Lar,

Spoken like a Flatlander! The image you posted *is not* a tessarect. It is the 3D "shadow" of a tessarect - sort of.

Take a deep breath.

If you make a wire frame cube (real world, using real wire) and hold it between a light and a piece of paper (plane) you will get a 2D shadow of the 3D wireframe cube. In fact, if you create a cube in DesignCAD and view it in wireframe you see this 2D "shadow" on the screen.

The picture you posted is actually a 2D shadow of the 3D shadow a 4D tessarect would project into the 3D universe.

In the "tessarect 1.jpg" picture below you see the "shadow" of a 3D cube (black) on the left. Then there is a stack of 8 colored cubes in a 3D cross shape - Salvidore Dali used this once in a painting. It is an "unfolded" tessarect. Just as you can unfold a cube into 6 squares on a sheet of paper, if unfolded a 4D tessarect would create a group of 8 cubes like this in 3D.

Just as we can fold the 6 2D squares to form a 3D cube, the 8 cubes can (theoretically) be folded into a 4D tessarect. The third figure shows the relationships of the 8 cubes in a tessarect - or what 3D shadows of the 8 cubes would look like. There is the black inner cube, six surrounding colored cubes, and an outer magenta cube. The figure at the right has all the parts assembled into the 3D shadow of a 4D tessarect.

BUT, in 4D all of these objects are cubes, all are the same size, and all are packed within the outer cube (actually, all 8 are "outer," "inner," "side" etc.). If you were standing in the black cube you could pass through any side into a colored cube except the magenta cube. Likewise, when you entered any of the 6 colored cubes surrounding the black cube that colored cube would become the "center" cube and you could access the 6 surrounding cubes but not the new outside cube. For example, if you went from the black cube to the dark blue cube you would no longer be able to access directly the light blue (cyan) cube.

The second image shows these things rendered - except on the third figure from the left I substituted a magenta outline for the surrounding cube.

Is this all gibberish? Well, Heinlein made a great sci-fi story using this "fiction." But mathemeticians figured out the nature of the tessarect long before Heinlein wrote his story.

The book "Flatlands" is an exploration of what the universe would look like to beings who experienced only two dimensions. Their universe would be a plane. But one day the shadow of a 3D cube fell on their universe, and all wondered what it meant. One bright fellow deduced that there was another third dimension and the shadow was the intersection of a 3D object composed of 6 squares - a cube - as it passed through their 2D universe.

Our pathetic ape senses limit us to experiencing only those aspects of the universe we need to "see" in order to feed, mate and survive here on earth. We can't see radio waves or X-rays - or even atoms. But by applying intelligent thought we have learned these things exist.

Likewise, if you try really hard, like the Flatlander you might be able to comprehend the nature of a 4D tessarect by looking at its 3D shadow. Don't be so quick to dismiss additional spatial dimensions as just fantasy. How would you, with your limited 3D senses, go about proving that additional dimensions are not possible?

Phil