All of these movies require QuickTime. Don't have QuickTime? Get it here.
This movie shows the failure of Thom's a_f condition between the ambient space and the z-axis, at the origin, for the Whitney Umbrella. There are 31 frames corresponding to level surfaces of f = y^2-x^3-zx^2. The blue planes are tangent to the level hypersurfaces, but the limit of these planes at the origin does not contain the z-axis.
The QuickTime movies below are intended to represent a 3-dimensional singularity sitting in 4-dimensional space. As the 4-th dimension, I use the ever-popular time dimension. Hence, each frame of the movie represents a cross-section of the time dimension.
The singularity here is the discriminant singularity of the 5-th degree polynomial, after it is put into normal form. The time coordinate is given by the coefficient in front of the term of degree 3.
These animations were created using Mathematica, and are three different views of the same singularity.
The complex curve defined by y^2-x^3 = 0 is a real 2-dimensional subspace of real 4-dimensional space. Using time as the 4th dimension,one can look at the real picture of this - in each frame, one sees a real curve in 3-space.
Another way to see the complex cusp is to project the four dimensions into 3-space. Below, we have movies showing the result of a series of such projections.
Now I'm really getting carried away. How can you see 5 dimensions?!? Well...we'll use time for the fourth dimension and color for the fifth. So, a different color represents a different color coordinate.