How the 51 minerals cluster
The six groups are not drawn by hand. They fall out of the data, by measuring how similar each mineral's 14-dimensional profile is to every other's, and letting the similar ones join up. This is that process, drawn as a tree.
A dendrogram is a family tree for data. Every mineral starts alone at the bottom. The two most similar are joined first, then the next two, and so on, until everything sits under a single root. The height at which two things join is how different they were when they merged: join low, and they were near-identical; join high, and the link is a stretch.
We use Ward's method, which at each step merges the two clusters that add the least new spread, so the groups come out compact and comparably sized. Scroll down to watch the tree build itself, from 51 lonely minerals up to one.
Read the finished tree from the bottom up. Coloured sub-trees are the six groups: each is a knot of minerals that merge cheaply among themselves, because their profiles rhyme. The grey links higher up are the expensive joins, where two unlike groups are forced together only because the tree must eventually have one root.
Promethium is the giveaway. It hangs off on its own and only joins anything near the very top, the clearest sign in the picture that it does not really belong with the commercial minerals at all.