Note: K-means does not assume an interpretation of the clusterings - in fact it is an [unsupervised][1] algorithm. The interpretations are a result of **human analysis** not something in the algorithm. 

For example, in the case of cats and dots one would most definitely chose k = 2 - which provides an easy interpretation. However, what would it mean if we set k = 1000. We no longer have a "clean" interpretation of the centroids. 

Note: how I keep saying "interpretation." The algorithm simply assigns a data point to a cluster and calls it a day. Humans then look at the results and try to **understand** them with an **interpretation**. 

Continuing with the example where k = 2. One could easily interpret "is cat" as "not dog" and "is dog" as "not cat." The idea here is that the data is **unlabeled** beforehand and humans try to fathom the results retrospectively by assigning the resulting clusters with an understandable label.

I hope this clarifies the issue.


  [1]: https://en.wikipedia.org/wiki/Unsupervised_learning