2x2x2x2, part 5
One problem with “flattening” an object into two dimensions is that it can be hard to understand what happens when you rotate the object. Let’s go back to our three dimensional case.
Suppose we turn every little cube red if it is on the right side of the big cube. Here is what this looks like as a 3D object:
In the image below I’m flattening this, so that the left part shows negative z (the little cubes in the back), and the right part shows positive z (the little cubes in the front):
If we rotate the cube in different ways in the 3D view, it’s always easy for us to see where the highlighted red cubes go. After all, we’ve been looking at rotating 3D objects all our lives.
But if we look at the flattened view, the same object rotated in differently ways can look a little strange:
These all represent the same shape rotated in various ways, but they don’t really look the same. I don’t have any answers for what to do about this. I think you just need to get used to the strangeness of it.
Because when we start rotating things in four dimensions, it’s going to get even stranger.