What is the smallest possible angle measure in which a figure can be mapped onto itself?

The smallest possible angle measure in which a figure can be mapped onto itself is the angle of rotation. This refers to the smallest angle you can rotate the figure around a central point such that it looks exactly the same as it did before the rotation.

For instance, consider a regular polygon; the angle of rotation that allows it to map onto itself is determined by the number of sides. For a square, this angle is 90 degrees, as rotating the square by 90 degrees results in the same configuration.

This concept is closely related to symmetry, where a figure can exhibit rotational symmetry if it can be rotated around a point and still appear unchanged. The other terms, while related, do not define the smallest angle of mapping. The order of symmetry pertains to how many times a figure can be rotated to look the same within a full rotation, while a full rotation is 360 degrees, which does not represent the smallest angle. Slicing angle isn't a standard term in this context. Therefore, the angle of rotation specifically captures the essence of the smallest measure for this mapping.