I am currently writing a research essay concerning Crystallographic theories applied to Virology. Do keep in mind that I am not a mathematician and I know very little about Coxeter groups in general. Please bear with me; I'm trying.
One of my sources,"Mathematical Virology: a novel approach to the structure and assembly of viruses" (Twarock 2006), draws a connection between the "six-dimensional root lattice 6D and the non-crystallographic Coxeter group H3". According to the paper, Twarock was able to use H3, and it's affine extension, to infer generalized lattices of the capsid structure.
What distinct a 'non-crystallographic' Coxeter group from a 'crystallographic' one?
Also, I believe "affine extension" to be an extension of an object that preserves the parallel relationship of its parts, sort of a scaling thing. Is this true?
And is it correct to assume that Coxeter Groups are part of Group Theory?
Thank you so much!
P.S. If anyone knows anything about what exactly a "six-dimensional root lattice 6D" entails, I'd be thrilled to find out.
Yes, Coxeter groups are a part of group theory. The crystallographic Coxeter groups are the Coxeter groups that correspond to symmetries of lattices ("crystal structures"). It's known exactly which finite Coxeter groups have this property (see Humphreys, which was also recommended in Will Jagy's answer), and $H_3$ isn't one of them.
The finite Coxeter groups are the simplest ones. The next simplest are the affine Coxeter groups, and there is a procedure for relating these to the finite ones. That's what "affine extension" refers to.