Final answer:
Without the specific number of sides of the polygon, we cannot definitively determine which rotations will map the polygon onto itself. Typically, a regular polygon maps onto itself with rotations that are multiples of its central angle. Relevant rotations depend on the polygon being an octagon or hexagon in the given choices.
Step-by-step explanation:
To determine which transformations carry a regular polygon onto itself, we must consider the properties of the polygon. A regular polygon will only map onto itself if it is rotated by multiples of the central angle, which is the angle subtended at the center of the polygon by one of its sides. Without knowing the specific number of sides of the polygon, we cannot definitively answer which rotations will map the polygon onto itself. However, general rules can be applied assuming the polygon has a corresponding number of sides to support the specified rotations.
For example, if the regular polygon is an octagon, it has 8 sides, and the central angle is 360 degrees divided by 8, which equals 45 degrees. Thus, a rotation of 45 degrees counterclockwise would map an octagon onto itself. If the polygon is a hexagon with 6 sides, each central angle would be 60 degrees, so both rotation of 60 degrees clockwise and rotation of 60 degrees counterclockwise would map a hexagon onto itself. Without knowing the specific polygon, we cannot assess if rotation of 36 degrees counterclockwise would map the polygon onto itself, as it would require that the polygon has 10 sides (a decagon).
The information provided about merry-go-rounds, cubic shapes, magnetic fields, and coordinate systems, while interesting, is not relevant to solving this problem.