Institut de Biologie StructuraleGrenoble / France

Contact person(s) related to this article / FARIAS ESTROZI Leandro

Symmetry-adapted Functions

The reconstruction of objects with point-group symmetry can be highly optimized by means of the Symmetry Adapted Functions (SAFs) (Navaza, 2003). SAFs are linear combinations of spherical harmonics which are invariant with respect to a given point-group symmetry.

The SAFs depend on the angular variables and each function is defined by only two integer parameters: ℓ (associated to the angular resolution) which only takes values allowed by the symmetry and µ (the multiplicity) which specifies the function for a given ℓ . Below you find the first 15 SAFs for the icosahedral group:

ℓ=6, µ=1 ℓ=10,µ=1
ℓ=12, µ=1 ℓ=15, µ=1
ℓ=16, µ=1 ℓ=18, µ=1
ℓ=20, µ=1 ℓ=21, µ=1
ℓ=24, µ=1 ℓ=25, µ=1
ℓ=26, µ=1 ℓ=27, µ=1
ℓ=28, µ=1 ℓ=30, µ=1
ℓ=30, µ=2

Note that some SAFs have a visually well defined triangulation number! The use of SAFs as a basis to represent point-symmetric structures is like using musical notation to represent music. It’s hard to represent noise in musical notation, isn’t it?