The klein bottle is relatively uninteresting. The technique which identified it is the interesting part here. The linked paper explains a method to take a high-dimensional point cloud and compute a 'bar code' which encodes fundamental geometric features of the solid of which the cloud is an approximation. Its a way of visualizing high-dimensional data without using dimensional reduction.
Does the point cloud approximate the shape, or does the shape approximate the point cloud? That distinction informed a recent discussion on the origin of the word "regression", which on the surface seems a weird term for curve-fitting, at least from my point of view.
I suppose the idea is that there is an underlying shape which the point cloud approximates. The fitted shape is then a statistical guess at the underlying shape. In which case it would be nice to see a proof that as more points are added the fitted shape eventually converges to the underlying shape.
"The full data set
consists of roughly 8,000,000 points in E9. By normalizing with respect to mean
intensity and restricting attention to high-contrast images (those away from the
origin), the data set is projected to a set of points M a topological seven-sphere
S7 ⊂ E8"
