Tesseract Terror

nope

Is ther something special about the doors?

You can leave by not making the 90 degree turn into fourth-dimensional space as you walk through the door.

Think of a 2-D being living on a square who would normally exit through one of the edges. If the square is one side of a 3-D cube, the being must make a 90-degree turn in leaving through the edge of its square in order to enter the adjoining square (which BTW is NOT the one he just left, but one of six different squares making up the cube). If it goes straight through the edge as usual it exits the cube.

'And He Built a Crooked House'
-Robert A Heinlien

Ummm... What if the tessaract was built out of a Klien Bottle?

I was also thinking about the analogy in 2D with a flatlander on a particular side. But can you call it a 90 degree angle with respect to a bounding cube of a tesseract. (And yes, I know it is not the room you just left as was portrayed on one of the Star Trek Next Generation episodes.)

Would you consider the door something of a transform function; a kind of intradimensional, 3D portal? If so, then we can't just break through the wall.

If the 2-D plane was wrapped around the cube (like wrapping paper around a gift), then the 2-D being could crawl around the edge into another square. But that's not how it works. The cube is in 3-space and one face happens to be congruent with a square on the 2-D plane. If the 2-D being crawl across the edge of the square as usual, it would exit it, and could only enter another face of the cube if it could make a 90 degree turn out of its normal space.

A 3-D being could not walk into a room outside of its 3-space without turning 90 degrees in a direction that is unimaginable to those of us whose minds are limited to the x, y, and z axes. If you were in the "inner" square in your diagram, you would walk out into 3-space, just as a 2-D being would remain in its own plane even after crossing the edge of a 3-D cube image projected onto its plane.

Terry

The mental image is quite a challenge. I was considering a hyper sphere before the tesseract but my brain started to overheat.

Oh yes. In the tesseract image, don't forget the bounding cube that does not appear succinctly. It is the one that is "inside out" and extends infinitely out into 3D space.