Fully-animated presentation of multi-dimensional cubes. Starts with a zero-dimensional cube (a point) and progresses through:
1D - line segment (complete at step 3)
2D - square (complete at step 6)
3D - cube (complete at 9)
4D - hypercube, tesseract, or octachoron (complete at 14)
5D - penteract or decateron (20)
6D - hexeract or dodecapetron (27)
7D - hepteract or tetradeca-7-tope. (35)
Notice that the lengths of the sides are animated, thus the faces are more often rectangular rather than square. However, many of the features of hyper-dimensional objects are more easily recognized if the sides are not equal. To display proper cubes, stop the animation and set the side lengths equal to each other.
The general formula for the elements of the progression is: Twice the previous element plus the previous element of the next lower progression.
Number each element by progression (p) and element number (e). Thus, the first element of the first progression (vertices) is p1e1, which equals 1.
p1e2 = 2(p1e1) + p0e1. There is no zeroth progression. Therefore, p0e1 = 0, and each element of the first progression is simply twice the previous element (1, 2, 4, 8, 16, 32, 64, ...).
Edges -- second progression.
p2e1 = 0. There are no edges in zero dimensions.
p2e2 = 2(p2e1) + p1e1 = 2(0) + 1 = 1
p2e3 = 2(p2e2) + p1e2 = 2(1) + 2 = 4
p2e4 = 2(p2e3) + p1e2 = 2(4) + 4 = 12
