Polytope

One special kind of polytope is the convex hull of a finite set of points. Roughly speaking this is the set of all possible weighted averages, with weights going from zero to one, of the points that lie at the vertices of the hull. When the points are in general position (are affine-linearly independent, no s-plane contains more than s of them), this defines an r-simplex (where r is the number of points).

For instance a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron (in each case here with interior, although the word simplex is also used to mean other variations). Note an r-simplex will not fit into an (r-1)-plane ((r-1)-dimensional space, if you prefer). Note also that any subset containing s of the r points defines a subsimplex, called an s-face. The 0-faces are just the vertices and the unique r-face is the simplex itself.

Now given any convex hull in r-dimensional space (but not in any (r-1)-plane, say) we can take linearly independent subsets of the vertices, and define r-simplexes with them. In fact you can choose several simplexes in this way such that their union as sets is the original hull, and the intersection of any two is either empty or an s-simplex (for some s < r).

For example, in the plane a square (convex hull of its corners) is the union of the two triangles (2-simplexes), defined by a diagonal 1-simplex which is their intersection?

In general, the definition (attributed to Alexandrov)is that an r-polytope is defined as a set with an r-simplicial decomposition. It is a union of s-simplices for values of s with s at most r, that is closed under intersection, and such that the only time that one of simplices is contained in another is as a face.

What does this let us build? Let's start with 1-polytopes. Then we have the line segment, of course, and anything that you can get by joining line segments end-to-end:

  *----*   *----*   *----*   *-*   *----*----*
                |   |    |    X         |
                *   *----*   *-*        *

If two segments meet at each vertex (so not the case for the final one), we get a topological curve, called a polygonal curve. You can categorize these as open or closed, depending on whether the ends match up, and as simple or complex, depending on whether they intersect themselves. Closed polygonal curves are called polygons.

Simple polygons in the plane are Jordan curves[?]: they have an interior that is a topological disk. And also a 2-polytope (as you can see in the third example above), and these are often treated interchangeably with their boundary, the word polygon referring to either.

Now we can rinse and repeat! Joining polygons along edges (1-faces) gives you a polyhedral surface, called a skew polygon when open and a polyhedron when closed. Simple polyhedra are interchangeable with their interiors, which are 3-polytopes that can be used to build 4-dimensional forms (sometimes called polychora), and so on to higher polytopes.

For a more abstract treatment, see simplicial complex.

See also Tesseract, Glome.