If you're dropped onto a planet with an exact map of whatever's on the surface, then by comparing the map to your surroundings, you learn something about where you are (and which direction you're facing). Can this surface structure be made arbitrarily uninformative using symmetry?
That is, for every $\epsilon>0$, is there a nonconstant function $f:S^2\to Y$ such that for all $t\in SO(3)$, there exists $u\in SO(3)$ such that $d(u,I)<\epsilon$ and $f\circ u = f\circ t$?
Here $S^2$ is the unit sphere in $\Bbb R^3$, $Y$ is any codomain, $I$ is the identity transformation, and $d$ is the standard metric on $SO(3)\subset M_{3\times3}(\Bbb R)$. The idea is that $f$ is $\epsilon$-uninformative if every possible orientation $t$ of the $f$-colored sphere is indistinguishable from some orientation $u$ that's within $\epsilon$ of the reference orientation.
I think a similar question is whether there exist isohedral spherical polyhedra with arbitrarily small faces (in terms of the ratio of the diameter of a face to the diameter of the polyhedron). Some web searching has led me to things like the classification of finite subgroups of $SO(3)$ and various lists of symmetric polyhedra, which often end with dodecahedral/icosahedral shapes. This leads me to suspect that the answer to my main question is no, but I'm not sure how the various types of polyhedral symmetry relate to the type of symmetry I've described.
You can have such an $f$ that works for all $\epsilon$ simultaneously, in fact. This is equivalent to saying that the subgroup $G\subseteq SO(3)$ consisting of rotations $u$ such that $fu=f$ is dense in $SO(3)$. So for instance, let $G$ be a countable dense subgroup of $SO(3)$ (say, the subgroup generated by a countable dense subset). Let $Y$ be the set of orbits of the action of $G$ on $S^2$ and let $f:S^2\to Y$ send each point to its orbit. Then $fu=f$ for all $u\in G$, but $f$ is not constant since each fiber of $f$ is countable.