# What kinds of data other than geographical are topologically spherical?

I'm trying to think of a data set that is essentially topologically spherical. It's easier to think of cylindrical datasets (two dimensions, one periodic) or toroidal datasets (two dimensions, both periodic).

Obvious candidates are geographical and astronomical, ground and sky; but I think the only thing spherical about the sky is its projection onto the ground, so it really just comes back to Earth.

I find it helpful to think about in terms of a circle's fundamental polygon:

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– Toph
Commented Jun 23, 2015 at 16:41

Any data consisting of a three element vector of numbers $v =\{x,y,z\}$ could be viewed as topologically spherical if, after converting to polar coordinates, there is only one datum for every value of the 3D angle $\theta$. Thus, every point in the data represents an exterior of a topological sphere. This is taking the term "topological" literally and formally.

There are also graph structures (acyclic) that have neighborhood structures that allows the graph to be mapped topologically to a sphere. For example, all graphs that can be mapped to Platonic Solids (i.e. vertexes to vertexes, edges to edges, 1:1) are topological spheres. This can include irregular neighborhood structures, too.

As far as practical examples of these, none come to mind immediately.

I'm not entirely sure what you mean by

I'm trying to think of a data set that is essentially topologically spherical.

but let me take a shot at answering the question I assume you intended.

I will assume you are either looking for examples of

1. A naturally occurring (whatever that may mean) dataset that is essentially spherical.

or

1. Examples where one takes a naturally occurring dataset, transforms it so that it is essentially spherical, and then, crucially, uses this transformed data to do or see something useful that could not easily be done/seen before the transformation.

If we don't asssume either of these two things, there are unlimited not-so-meaningful answers to your question. For example, one could arbitarily construct a data generating process that results in an essentially spherical dataset (I'm sure scikit-learn has some function to do this). One implementation could do this by generating vectors (in any dimension) by sampling a gaussian distribution (in aforementioned dimension), and then look only at those vectors whose norm is in the top 10% of the norms of all of the generated vectors. This yields a rouhly spherical dataset (in said dimension). Let's ignore answers of this sort, and get back to the examples I assume you are looking for.

One way in which spherical data tends to come up in a useful way in my experience is in representing the possible directions in the vector space in which your data lives. For example, say you are studying a dataset which consists of 3 dimensional vectors and these vectors do not appear to resemble a two dimensional sphere topologically. Well, one can certainly try to transform the data so that the non-zero vectors do appear to resemble a sphere by performing the following transformation to each nonzero vector:

$$\vec{x} \mapsto \dfrac{\vec{x}}{|\vec{x}|}$$

This transformation will convert each nonzero vector to a vector living on the unit sphere (in one dimension less than the dimension of the vectors), so if the vectors are "fairly dense and well-distributed" in their directions, the result would resemble an "essentially spherical" dataset. This can be a useful intermediate step to visualize the directions your data spans, before using that direction information to calculate things like the angle between different vectors, or the mean angle between representatives from different clusters of vectors.

I think there are many other useful transformations that yield examples of spherical data sets in the sense of the description # 2) above.