What's a good way to mentally visualize n dimensions in a k means I've been using k-means to do some clustering and one of the ideas I'm struggling with is the n dimensions aspect. If I were clustering housing prices vs sq. feet its just a simple 2d graph. That I can visualize and it makes sense.
Where it gets difficult is when it's more than 3 features. If I understand properly, each feature is plotted in it's own dimension. How does something with 3+n dimensions look, conceptually? What's a good way for me to think about how the data is represented?
I know this question is a bit vague (which is a result of my lack of understanding in this area), but any help is appreciated. 
 A: It is really hard to plot more than 3 dimensions. Maybe you can afford visualise a fourth dimension if one of those is time, so the end result is a video.
In practice it makes sense to use a dimensionality reduction technique. For example PCA, or just eigendecomposition. Using these or many of the similar techniques you project the $n$ dimensions to 2 or 3 that can be handled by humans. 
The trade off is that depending on your data that projection may be really bad and hide what's really going on with the data. There are ways to assess the quality of these visualisations. For example by checking out the relative values of the first two or three eigenvalues compared to the rest of the them.
A: One way I have used in the past is to compare bivariate scatter plots. Obviously you're still limited dimensionally, but you can investigate about five or six dimensions at a time.
For the figure below, I took data in four variables and put them in seven clusters. Then I used a very nice ggplot2 function ggpairs (from this answer), coloring on the predicted clusters. The margins show the density functions/boxplots of the clusters. 

I would never dream of publishing this figure, but it was surprisingly helpful in our analysis.
