As I understand it, the process of K means clustering takes a set of sample points with k arbitrary centroids and uses Euclidean distance to classify the points closest to centroids to k groups.
What I am unable to understand is a point in the cartesian plane has only an x and y coordinate and so, amongst a given dataset we can only chose 2 independent variables and plot the points and proceed with the algorithm. However, there might be many more independent variables which could influence classification, for example, if we are trying to classify dogs based on their breeds using physical attributes such as size of ears, radius of eyes, body weight, length of legs, lifespan and so on. I'm not sure how this problem is resolved in K clustering.
Are the two variables with maximum information gain considered or are the points plotted in an n-dimensional space where each axis defines each attribute.
Could someone provide clarity on this issue. Thanks for any help