The icosahedral structure is extremely common among viruses. The number and arrangement of capsomeres in an icosahedral capsid can be classified using the "quasi-equivalence principle" proposed by Donald Caspar and Aaron Klug. Like the Goldberg polyhedra, an icosahedral structure can be regarded as being constructed from pentamers and hexamers. The structures can be indexed by two integers h and k, with and ; the structure can be thought of as taking h steps from the edge of a pentamer, turning 60 degrees counterclockwise, then taking k steps to get to the next pentamer. The triangulation number T for the capsid is defined as:
Icosahedral capsids contain 12 pentamers plus 10(T − 1) hexamers.  The T-number is representative of the size and complexity of the capsid proteins: all know viruses with T > 7 require auxiliary proteins for assembly. Geometric examples for many values of h, k, and T can be found at List of geodesic polyhedra and Goldberg polyhedra. Although this classification can be applied to the majority of known viruses, exceptions are known including the retroviruses where point mutations disrupt the symmetry.
T-numbers can be represented in different ways, for example T = 1 can only be represented as an icosahedron or a dodecahedron and, depending on the type of quasi-symmetry, T = 3 can be presented as a truncated dodecahedron, an icosidodecahedron, or a truncated icosahedron and their respective duals a triakis icosahedron, a rhombic triacontahedron, or a pentakis dodecahedron.
An elongated icosahedron is a common shape for the heads of bacteriophages. Such a structure is composed of a cylinder with a cap at either end. The cylinder is composed of 10 triangles. The Q number, which can be any positive integer
, specifies the number of triangles, composed of asymmetric subunits, that make up the 10 triangles of the cylinder. The caps are classified by the T number.
3D model of a helical capsid structure of a virus
Many rod-shaped and filamentous plant viruses have capsids with helical symmetry. The helical structure can be described as a set of n 1-D molecular helices related by an n-fold axial symmetry. The helical transformation are classified into two categories: one-dimensional and two-dimensional helical systems. Creating an entire helical structure relies on a set of translational and rotational matrices which are coded in the protein data bank. Helical symmetry is given by the formula P = μ x ρ, where μ is the number of structural units per turn of the helix, ρ is the axial rise per unit and P is the pitch of the helix. The structure is said to be open due to the characteristic that any volume can be enclosed by varying the length of the helix. The most understood helical virus is the tobacco mosaic virus. The virus is a single molecule of (+) strand RNA. Each coat protein on the interior of the helix bind three nucleotides of the RNA genome. Influenza A viruses differ by comprising multiple ribonucleoproteins, the viral NP protein organizes the RNA into a helical structure. The size is also different the tobacco mosaic virus has a 16.33 protein subunits per helical turn, while the influenza A virus has a 28 amino acid tail loop.