Question

2. Consider an n-cube, where n = 3. Is it possible to draw this on a two dimensional plane where the lines do not cross? If it is possible, draw it. If not, prove that it isn't possible.

Answer 1

Best thing to do, to build up a good intuitive sense for this, is to first analyze what happens when we go from 0 to 1 dimension (point to line), from 1 dimension to 2 (line to square), and from 2 to 3 dimensions (square to cube).

Since we already have good intuition for these transitions, we can dissect what happens during these transitions, regarding boundaries, number of faces, volume, and extrapolate from there, towards the fourth dimension.

We can drag a point along to form a 1 dimensional line;

Its volume is length L, it has 2 faces, which are formed by 2 points.

Now lets drag the line to form a 2 dimensional square;

Its volume is area L2L2, it has 4 faces, which are formed by 4 lines.

Now lets drag the square to form a 3 dimensional cube;

But we can also take 2 of its faces, and connect the corners (marked red);

Its volume is L3L3, it has 6 faces, which are formed by 6 squares.

Now lets drag the cube to form a 4 dimensional tesseract;

But we can also take 2 of its faces, and connect the corners (marked red);

Its volume is L4L4, it has 8 faces, which are formed by 8 cubes.