Butler's result is an upper bound, that the packing-covering ratio is $\leq 2 + o(1)$. I don't think that an analogous lower bound has been established.

As a response to your comment "Because neither NAND or NOR are associative I was fiddling with trying to come up with a more than two parameter Boolean universal gate that is associative, and considering going into multivalued or ternary logic to achieve that", the group $A_5$ works -- see this discussion of Barrington's theorem: crypto.stanford.edu/~dabo/pubs/papers/barrington.html

The gcd function is only called relatively rarely; most loop iterations terminate before reaching this point because either x or y turns out to be irrational.

The corresponding Coxeter group is paracompact and hyperbolic. Apparently it's called $\overline{L_5}$ and it's mentioned here: en.wikipedia.org/wiki/…

@M.Winter I used this answer: mathematica.stackexchange.com/a/179755 (and they're backticks, not single-quotes)