The fourth dimension isnít another place, itís another direction to move along. We know about left, right, front, back, up, and down, but there are other possibilities. Time, for example, extends from past to future. Suppose that a cube suddenly appears floating in front of you. It stays there for one unit of time and then disappears. In the first instant of its existence, there is a cube. At the last moment of its existence, there is also a cube, but displaced in time from the first one. In between, each face of the early cube serves as the oldest face of another cube that extends through time to the corresponding youngest face of the late cube. The whole array forms a tesseract (or hypercube), a four-dimensional object bounded by eight cubes. Of course, we can only see the part of the tesseract thatís now, that happens to exist at the moment weíre looking at it.
I wondered, what does a tesseract look like when it spins? Click here to find out.
A simplex-n (or n-simplex) is the simplest figure that can exist in n dimensions. Simplex-2, residing in the 2-dimensional plane, is a plain old plane triangle. Simplex-3 is a tetrahedron, or triangular pyramid. Simplex-4 is a pentatope, a 4-dimensional object bounded by 5 tetrahedra. Click here to watch a tetrahedron spin in 3-space and 4-space, and a pentatope rotate in 4-space.