A small black and white image of D4 transformations acting on a squarecubeLife.org

The action of D4 rotational symmetry on a magic matrix

animation of D4 transformations acting on a square

Through 4 rotations and one reflection (or vice-versa), a triangular one-eighth segment of a square will cover the entire surface area of the same square. This is known as the dihedral group of symmetries, and in 4-sided figures occurs in those that are regular and symmetrical along two axes at 90º to each other, like… squares.

The diagram above (see large version) consists of stills from an animation illustrating these 4 rotations and one reflection.

When this simple procedure is applied to a magic square matrix, it can be shown how the process of rotation and reflection demonstrates that the unique positions available to any number in a magic matrix must occur within the triangular segment, and (incidentally) why rotation and reflection create trivial duplicate magic squares.