Geometry of Virus


This semester I am taking a course about protein structures. Here is a quick intro to proteins:

Though I have taken some other biology courses in past years, I found this course very much relatable to mathematics. Proteins are made up of “amino acids”. Though, chemistry allows large number of possible structures for amino acids (considering steric hindrance etc.), nature uses only 20 unique amino acids to make billions of different proteins. In my opinion, these 20 amino acids are “axioms” of protein building just like the 5 axioms of euclidean geometry.

Using just 20 amino acids we can get a large variety of protein structures, just like creating any kind of shape in euclidean space using just 5 axioms. Even more fascinating is the existence of “Quasisymmetry in Icosahedral Viruses”. An awesome article explaining this is available here. Note that, the term “triangulation number” stated in that article was not borrowed from mathematics. It’s a term used to study symmetries in icosahedral viruses and refers to “the square of the distance between 2 adjacent 5-fold vertices.”


200 Icosahedral Viruses from the PDB (source:

Moreover, the structures which don’t conform to classic quasisymmetry are similar to Escher print and Penrose tiling, as visible in following picture:

If you are interested in doing a fun activity, you may refer to:


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