  We show that specific higher dimensional shape information of point cloud
data can be recovered by observing lower dimensional hierarchical clustering
dynamics. We generate multiple point samples from point clouds and perform
hierarchical clustering within each sample to produce dendrograms. From these
dendrograms, we take cluster evolution and merging data that capture clustering
behavior to construct simplified diagrams that record the lifetime of clusters
akin to what zero dimensional persistence diagrams do in topological data
analysis. We compare differences between these diagrams using the bottleneck
metric, and examine the resulting distribution. Finally, we show that
statistical features drawn from these bottleneck distance distributions detect
artefacts of, and can be tapped to recover higher dimensional shape
characteristics.
