The capsid is the protein coat that surrounds the genetic material (DNA or RNA) of a virus, providing protection and facilitating the virus’s interaction with host cells. The arrangement of proteins in the capsid determines its symmetry. There are three main types of symmetry observed in virus capsids: icosahedral, helical, and complex.

1. Icosahedral Symmetry:
   • Icosahedral symmetry is characterized by a roughly spherical shape with 20 equilateral triangular faces. Each face is an equilateral triangle and every vertex of the icosahedron is formed by 5 triangular faces. The icosahedral capsid is made up of identical protein subunits called capsomers. Each capsomer contributes to the overall symmetry, and the arrangement allows for efficient packaging of the viral genetic material. Many viruses, such as adenoviruses, polioviruses, and herpesviruses, exhibit icosahedral symmetry.Icosahedron: a geometric solid with 20 faces, 30 edges and  12 vertices.
2. Helical Symmetry:
   • Helical symmetry is characterized by a rod-like or filamentous structure. The capsid proteins arrange themselves in a helical pattern around the viral genetic material. This structure is often observed in viruses that infect plants (e.g., Tobacco Mosaic Virus), animals (e.g., influenza virus), and bacteria (e.g., bacteriophages). The length of the helix is determined by the size of the viral genome.
3. Complex Symmetry:
   • Complex symmetry refers to viruses that do not strictly fit the icosahedral or helical categories. These viruses may have additional structures, such as tail fibers or other elaborate features. Bacteriophages (viruses that infect bacteria) often exhibit complex symmetry. Tailed bacteriophages, for example, have an icosahedral head connected to a tail-like structure.

Understanding the symmetry of the capsid is crucial for studying the biology of viruses and for developing antiviral drugs and vaccines. The symmetry influences the stability and assembly of the viral particle, as well as its ability to interact with host cells during the infection process.

It’s important to note that some viruses may exhibit a combination of symmetries. For instance, some viruses with complex symmetry may have an icosahedral head combined with a helical tail. The study of virus capsid symmetry has been greatly facilitated by advanced imaging techniques, such as X-ray crystallography and cryo-electron microscopy, which allow scientists to visualize and analyze the three-dimensional structures of viruses at high resolution.