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

The action of D4 rotational symmetry on a magic matrix

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 90° to each other, like… squares.

The diagram above (it links to a larger image) consists of stills from an animation illustrating these 4 rotations and one reflection (greyscale animated gif | QuickTime colour movie.

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.