Continuous symmetry measures

A continuous symmetry measure (CSM) quantifies how much a coordination polyhedron deviates from an ideal reference geometry. A CSM of zero indicates perfect agreement; larger values indicate greater distortion.

The CSM is computed using the approach of Pinsky and Avnir [1], which finds the optimal rotation, scaling, and vertex permutation to minimise the distance between the observed and reference vertex positions via SVD-based Procrustes analysis.

Computing symmetry measures

Every CoordinationPolyhedron has a symmetry_measure property that returns the CSM against all reference geometries defined for its coordination number:

poly = config.polyhedra[0]
poly.symmetry_measure
# {'Octahedron': 1.2e-30, 'Trigonal prism': 16.7, ...}

The keys are geometry names and the values are CSM values.

Best-fit geometry

The best_fit_geometry property returns the reference geometry with the smallest CSM:

poly.best_fit_geometry
# {'geometry': 'Octahedron', 'symmetry_measure': 1.2e-30}

Available reference geometries

The following reference geometries are available, grouped by coordination number:

CN	Geometries
4	Tetrahedron
5	Trigonal bipyramid, Square pyramid
6	Octahedron, Trigonal prism
7	Pentagonal bipyramid, Square-face capped trigonal prism, Face-capped octahedron
8	Cube, Square antiprism, Square-face bicapped trigonal prism, Triangular-face bicapped trigonal prism, Dodecahedron with triangular faces, Hexagonal bipyramid, Bicapped octahedron (opposed cap faces), Bicapped octahedron (cap faces with one atom in common), Bicapped octahedron (cap faces with one edge in common)

Reference coordinates are defined in polyhedral_analysis.reference_geometries.

Using SymmetryMeasure directly

For lower-level control, you can construct a SymmetryMeasure and compute the CSM against custom vertex coordinates:

from polyhedral_analysis.symmetry_measure import SymmetryMeasure

sm = SymmetryMeasure.from_name('Octahedron')
csm_value = sm.minimum_symmetry_measure(vertex_vectors)

The minimum_symmetry_measure() method searches over all symmetry-inequivalent vertex permutations, using the point-group symmetry of the reference geometry to reduce the number of permutations that must be tested.