How many unique shapes can you get by unfolding a paper cube?
You can only cut along edges, and the shape must be in one piece and flat. By unique, I mean rotations and reflections don't count.
This problem can be analogized to four dimensions as well. How many unique 3dimensional shapes can be made by "unfolding" a 4dimensional hypercube into 8 cubes? This problem is significantly more difficult than the first.
(In reply to
4d??? by VeeLiem Veefessional)
Most people find the concept hard to grasp. The best way to understand is to use analogies. A line segment is to a square as a square is to a cube as a cube is to a 4d hypercube (also known as a tesseract).
Take a square. One way to look at it is as 2 parallel line segments, with the ends connected by 2 more congruent line segments.
Take a cube. One way to look at it is as two parallel squares, with the corners connected by line segments congruent to the sides of the squares
So a tesseract would be two parallel cubes with all the vertices connectedby line segments congruent to the sides of the cubes.
Another way of looking at a cube is this:
________
\ / This is a central projection.
 \__/  The smaller square inside is actually
    the same size as the larger one,
 __  but it apears smaller because it is
 / \  further away. Notice the shape of the
/____\ six square faces. They are distorted.
Similarly, a hypercube can be seen as a small cube inside a bigger one, with all the vertices connecting. It has 8 "faces," but the faces are 3d cubes, not squares. They, too will seem distorted because we are flattening the tesseract into 3 dimensions.
___________________________________________
I can't find a better way to explain the unfolding of a tesseract than this animation.

Posted by Tristan
on 20040731 11:47:53 