The Incongruity of Counterparts

Extract

When a body is turned through 180°, reversal occurs in two dimensions. If it is turned about the vertical axis, then left and right sides change places, and also front and back. If it is turned about the horizontal axis which runs from side to side, then top and bottom change places, and also front and back. If, finally, it is turned about the horizontal axis which runs from front to back, then left and right change places as well as top and bottom. All this appears to follow from the properties of three-dimensional space and of rigid bodies. If XOX' is the axis of rotation of a three-dimensional structure, then there are two axes YOY' and ZOZ' at right angles to XOX', and if one of these moves round 180°, then so must the other. It follows that a body is always reversed in two dimensions whenever it is turned about an axis in the third. It follows also that there are always two ways of turning a body in space so as to reverse it in a given dimension: namely by turning it about either of two axes perpendicular to the given dimension.