vectors of Cartesian coordinates are Here the symbol means that each coordinate is either equal to or to . This unit hypercube is also the cartesian product . Any unit hypercube has an edge length of and an -dimensional volume of . The -dimensional hypercube obtained as the convex hull of the points with coordinates or, equivalently as the Cartesian product is also often considered due to the simpler form of its vertex coordinates. Its edge length is , and its -dimensional volume is . Faces Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension admits facets, or faces of dimension : a (-dimensional) line segment has endpoints; a (-dimensional) square has sides or edges; a -dimensional cube has square faces; a (-dimensional) tesseract has three-dimensional cubes as its facets. The number of vertices of a hypercube of dimension is (a usual, -dimensional cube has vertices, for instance). The number of the -dimensional hypercubes (just referred to as -cubes from here on) contained in the boundary of an -cube is , where and denotes the factorial of . For example, the boundary of a -cube () contains cubes (-cubes), squares (-cubes), line segments (-cubes) and vertices (-cubes). This identity can be proven by a simple combinatorial argument: for each of the vertices of the hypercube, there are ways to choose a collection of edges incident to that vertex. Each of these collections defines one of the -dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the -dimensional faces of the hypercube is counted times since it has that many vertices, and we need to divide by this number. The number of facets of the hypercube can be used to compute the -dimensional volume of its boundary: that volume is times the volume of a -dimensional hypercube; that is, where is the length of the edges of the hypercube. These numbers can also be generated by the linear recurrence relation , with , and when , , or . For example, extending a square via its 4 vertices adds one extra line segment (edge) per vertex. Adding the opposite square to form a cube provides line segments. Graphs An n-cube can be projected inside a regular 2n-gonal polygon by a skew orthogonal projection, shown here from the line segment to the 16-cube.