OpenStudy (anonymous):

In page 11 of the textbook (3rd edition), he explained that " (1,1,1,1) has length of sqrt(1^2 + 1^2+1^2+1^2) = 2. This is the diagonal through a unit cube in four-dimensional space. Here, I am wondering what is "the unit cube in four-dimensional space". I know that it is nearly impossible to visualize the dimensions bigger than 3. But, is there any intuitive explanation that give insights on "the unit cube in four-dimensional space?

4 years ago
OpenStudy (anonymous):

It's by extension/analogy from lower dimensions, and a matter of language use. But you still have to have a recipe for constructing one in an arbitrary number of dimensions. Take an interval of length one along the real number line ( a 1D space ), that's unit cube for that space. It's bounding vertex points are 0 and 1 say. Next go to the Cartesian plane ( a 2D space ) with a square of side length one, that's a unit cube for that space. It's bounding points could be listed as (0,0) (0,1) (1,0) (1,1). Now proceed to 3D and you make a cube of side length one and that's what we normally call and think of as a 'unit cube'. The point listing might be (0,0,0) (0,0,1) (0,1,0) (0,1,1) (1,0,0) (1,0,1) (1,1,0) (1,1,1). Now go to 4D space ( don't bother visualising it ) and list the points. Use the pattern already defined. As you can see, the number of vertices goes like 2 raised to the dimension of the space, so we'd expect 16 here. They will be (0,0,0,0) (0,0,0,1) (0,0,1,0) (0,0,1,1) (0,1,0,0) (0,1,0,1) (0,1,1,0) (0,1,1,1) (1,0,0,0) (1,0,0,1) (1,0,1,0) (1,0,1,1) (1,1,0,0) (1,1,0,1) (1,1,1,0) (1,1,1,1). I know that I have expressed them correctly, because at each dimension I expect vectors to have the same number of components as dimensions ( by definition ) and I have used a 'binary' encoding method to enumerate them. The remaining question is which vertex is related to which ( because you want to connect with lines ). Firstly, how many vertices that should be ie, how many 'neighbours' does a vertex have? You can see that goes like the number of dimensions ( think about it ), so in 4D each vertex has four neighbours to connect to it with a line. The method is, for a given vertex, to connect it to others that have co-ordinates only differing along one dimension. So taking (0,0,0,0) then it has neighbours (0,0,0,1) (0,0,1,0) (0,1,0,0) (1,0,0,0). So now at least programmatically, alas not so intuitively, you can go ( only limited by exhaustion ) to any dimension.

4 years ago