  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 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 ${\cal M}$ at some large
$D_{cr}$, which we interpret as the manifestation of "black days" and which
occurs in a narrow interval around $D_{cr}$. We found that increasing the
number of sequences within rather wide interval below $D_{cr}$ leads to a
plateau in the clustering on ${\cal M}$. 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 $1\ll N<D$. Our findings
revealed that a transition to a single-component cluster occurs at a critical
dimensionality, $D_{cr}$, 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
$\nabla^2_D$ in the limit $D\to\infty$, we conjectured about the topology of a
target space representing the hyperspherical layer.
