence forms an arithmetic progression with:", "- First term ( a = 0^\circ )\n- Common difference ( d = 30^\circ )\n- Number of terms ( n = 12 )", "Such symmetry patterns are not only elegant but also fundamental to understanding symmetry groups in geometry — important concepts in mathematics, physics, and even art and design.", "### Calculating the Sum of Rotations", "The sum of an arithmetic sequence can be powerful in mathematical analysis. The sum ( S ) of the first ( n ) terms is given by:", "[\nS = \frac{n}{2} \ imes (\ ext{first term} + \ ext{last term})\n]", "For the dodecagon, this becomes:", "[\nS = \frac{12}{2} \ imes (0^\circ + 330^\circ) = 6 \ imes 330^\circ = 1980^\circ\n]", "So, the total of all rotational symmetries (when combined in angular terms) amounts to 1980 degrees.", "This numeric insight helps quantify the symmetry’s scale and reinforces why the dodecagon holds group-theoretic significance — particularly within the dihedral group ( D_{12} ), which describes all symmetries (rotations and reflections) of the shape.", "### Why This Matters", "Understanding rotational symmetry helps in fields ranging from crystallography to architectural design and computer graphics. The regular dodecagon’s perfect 12-fold symmetry makes it a favorite in tiling, ornament design, and even digital modeling.", "By recognizing the structure of its symmetry—encoded in this arithmetic sequence—we uncover the deep mathematical order embedded in geometric form.", "---", "In summary, the rotational symmetry of a dodecagon — with 12 evenly spaced rotational symmetries from ( 0^\circ ) to ( 330^\circ ) in 30° increments — forms an arithmetic sequence whose sum is 1980°. This elegant mathematical property reflects the harmony inherent in regular polygons and underscores their importance in both theoretical and applied contexts."]