Measure polytope
In geometry, a measure polytope is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed convex figure consisting of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other.
A point is a measure polytope of dimension zero. If one moves this point one unit length, it will sweep out a line segment, which is the measure polytope of dimension one. If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a two-dimensional square. If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a three-dimensional cube. This can be generalized to any number of dimensions. For example, if one moves the cube one unit length into the fourth dimension, and it generates a hypercube or tesseract.
A measure polytope is one of the few families of regular polytopes that is represented in any number of dimensions.
A measure polytope of dimension n has 2n "sides" (a 1-dimensional line has 2 end points; a 2-dimensional square has 4 sides or edges; a 3-dimensional cube has 6 faces; a 4-dimensional tesseract has 8 cells). The number of vertices (points) of a measure polytope is 2n (a cube has 23 vertices, for instance).
The number of m-dimensional measure polytopes on the boundary of an n-dimensional measure polytope is
For example, the boundary of a 4-dimensional hypercube contains 8 cubes, 24 squares, 32 lines and 16 vertices.
The dual polytope of a measure polytope is called a cross-polytope.