Let's consider a hypersphere of radius
r inscribed in a hypercube with sides of length
2r. Than take the ratio of the volume of the hypersphere of radius
r to the hypercube with side length
l = 2r. We observe the following trends.
In two dimensions we have
and in three dimensions we have
For a general case, as the dimensionality
d increases asymptotically, we get
This means that as the dimensionality increases, the most of the volume of the hypercube is in the corners, whereas the center is essentially empty. In very high dimensions the figure looks like porcupine, as illustrated in the figure below (figure is taken from Zaki, 2013).
Subfigure (a) represents hypersphere inscribed withih hypercube in two dimensions. Subfigures (b) and (c) represent the same concept in three and four dimensions, respectively. In
d dimensions there are
2^d corners and
2^(d - 1) diagonals. The radius of the inscribed circle reflects the difference between the volume of the hypercube and the inscribed hypersphere in
My question: How to start with reproduction of this figure? Should I use Metapost or TiKZ? Thanks in advance for any pointers or suggestions?