Exploratory data analysis uses techniques of projections and slicing to investigate high-dimensional data sets from social sciences as well as from the physical and biological sciences. Long before students are introduced to the notions of critical point theory, they can already understand and appreciate slicing phenomena that relate different dimensions. It is easy to carry out the actual experiments and see that there are positions where the slice yields a pair of circles. Less obvious is the slice that consists of two interlocked circles. Again, a good way to see this would be to experiment with a transparent inner tube filled halfway with colored liquid.
Geometry can be a surprising observational science.