The rotations that carry a regular pentagon onto itself can be classified into two main types: rotations of 72 degrees and rotations of 144 degrees. A rotation of 72 degrees will rotate the pentagon by one-fifth of a full turn, while a rotation of 144 degrees will rotate the pentagon by two-fifths of a full turn. These rotations preserve the shape and orientation of the pentagon, allowing it to appear identical when rotated.
To carry a regular pentagon onto itself, there are three possible rotations. The first is a rotation of 72 degrees, which involves rotating the pentagon by one-fifth of a full turn. The second is a rotation of 144 degrees, which involves rotating the pentagon by two-fifths of a full turn. The third is a rotation of 216 degrees, which involves rotating the pentagon by three-fifths of a full turn. These three rotations can be repeated to create different transformations of the regular pentagon onto itself.
To summarize, a regular pentagon has five-fold rotational symmetry, meaning it can be rotated by angles of 72 degrees, 144 degrees, 216 degrees, and 288 degrees to carry it onto itself. These are the only rotations that will exactly align the pentagon's vertices back to their original positions. Each rotation corresponds to a different number of complete turns around the center of the pentagon, with 72 degrees representing one turn, 144 degrees representing two turns, and so on.
