Rotoinversion

Rotoinversion axes are compound symmetry operations that combine rotation with inversion through the centre of the crystal in a single step. ^[1] The notation uses a capital “A” with a bar over it and a subscript - so Ā4 denotes a 4-fold rotoinversion. There are five rotoinversion axes: Ā1, Ā2, Ā3, Ā4, and Ā6. Three of them turn out to be equivalent to combinations of simpler symmetry elements already encountered; one - Ā4 - is unique and cannot be reproduced by any simpler combination.

The Five Rotoinversion Axes

Ā1 (1-fold rotoinversion): Rotation by 360° followed by inversion. This is trivially equivalent to simple inversion, so Ā1 = i. ^[1]

Ā2 (2-fold rotoinversion): Rotation by 180° followed by inversion. The result places a point directly below the original, and the operation turns out to be equivalent to a mirror plane perpendicular to the rotation axis - so Ā2 = m. ^[1]

Ā3 (3-fold rotoinversion): Rotation by 120° followed by inversion, repeated three times. The net result is equivalent to a conventional 3-fold rotation axis combined with inversion centre - so Ā3 = A3 + i. ^[1]

Ā4 (4-fold rotoinversion): Rotation by 90° followed by inversion. This operation produces four faces or sets of faces at 90° to each other that alternate between being upright and upside down. ^[1] This is the only rotoinversion axis that cannot be broken down into simpler symmetry elements - Ā4 is genuinely new and is responsible for the distinctive appearance of crystals in certain tetragonal crystal classes.

Ā6 (6-fold rotoinversion): Rotation by 60° followed by inversion. The result is equivalent to a 3-fold rotation axis perpendicular to a mirror plane - so Ā6 = A3 + m. ^[1]

Why Rotoinversion Matters

Four of the five rotoinversion axes reduce to combinations of simpler operations - but Ā4 cannot be duplicated with any combination of simpler symmetry elements. ^[1] This matters for mineral symmetry because Ā4 appears in several crystal classes - particularly in the tetragonal system - and describes a type of symmetry that cannot be identified by looking for mirrors and rotation axes alone. A crystal with Ā4 symmetry but no mirror plane and no conventional 4-fold axis belongs to a different crystal class than one with A4, and the two have different physical and optical properties as a consequence.

References

1. Nesse, W. D. (2017). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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