Polytope

The terminology surrounding polygons, polyhedra, and polytopes is confusing, and different people often mean different things with the terms. Here polytope is being taken to mean polygons, polyhedra, and higher dimensional versions of the same. There is a very nice formal definition in terms of decompositions, thanks (I think) to Alexandroff, that I'd like to include.

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 same; the points are called the vertices of the hull. Of particular importance is the case where the points are all linearly independent (that is, no s-plane contains more than s of them), in which case the convex hull is called 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. 0-faces are just the vertices and the one 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 subset of the vertices, and define r-simplexes with them. In fact you can choose several simplexes in this way such that they union up to give the original hull, and the intersection of any two is either empty or an s-simplex (s<r). To feebly try and draw a picture:

  ******
  *...**
  *..*.*
  *.*..*
  **...*
  ******

See, the square (with interior would be the convex hull of its corners) is the union of the two triangles (2-simplexes), which intersect at a line segment (1-simplex)? See? Ah well. This sort of thing is called an r-simplicial decomposition, and forms the basis for a general definition: an r-polytope is anything with an r-simplicial decomposition. s-faces of the simplexes that intersect and lie in a common s-plane union to give the s-faces of the polytope.

Ok, more than enough formality. 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         |
                *   *----*   *-*        *

Now, if we restrict ourselves to forms with two segments meeting at each corner (so not the last), 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 are especially interesting because, as Jordan curves, they have a single distinct interior. This is, as it turns out, 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 two to an edge (1-face) 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.

Later I plan to add some details here about regular polytopes, but this should be fine for now.