Quantity of Vertices, Edges, Etc In Higher Dimensional Cubes

[Richard Craig]

Has anyone came up with a way to find the number vertices, lines or face in different dimensional cubes? I'm would most interested in a non-recursive equation.

 

[andrewkirk]

The vertices are easiest. The vertices of the n-dimensional cube ##\{\mathbf{x}\in\mathbb{R}^n\ |\ \forall i: x_i\in[0,1]\}## are the n-tuples ##(x_1,x_2,...,x_n)## in which every element is 0 or a 1. How many such n-tuples are there?

 

[mathwonk]

the easiest way to see it seems recursive, but you might be able to write a non recursive formula. I.e. just cross an n cube with an interval to get an n+1 cube and look at what happens to the various simplexes. e.g. you double the number of vertices, but in addition to doubling the number of edges you also transfiorm each vertex into a new edge,... so a 3 cube has twice as many vertices as a square, and 3 times as many edges,... But maybe you already see this.