This demo is actually fairly close to showing you a hypercube.
The six rooms in the house form six cells of the hypercube. You can make one more cube by joining up the tops of the walls: that cell is always above you. And finally, there would be one more cell below you. (To be a bit more precise: the six rooms of the house would have to be extended to be twice as tall. The tops of the walls are at coordinates (+-1, 1, +-1, +-1), but right now the bottoms of the walls are at (+-1, 0, +-1, +-1). You have to remove the floor and make the walls go all the way down to (+-1, -1, +-1, +-1).
You can do this for any 4D polytope. Taking it down a dimension, the layout of the house shows how to put a cube on a 2-sphere. I could have done it for any other 3D polyhedron. Give the sphere and the polyhedron the some centre point, and then project all the lines of the polyhedron onto the sphere. Each face of the polyhedron becomes a 2-D region on the sphere (in the case of the house, each room is a square). Going a dimension up, you can centre the 3-sphere and the polytope on the same point, and do the same kind of projection. Every cell of the polytope turns into one 3-D region of the 3-sphere.
The six rooms in the house form six cells of the hypercube. You can make one more cube by joining up the tops of the walls: that cell is always above you. And finally, there would be one more cell below you. (To be a bit more precise: the six rooms of the house would have to be extended to be twice as tall. The tops of the walls are at coordinates (+-1, 1, +-1, +-1), but right now the bottoms of the walls are at (+-1, 0, +-1, +-1). You have to remove the floor and make the walls go all the way down to (+-1, -1, +-1, +-1).
You can do this for any 4D polytope. Taking it down a dimension, the layout of the house shows how to put a cube on a 2-sphere. I could have done it for any other 3D polyhedron. Give the sphere and the polyhedron the some centre point, and then project all the lines of the polyhedron onto the sphere. Each face of the polyhedron becomes a 2-D region on the sphere (in the case of the house, each room is a square). Going a dimension up, you can centre the 3-sphere and the polytope on the same point, and do the same kind of projection. Every cell of the polytope turns into one 3-D region of the 3-sphere.
I wish I had had time to include that.