A geometrical model for the double octahedral group based on a genus two Riemann surface

Abstract

Geometrical models of double point groups are provided by double-sheeted Riemann surfaces of nonzero genus. This avoids the geometrically confusing picture of the doubling operation R as a rotation by 2π (i.e. 360°), which should instead be the identity operation E. In this manner a Riemann surface of genus 2 doubly covering a sphere and platonically tessellated b 16 equilateral triangles provides a geometrical model for the double octahedral group ²O_h. Stretching this Riemann surface along one axis to convert the 16 equilateral triangles to isosceles triangles and the underlying sphere to a prolate ellipsoid provides a model for the ²D_4h double group arising from the Jahn-Teller elongation of the regular octahedron into a prolate tetragonal bipyramid.