Black-and-White Symmetry, Magnetic Symmetry, Self-Duality and Antiprismatic Symmetry: The Common Mathematical Background

Seemingly distant areas of symmetry, such as mentioned in the title, are closely related to each other on an abstract level. The unified treatment is made possible by some common features of the symmetry groups used in the mathematical description of these types of symmetries. The key property is the way as a two-element group generated by an involutory transformation can extend a geometric symmetry group. The close relationship between the first two kinds of symmetry is well known. The actual physical objects of the one kind whose symmetry is to be described mathematically are magnetic crystals, or a bit more precisely, “crystals with magnetic structure” (LANDAU and LIFSHITZ, 1986). The symmetry in stricter sense of any crystal is in essence a geometric symmetry, namely invariance with respect to a congruent transformation (that is, the symmetry transformation may be rotation, reflection, translation or combinations of these). However, in a magnetic crystal one has to take into account a non-geometric symmetry transformation—time reversal. Pure time reversal does not affect the geometric structure of the crystal; i.e. the spatial position of the particles remains the same. If the constituent particles, however, carry a magnetic moment, this momentum is reversed. To carry over the crystal to a state indistinguishable from the original, time reversal must be followed by a pure geometric transformation. To be precise, it must be followed or preceded by that—the actual order is unessential, whatever is the pure geometric symmetry. This commutativity is one of the key properties of such a special kind of transformation and in what follows it will be referred to as property ( ). It is quite a natural idea to conceive the two opposite directions of the magnetic moment as if the particle in question were painted black or white. In this approach the crystal is considered as a symmetrical material pattern which is coloured symmetrically by two colours. Then time reversal is interpreted as colour reversal, i.e. “black particles” are painted over to “white” and vice versa. This idea was already raised in 1929 by Heesch as well as by Hermann (JANSSEN, 1973; LANDAU and LIFSHITZ, 1986; GRÜNBAUM and