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This is a bit of a strange question; I am not sure of some of the correct terminology and so there may be a solution already that I cannot find; I apologize in advance if this is the case. The bumpy sphere $1+a\sin n\theta\sin m\varphi$ is not symetrical about the bumps and has points at the poles where the surface becomes 'bunched up'. I have tried (purely from interest) to find an $r(\theta,\varphi)$ defining a bumpy sphere that is symmetrical about each of the bumps but failed. I have tried to formalize my question as follows:

For what $n\ge3$ is the following definition of a function $f:S\rightarrow\mathbb{R}$ where $S$ is the unit sphere possible?

  1. Let $b_k,\;k\in{1,2,...,n}$ be points on $S$ (not all on a great circle) such that for all $(b_i,b_j)$ any rotation $R$ satisfying $Rb_i=b_j$ has the property that $R\{b_k\}=\{b_k\}$.
  2. Let $f:S\rightarrow\mathbb{R}$ be a non-constant function such that for all $(b_i,b_j)$ any rotation $R$ satisfying $Rb_i=b_j$ has the property that for all $p\in S$ we have $f(p)=f(Rp)$. In some sense, $f$ should also be smooth.
  3. Ideally I would like the following to hold: For any $b_k$ let $\operatorname{close}(b_k)$ be the set of all $b_i$ that are minimally distant along the sphere from $b_k$; then for all $b_k$ we should have that for any $b_i\in\operatorname{close}(b_k)$ the graph of $f$ on the minor arc of the great circle connecting $b_i$ and $b_k$ should have global maxima at its endpoints and a unique local minimum between them.

What I am trying to require by the first point is that we have points equally distributed on $S$ such that their arrangement is preserved by any rotation mapping one to the other. I'm sure this is possible for $n=6$ but I'm not sure about other $n$; this unanswered question asks about this and this question may be relevant. The second point is trying to find a smooth non-constant function that also satisfies this rotation invariance; again I do not know if this is possible. The third point is trying to establish that $f$ should be maximized at the $b_k$ and minimized in between. The picture in my head is of a sine-ish function on the sphere where the maxima are at the $b_i$ and the function is symmetrical about the $b_i$ (the connection to the bumpy sphere is that we could translate and scale $f$ to get an $r(\theta,\varphi)$). Apologies if I haven't been clear.

My questions are: For which (if any) $n$ is the given construction possible? If we remove requirement 3 does it become possible? Is there any way of relaxing the conditions slightly to create an $f$ that corresponds to the idea I have described? Can any such function for some $n$ have a simple representation, say in terms of trigonometric functions?

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    It sounds like the first requirement necessitates that you get subgroups of the rotation group. There are only a [few different ones](https://groupprops.subwiki.org/wiki/Classification_of_finite_subgroups_of_SO(3,R)). A requirement that any vertex can be carried into any other would further imply that you can only have bumps at the vertices of the platonic solids. – 2017-02-09
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    It appears that this article https://arxiv.org/abs/1701.01348 gives functions which are invariant under octahedral, tetrahedral, etc. groups. – 2017-02-09
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    @NeuroFuzzy That makes sense. Thanks for the link; I'll have a closer read of it later. – 2017-02-09

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