Hello there. I've been studying the geometry of the fourth dimension for maybe a year or so now. It's really interesting. I'm not good at the mathematical aspect, but I can make sense of most 4D shapes at this point. However, when I became able to understand what 4D objects really looked like, I discovered a few things that surprised me. Most of you will probably have already known this, but whatever.

First off, from what I can tell, most perspective projections on almost all websites with information on tesseracts is incorrect, all because of a simple error in the pictures. See, almost all pictures of the tesseract depict the tesseract as a large cube with a smaller, internal cube inside it, such as this:

Now, #1. The "center" cube is too small to make this an actual tesseract. What this projection is telling me is that this figure is actually a hyper-rectangular prism. If it were an actual tesseract, the smaller cube would likely be bigger. But now on to the bigger problem with this picture:

#2. The lines connecting to smaller cube to the bigger cube are projected completely wrong. This aggravates me because it makes tesseracts much harder to understand, and also because this mistake is made in nearly every picture of this kind of projection of the tesseract. I will now explain why this is wrong. The lines that connect the two cubes are projected as leading INSIDE the bigger cube itself. The correct way to show these lines would be to put them BEHIND the larger cube instead of leading to the inside of it, like in this picture:

If you look at how the far cube is shown now, you might be able to make more sense of the tesseract's actual structure. To aid with this correct projection, I think of the two front and back cubes as completely flat, like paper. These cubes are then connected by lines to and from all of each cube's vertices. Of course, I know that 3-cubes aren't completely flat in 4D, retaining some kind of 3D depth. But that's about it for this kind of projection. Of course, I could be wrong about this entire thing. I could be looking at it the wrong way. But simply making inferences from lower dimensions seems to yield more logical results, so tell me what you guys think of this.

Also, I found a way to show an opaque tesseract. This has been found out before, but it's helped me with my studies a whole lot. Basically it shows that the tesseract itself is little more than a cube with cubes for sides.

http://img28.imageshack.us/img28/8785/imsuchagenius.png

Ignore the rather gloaty filename.

Now, I figured this out by doodling an opaque cube and then spontaneously getting the idea to "lift" each square side of the cube to the third dimension, successfully making an opaque tesseract. This outcome can also be reached by simply connecting two opaque cubes by their vertices.

Questions or comments anyone? I'd like to know what you fellows think of this.