Ordering tuples over (0,1)

Question

To understand the natural ordering of $\{0,1\}^N$ I first thought about $\{0,1\}^3$ which has this Hasse diagram:

There's something interesting going on in the second and third rows. In 3-D it would attach the corners of ▲ to the corners of ▼. But I'm missing if there's a familiar structure (symmetric group?) that can help me decompose the more complicated natural ordering of $\{0,1\}^N$.

Answer 1

It's simply an hypercube, in which every point of the sequence gives you the $n$-th coordinate