The author clearly didn't read the page of the math paper he posted in trying to argue his point. It says, and I quote: Stated informally, the k-means procedur

The author clearly didn't read the page of the math paper he posted in trying to argue his point. It says, and I quote: Stated informally, the k-means procedure consists of simply starting with k groups each of which consists of a single random point, and thereafter adding each new point to the group whose mean the new point is nearest. Admittedly, it's not the prettiest English sentence over written, but it's just as plain and simply stated as the author of this article. The article itself is interested in proving asymptotic guarantees of the algorithm (which the author of the article seems to completely ignore, as if it were not part of machine learning at all). Of course you need mathematics for that. If you go down further in the paper, the author reverts to a simple English explanation of the various parameters of the algorithm and how they affect the quality of the output. So basically the author is cherry-picking his evidence and not even doing a very good job of it.