Here's a great video by Sixty Symbols -- it's all about dimensions.
This video really only skims the surface of Flatland. For the full experience, you can read the book for free.
The visualization of the hypersphere in the video was generated using the following equation:
x^2 +y^2 +z^2 +t^2 = 1
This describes a set of points at fixed distance from a central point (just as the equation for a circle is x^2 +y^2= 1 and for a sphere is x^2 +y^2 +z^2 = 1)
We subtracted t^2 from both sides to get
x^2 +y^2 +z^2 = 1 – t^2
We then treated t as a time variable and ran the model over the interval time t = -1 to time t =1. Try plugging in different values between -1 and 1 for t, and you’ll see that this gives a series of spheres that start as a point, grow into a unit sphere (of radius 1) and shrink back into a point, as shown in the video.
However, this is not the only way of visualizing 3D cross sections of a hypersphere. Another visualization may be seen in this animation. This was animated by using a different, more complicated variable for time and gives you some sense of the complexity of the hypersphere.
This article, written by Professor Thomas Banchoff, gives some historical background – both from a social and a mathematical perspective--into the circumstances surrounding Edwin Abbott when he wrote Flatland.
Here's a YouTube video that explores the tenth dimension.