Rotoinversion, also called improper rotation, is a crystallographic symmetry operation that combines a rotation about an axis with an inversion through a centre point lying on that axis. The rotation and the inversion together form a single compound operation; performing the two in sequence yields a transformation that may not be reproducible by any combination of pure rotations and pure mirror reflections alone, and which is therefore counted as a distinct elementary symmetry operation. Rotoinversion is one of the operations used to derive the thirty-two crystal classes that organise crystalline solids by symmetry, and the rotoinversion-bearing classes are central to the systematic mineralogy of gem species.

Designation and operation

Rotoinversion axes are designated by an integer fold number with an overbar — written 1^—, 2^—, 3^—, 4^—, and 6^—, often read aloud as bar one, bar two, and so forth. A rotoinversion of fold n applies a rotation through 360°/n about the axis followed immediately by an inversion through the centre point on the axis. The 1^— operation reduces to a pure inversion; the 2^— operation reduces to a mirror reflection across the plane perpendicular to the axis; the 3^—, 4^—, and 6^— operations are independent symmetry operations not reducible to simpler forms.

The 4^— operation, in particular, generates a symmetry pattern that combines aspects of a 4-fold rotation with an inversion centre but that cannot be reproduced by 4-fold rotation alone or by any single mirror reflection. The tetragonal scalenohedral class, characterised by 4^— symmetry, includes the gem species chalcopyrite and certain zircon morphologies. The 3^— operation defines the trigonal-rhombohedral class containing dolomite and tourmaline-related morphologies; 6^— defines the trigonal-dipyramidal class.

Equivalence to other operations

Two of the rotoinversion operations are equivalent to simpler operations. The 1^— rotoinversion — a rotation through 360° (which is no rotation) followed by an inversion — is geometrically identical to a pure inversion through the centre point. A crystal with 1^— symmetry therefore possesses a centre of symmetry. The 2^— rotoinversion — a rotation through 180° followed by inversion — is geometrically identical to a mirror reflection across the plane perpendicular to the rotation axis. A crystal with 2^— symmetry therefore possesses a mirror plane.

This equivalence permits the alternative notation system based on rotations and mirror reflections (the Hermann-Mauguin notation in its mirror-explicit form, or the Schoenflies notation), which omits 1^— and 2^— in favour of the equivalent inversion centre and mirror plane. Both notation systems describe the same set of thirty-two classes; the choice of notation is a matter of convention and convenience for the application.

Relevance to gemmology

The rotoinversion-bearing classes include several gem species of consequence. Tourmaline, in the trigonal-pyramidal class with a 3-fold rotation axis but no rotoinversion, is hemimorphic — its top and bottom terminations differ — and therefore pyroelectric and piezoelectric, properties exploited in some technological applications and observed in routine handling of tourmaline crystals. Quartz, in the trigonal-trapezohedral class, also lacks rotoinversion symmetry, is hemimorphic at the level of its tetrahedral substructure, and therefore exhibits enantiomorphism (left- and right-handed crystal forms) and the related optical-rotation phenomenon for which quartz is well known.

By contrast, gem species in centrosymmetric classes — those carrying a 1^— centre of symmetry — cannot be optically active and cannot be pyroelectric or piezoelectric. Diamond, garnet, spinel, and most other cubic gem species fall into centrosymmetric classes, as do the orthorhombic species peridot and topaz. The presence or absence of rotoinversion symmetry, and specifically of a 1^— centre of symmetry, is therefore a determinant of optical and physical behaviour relevant to gemmological identification and to the systematic mineralogy of gem materials.

Further reading

• GIA Gems & Gemology — crystallography and optical-properties reference
• International Gem Society — crystal systems and classes
• GIA gem encyclopaedia — optical properties