Dosymetrii

Dosymetrii is a term used in some branches of geometry and theoretical discussion to describe a class of structures that exhibit two independent symmetry operations, each of which is its own inverse (an involution). In practice, these are objects or systems that admit two distinct reflections or similar actions that square to the identity. The two symmetries may act in the same space (for example, two mirror planes in a plane or three-dimensional object) and often generate a finite symmetry group.

In a geometric setting, the most common example is a shape that has two mirror symmetries, such as a rectangle in the plane, which has both a vertical and a horizontal axis of symmetry. The symmetry operations of these two reflections, together with the identity and their product (a 180-degree rotation in this case), form a four-element group. In higher dimensions or in tiling and molecular contexts, similar configurations of two order-two symmetries give rise to analogous four-element symmetry groups, sometimes referred to collectively as a Klein four group.

Etymology reflects the concept: dosymetrii derives from the Greek dos (‘two’) and symmetria (‘symmetry’). The term is used sporadically in mathematical and theoretical discussions and is not universally adopted as a standard label in core textbooks.

See also: symmetry, involution, dihedral group, Klein four group, geometry.