We propose "breathing $k$-means", a novel approximation algorithm for the $k$-means problem. After seeding the centroid set with the well-known $k$-means++ algorithm, the new method cyclically increases and decreases the number of centroids in order to find an improved solution for the given problem. The $k$-means++ solutions used for seeding are typically improved significantly while the extra computational cost is moderate. The effectiveness of our method is demonstrated on a variety of $k$-means problems including all those used in the original $k$-means++ publication. The Python implementation of the new algorithm consists of 78 lines of code.