Some forms are chiral, meaning asymmetrical, others preserve mirror symmetry, some rotational symmetry and some preserve both. The number of rotations or faithful mirrorings gives the number of symmetrical elements. So a dodecahedron, 12 pentagon (5) faces, has a rotational symmetry of 60 elements (12x5).
Like regular taxonomy, most forms can be put into categories, though there are a few outliers. E8 is one of them. Also called Baby Monster, it is preserved by over 13 billion mirror symmetries, and is incredibly beautiful. This was all generously told to me through the medium of Cabinet Magazine, by Margaret Wertheim, who is a personal hero of mine. (Also my crocheting inspiration, which we will come to another time)
And here is the hook, or as Margaret prefers, the spooky thing:
"the Monster may encode the structure of our universe. There is tantalising evidence that this extraordinary symmetry group may lie at the heart of physical reality as explained by string theory. In some versions of string theory, our universe is described as having twenty-six dimensions and there are mystically eerie resonances between this version of the theory and the formal characteristics of the Monster. There are whole columns of numbers in the Monster’s character table that mirror properties of the mathematics that can also be used to describe the stringy structure of spacetime."