It is interesting to notice how much importance we put on shadowing. Drawings on paper look okay without shadowing, but when shadows are added our brain begins to register the images and allow us to percieve the three dimensional qualities of the dipicted object much better. One of the most important aspects of our vision is the presence of shadows. Why does a photograph look so real to us when it is simply a two dimensional picture of the three dimensional world? It's because our retinas are basically curved walls that receive a projected image of the world around us just like a roll of cammera film recieves a projected image of the world through its lense, and shadows are an integral part of the world around us. It would be very interesting to see the world around us lacking any shadow.

Somewhere between retina and brain the two dimensional recordings of our eye are manipulated into perception of three dimensions. It is a difficult task that improves with developement and experience (a five year old won't be as good at it as a 25 year old). The practice today is to project four dimensional objects like a hypercube down to the two dimensional computer screen. The benefits the computer offers are quite high, but a two dimensional representation of a 4 dimensional object poses problems for our retina and brain when we try and decipher the image and percieve what is going on. It's easier to understand (most of the time) what's going on when the represented objects like hypercubes are three dimensional, so my question is when will a three dimensional "t.v." projector come about; or is it even worth while? Would it make that much of a difference in understanding? Who remembers R2D2's Princess Lea projection in StarWars? What about the thought of an ancient illusionist who with a flip of his finger could create a dancing spirit on the table in front of him? These are both examples of three dimensional "video projections", a projection higher than those at a movie theater.

If we were able to thus project four dimensional hypercubes into a three dimensional medium (using computers lets assume), then instead of having two perpundiculer lines from a vertex and two skew ones (representing the edges in the third and fourth dimensions like we get in the projection to the plane) we would have three perpundicular lines from a vertex with only one skewed line representing the edge in the fourth dimension. Our brains would then have less "abstractness" to decipher; it would take the unconcious strain off our brain, allowing us to better focus on how the three dimensional "shadow" of the hyper cube is a product of the real four cube.

EXERCISE: Turn all the lights out and light a candle on a desk close to a wall. Take objects around and check out the projections you get on the wall in front of you. Cubes are interesting (get hexagons), as well as octagons (diamonds that turn to hexagons as you rotate it), and other types of dice (20 sided, 12 sided). Another good thing is a paper towel roll. Here, with a change of angles, you can get a feel for how perspective works (the end closer to the candle will look much larger on the wall than the other end). When you move the roll so it is projected on the cieling (if you've really absorbed yourself), the magnitude of the new projection can be quite amazing.