[flaubere]

Here's a visualisation that helped me think about the 4-cube:

Cut the 3-cube with a plane which is diagonal to all axes, eg x + y + z = c. Start at a corner and take sequential sections. First you get a small equilateral triangle, then a bigger and bigger one, until the cut goes between 3 vertices of the cube. Next you get truncated equilateral triangles, with bigger and bigger truncations. In the center of the cube the size of the truncations matches the remaining edges and you get an regular hexagon. Then the whole thing in reverse as half the sides get smaller and smaller, until you're back to triangles.

If you're not sure what it looks like at any point, you can easily solve the intersection of x + y + z = c and the equation of one face of the cube (x or y or z = 0 or 1).

Now do it for the 4-cube. Important observations: 1. again you can solve algebraically, either for the 3-cubes which bound the 4-cube or the 2-squares which bound them 2. you can also just try and imagine the intersection with the 3-cubes, since it will be one of the shapes you thought about in the previous exercise (x + y + z + w = c && w = 1 => x + y + z = c - 1) 3. c goes between 0 and 4, with [0, 2] symmetrical to [2, 4]. There are two 'regions' of behavior, c in [0, 1] and c in [1, 2], with the type of shape only changing when the plane intersects with vertices.