I had mentioned Bernhard Riemann in my last blog entry. I want to here mention a tool he used to help with the analysis of higher dimension geometry. It is referred to as "stereographic projection". It is kind of like looking at the shadow of an object of a higher dimension by viewing its "shadow" on a lower dimension. However, it is more sophisticated than that because it actually constructs the projected shapes using mathematical logic that lends itself to mathematical proof.
I would probably do a terrible job trying to demonstrate this for you; so I will include here a link on the Internet that can do that job better than I can. http://www.dimensions-math.org/Dim_reg_AM.htm. This link takes you on a journey from our 3-D world to the worlds of higher dimensions. If you like, you can scroll down on the main page of the site mentioned above and just view the presentation starting with "dimensions_3". This particular movie will open the door to the higher dimensions by showing you some of the characteristics of the 4th dimension. Of course, if you have time you will want to view all the movies on the site. By the way, if you view the dimensions_3 movie first, do not be confused by the mention of lizards. The lizards are a part of the dimensions_2 movie explaining how hard it would be for 2-dimensional creatures to understand our 3-dimension world.
The dimension_3 movie will first introduce you to a 4-dimensional object they refer to as a "simplex" using the general term that includes all polyhedroids. Polyhedroids are figures in 4-dimensional space that are composed of sets of polyhedrons. The polyhedroid introduced is called a pentahedroid, because it has five sets of polyhedron interiors--a polyhedron at every one of its five vertexes. This 4-dimensional creature is the 4th dimension counterpart of a tetrahedron. The movie shows you how to extrapolate, using the concept of the tetrahedron, by following the same procedure for drawing a tetrahedron in our space but extending the drawing into a higher space that we can't really see or image in our minds. The difference is that there will 5 vertexes with the 5th vertex extending into the 4th dimension.
If you watch dimension_2, you will experience by example what it would be like to be a 2-dimensional creature living in a two-dimensional space that is "visited" by a 3-dimensional object passing through its space. Of course, you are a 3-dimensional creature and it is desirable to know what a 4-dimensional object might look like passing through our 3-dimensional space. This is accomplished in dimension_3 by showing you how the pentahedroid looks passing through our space. We see only the "slices" that our space embodies. It is a really challenging mental experience. The earlier movie makes it easier for you to make the mind leap, however.
From there you are shown other 4-dimensional objects and how they would be viewed. It is a masterpiece of animation combined with human logic. I hope you enjoy viewing the movies as much as I did.