Math behind everyday life: on distribution of "black days" and beyond

In our daily lives, we encounter numerous independent events, each occurring with varying probabilities over time. This letter delves into the scientific background behind the inhomogeneous distribution of these events over time, often resulting in what we refer to as ``black days'', where multiple events seem to converge at once. In the first part of the work we have performed an analysis involving D independent periodic and random sequences of events. Employing the Uniform Manifold Approximation and Projection (UMAP) technique, we observed a clustering of sequences of events on a 2D manifold M at some large Dcr, which we interpret as the manifestation of ``black days'' and which occurs in a narrow interval around Dcr. We found that increasing the number of sequences within rather wide interval below Dcr leads to a plateau in the clustering on M. In the second part of the work we examined in detail clustering patterns of independently distributed N points within the corners of a D-dimensional cube when 1 N<D. Our findings revealed that a transition to a single-component cluster occurs at a critical dimensionality, Dcr, via a nearly third-order phase transition. Finally, we have addressed the question ``How the infinite-dimensional space could look like?'' Considering the eigenvalue problem for the hyperspherical Laplacian ∇²D in the limit D→∞, we conjectured about the topology of a target space representing the hyperspherical layer.