K-Means Stepper: same data, two confident truths

Sixty support tickets, plotted so similar tickets sit near each other. You choose how many clusters exist and where the search starts; the algorithm does the rest, one honest step at a time, and it will converge proudly from anywhere. Run Start A, then Start B. Watch the middle clump change sides.

The map · × marks a centroid

Grey documents are unassigned. Assign colours every document by its nearest centroid (shape follows colour); Move drags each centroid to the mean of its documents. Nothing else ever happens.

The choices the algorithm never makes

How many clusters exist (your claim, not its discovery)
Where the search starts
Iteration0
Documents that switched cluster–
Total distance (what it is minimising)–
Pick a start, then Step.

The clusters are not in the data; they are in the choices. Start A and Start B both end with "converged", with totals 2% apart, disagreeing about ten documents; k=5 happily shatters a coherent blob and never says "too many". The algorithm answers "how do these divide into k?", never "are there k?". Orange will hand you this same converged answer in one widget, no stepping, no second start; now you know what it did, what it optimised, and what it never checked. Practitioners run many starts and keep the best total (and elbow plots and silhouette scores help pick k), but every one of those tools just moves the human judgement one level up. The insight starts when someone names a cluster and the name survives contact with the documents.