This is a fascinating question. If I understand it. (I presume that exp(pi/5) and exp(3pi/5) in Adam's blog post refer to exp(i*pi/5) and exp(i*3pi/5). Right?) To get pi/2 + 2/pi, it would be necessary to see first whether the generators of the dense subgroup of SO(3) can be said to approach a limiting density, preferably the uniform one. Given the two generators, say g and h, of the dense subgroup, a natural thing to try would be the limit of the relative densities of the finite set of group elements of wordlength N, as N -> oo. Let S(N) be the set of all group elements expressible by words of length <= N in g and h. Then the "relative density" condition means that for any two open sets U and V of SO(3), the ratio #(U \int S(N)) / #(V \int S(N)) should approach volume(U) / volume(V) as N -> oo. IF this is true, it probably is one way to get pi/2 + 2/pi just from the dense subgroup. In case it approaches a limiting relative density (for any two open sets U and V) that is NOT the ratio of their volumes, there is the dense subgroup. --Dan On Jul 28, 2014, at 2:59 PM, Bill Gosper <billgosper@gmail.com> wrote:

Adam's blog http://cp4space.wordpress.com/ currently has "<s, t : s^10 = t^3 = (st)^2 = 1>

This is a surprisingly exciting group.

Firstly, it is straightforward to verify from its presentation that the group is a hyperbolic *Dyck group*, *i.e.* the orientation-preserving symmetries of the kaleidoscopic tiling of the hyperbolic plane by triangles with interior angles (π/2, π/3, π/10).

[...]

In other words, it is the group generated by the following Möbius transformations:

- s(z) = z exp(i pi/5); - any t with the property that t(t(t(z))) = s(t(s(t(z)))) = z for all z.

Now, this is a ring-theoretic definition, so we can apply a ring automorphism and replace exp(pi/5) with exp(3pi/5) in the above definition without changing the resulting abstract group. However, this is entirely different from a topological perspective, since we have a group of spherical symmetries — rotations — rather than hyperbolic symmetries. So we can regard this as a dense subgroup of SO(3) generated by two rotations."

Question: Does this have a "group volume", and can it show the expected magnitude of an element of SO(3) is pi/2 + 2/pi?